Options, Futures and Exotic Derivatives (Frontiers in Finance Series)

O-PTIONS, FUTURES

AND

Exovic ERIVATIVES Theory, Application

and Practice

M. Bellalah

E. Briys

Université du Maine

Lehman Brothers

F. de Varenne

H.M. Mai

FFSA

MONEP

OM A

jOHN WILEY & SONS Chichester

·

New York

-

Weinheim

Brisbane

Singapore

Toronto

O.

Mr

Boll OPTIONS,

UTURES

Exovic DERIVATIVES

*

Copyright

@ 1998 by John Wiley

& Sons Ltd, Baffins Lane, Chichester, West Sussex POl9 lUD, England National International

01243 779777

(+44)1243779777

e-mail (for orders and customer service enquiries): cs-books Visit our Home Page on http://www.wiley.co.uk

On teiItS

@wiley.co.uk

or http://wwwwiley.com The authors have asserted their right under the Copyright, Designs and Patents Act, 1988, to be identifiedw&e authors of this work. Reprinted

July 1998, October 1999

All rights reserved. No part of this publication rnay be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of licence issued by the Copyright a Licensing Agency, 90 Tottenham Court Road, London, UK WlP 9HE, without the permission in writing of John Wiley and Sons Ltd., Baffins Lane, Chichester, West Sussex, UK POl9 lUD.

About the A

CHAPTER

1: FINANCIAL yggy

MAHi(ETS, INNOVATION ANÐ TRADING

Chapter Outline Introduction 1.1 Do Firms Care about Finance? 1.1.1 Finance is a fun game to play, hard to win 1.1.2 Some commonly encountered pitfalls 1.1.3 Some good reasons for hedgmg 1.2 How to Implement Strategic Financial Risk Management 1.3 Trading Mechanisms in Securities Markets

WlLEYVCH Verlag GmbH, Pappelallee 3, D-69469 Weinheim, Germany Jacaranda

XX

Foreword

Other Wiley Editorial Ogces John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, USA

XVI

Preface

Wiley Ltd, 33 Park Road, Milton,

.

Queensland4064,

Australia

John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons (Canada) Ltd, 22 Worcester Road, Rexdale, Ontario M9W lLl, Canada Library of Congress Cataloging-in-Publication Data Options, futures, and exotic derivatives : theory application and practice / authors, E.C. Briys [et al.]. p. cm. - (Wiley frontiers in finance) Includes bibliographical references and index. ISBN 0-471-96909-5 (cloth). ISBN 0-471-96908-7 (pbk.) 1. Options (Finance). 2. Futures. 3. Derivative securities. L Bnys, E.C. II. Senes. ...

HG6024.A30653 332.63'228-dc2l

.

1998

1.4 Trading Option Contracts 1.4.1 Options and synthetic positions 1.4.2 The basic option strategies 1.4.3 Trading straddles and strangles 1.4.4 Trading spreads 1.4.5 Trading ratios 1.4.6 Conversions and reversals 1.4.7 Trading a box spread

Summary Points for Discussion

2 2 5 7 9 13 15 15 16 20 2I 22

23 24 25 25

97-44712 CIP

CHAPTER British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library 0-471-969095 (cased) 0-471-96908-7 (paperback) Typeset in 10/12 pt Times by Keytec Typesetting Ltd, Bridport, Dorset Printed and bound in Great Britain by Bookcraft (Bath) Ltd., Midsomer Norton, Somerset This book is printed on acid-free paper responsibly manufactured from sustainable forestation, in which at least two trees are planted for each one used ISBN

2: THE DYNAMICS OF ASSET PRICES: ANALYSIS AND A PP LI C AT IO NS

Chapter Outline Introduction 2. I Continuous Time Processes for Asset Price Dynamics 2.1.1 Asset price dynamics and the Wiener process Wiener orocess 2.1.2 Asset nrice dvnamics and the eeneralized

27 27 27 28 28 31

CONT

2.1.3 Asset price dynamics and the Itô process 2.1.4 The log-normal property 2.1.5 The distribution of the rate of return 2.2 Itõ's Lemma and its Applications 2.2.1 Intuitive form 2.2.2 Mathematical form 2.2.3 Generalized Itö's formula 2.3 Taylor Series, Itõ's Theorem and the Replication Argument 2.3.1 The relationship between Taylor series and Itö's differential 2.3.2 Itõ's differential and the replication portfolio 2.3.3 Itô's differential and the arbitrage portfolio 2.3.4 Why are error terms neglected? 2.4 Forward and Backward Kolmogorov Equations Summary Points for Appendix Appendix Appendix

CHAPTER

Discussion 2.A: Introduction to Diffusion Processes 2.B: The Conditional Expectation 2.C: Taylor Series

.

.

35 36

39 40 40 41 42 43 44 45 45 45 46

ASSET

Chapter Outlme

.

31 33 33 34

47

3: APPLICATIONS TO ASSET AND DERIVATIVE PRICING IN COMPLETE MARKETS

Introduction 3.1 Characterizations of Complete Markets 3.2 Pricing Derivative Assets 3.2.1 The problem 3.2.2 The partial differential equation method 3.2.3 The martingale method 3.3 Numerical Analysis and Simulation Techniques 3.3.1 Introduction to finite difference methods 3.3.2 Application to European calls on non-dividend 3.3.3 Simulation methods

ITS

.

paying stocks

Summary Points for Discussion Appendix 3.A: The Change m Probability and the Girsanov Theorem 3.A.l The equivalent probability 3.A.2 The Girsanov Theorem Appendix 3.B: Resolution of the Partial Differential Equation Appendix 3.C: Approximation of the Cumulative Normal Distribution Appendix 3.D: Introduction to Numerical Analysis 3.D.1 The heat transfer equation 3.D.2 Some numerical schemes Appendix 3.E: An Algorithm for a European Call Appendix 3.F: Leibniz's Rule for Integral Differentiation

49

49 49 50 51 51 52 53 56 57 59

61 62 63

63

CONTENTS

ER 4: ASSET PRICING IN COMPLETE NUMERAIRE AND TIME

MARKETS: CHANGING 73 73 73 74 76 78 78 8l 83

Chapter Outline Introduction 4.1 Assumptions 4.2 Valuation in the Black-Scholes Economy 4.3 Changing Numeraire and Time 4.3.1 The change of numeraire 4.3.2 The time change

Summary Points for Discussion Appendix 4.A: Application

83

5: ANALYTICAL

CHAPTER

84

to Barrier Options EUROPEAN

OPTION

PRICING MODELS

87

Chapter Outlme Introduction

87

5.1 Precursors of the Black and Scholes Model 5.1.1 Bachelier formula 5.1.2 Sprenkle formula 5.1.3 Boness formula 5.1.4 Samuelson formula 5.2 Black and Scholes (B-S) Model 5.2.1 The model 5.2.2 Applications 5.3 Black Model 5.3.1 The model 5.3.2 Applications 5.4 Garman and Kohlhagen, and Grabbe Models 5.4.1 The model 5.4.2 Applications 5.5 The Merton, Barone-Adesi and Whaley Model and its Applications 5.5.1 The model 5.5.2 Application of the model

88

-

87

Summary Points for Discussion

88 89 90 90 92 92 99 105 105 108 l 10 111 l 14 l 14 114 116 , 1=0 121

63

63 64

CHAPTER

6: MONITORING

POSITIONS

68 69 69

69 70 71

Chapter Outline Introduction 6.1 Option Price Sensitivities 6.l.l The delta 6.1.2 The gamma

AND MANAGEMENT

OF OPTION 123 123 123 124 125 126

CONTENTS

y¡¡¡

6.1.3 The theta 6.1.4 The vega 6.1.5 The Rho 6.1.6 Elasticity 6.2 Monitoring and Managing an Option Position in Real Time 6.2.1 Simulation and analysis of option price sensitivities 6.2.2 Monitoring and adjusting the option position in real time 6.3 The Characteristics of Volatility Spread Summary Points for Discussion Appendix 6.A: Greek-Letter Risk Measures in Analytical Models 6.A.1 B-S model .

.

127 127 128 128 129 129 134 149

,

150 150 151

6.A.2 Black's model 6.A.3 Garman and Kohlhagen's model 6.A.4 Merton's and Barone-Adesi and Whaley's model Appendix 6.B: The Relationship between Hedging Parameters Appendix 6.C: The Generalized Relationship between the Hedging Parameters

CHAPTER 7: EXTENSION TO AMERICAN OPTIONS: DIVIDENDS AND EARLY EXERCISE Chapter Outline

157 157 157 158 159 161

lntroduction 7.1 The Valuation of American Options: the General Problem 7.1.1 Early exercise of American calls 7.1.2 Early exercise of American puts 7.2 Valuation of American Commodity Options, Futures Options with Continuous Distributions 7.2.1 American commodity options 7.2.2 American futures options 7.2.3 Capped variable loan commitments 7.3 Valuation of American Options with Discrete Cash Distributions 7.3.1 Early exercise of American options 7.3.2 Compound option approach 7.3.3 Valuation of American options with dividends 7.3.4 Applications

Summary Points for Discussion Appendix 7.A: Simulation Results for American Cap Options 7.A.l Effect of a change in the underlying asset price and time to maturity 7.A.2 Effect of a change in the underlying and time to maturity CHAPTER

8: GENERALIZATION

Chapter Outline Introduction

.

151 152 152 153 154 154

163 163 166 168 170 170 171 175 176 177 177 178 178

asset price volatility

TO STOCHASTIC INTEREST

179

RATES

181 181 18 I

CONTENTS

ix

8.1 Derivation of Merton's Model 8.1.1 The model 8.1.2 Applications Model ; 8.2.1 The model for equity options 8.2.2 The model for bond options 8.2.3 Chen's correction 8.3 Pricing of Bonds and Interest Rate Options 8.3.1 Instantaneous interest rates under certainty 8.3.2 Instantaneous interest rates under uncertainty 8.3.3 Interest rate processes and the pricing of bonds and options

8.2 Rabinovitch's

·

Summary Points for Discussion CHAPTER

9: PRICING CORPORATE

BONDS

182 182 185 186 186 189 190 190 190 191 192 195 195 197

Chapter Outline Introduction 9.1 The Traditional Contingent-Claims Modeling 9.1.1 Assumptions 9.l.2 The pricing of corporate debt 9.l.3 The pricing of corporate spreads 9.2 The Limits of the Traditional Approach 9.2.1 The three weaknesses 9.2.2 Recent contributions 9.3 The Longstaff-Schwartz Model 9.4 The Briys-de Varenne Model 9.4.1 The model and its assumptions 9.4.2 The valuation of risky zero-coupon bonds 9.4.3 The valuation of corporate spreads 9.4.4 The interest rate elasticity of corporate bonds

197 197 198 198 200 202 203 203 205 205 207 208 210 212 219

Summary

224

Points for Discussion

224

CHAPTER

10: PRICING

INSURANCE

LINKED

BONDS

Chapter Outline Introduction 10.1 Natural Hazards, Insurance Risks and Insurance Linked Bonds 10.2 The Structure of Insurance Linked Bonds 10.3 A Simple Pricing Model of Insurance Nature Linked Bonds 10.3.1 The valuation model 10.3.2 The valuation of insurance linked spreads 10.3.3 The duration 10.3.4 Time-series properties

Summary Points for Discussion

225 225

225 226 228 230 230 233 235 238 239 239

COMTENTS

x

Appendix Appendix CHAPTER

10.A: Extension to Stochastic Interest Rates 10.B: Computation of the First Passage Time Distribution

240 240

11: FURTHER

GENERALIZATION TO JUMP PROCESSES, STOCHASTIC VOLATILITIES AND INFORMATION

COSTS

241 241

and the Constant Elasticity of Variance (CEV) 11.1 The Jump-Diffusion Model l 1.1.1 The jump-diffusion model 11.1.2 The constant elasticity of variance (CEV) diffusion process 1l.2 The Hull and White Model 11.3 Stein and Stein's Model 11.4 Generalization to Stochastic Volatilities 11.4.1 Heston's model 11.4.2 Hoffman, Platen and Schweizer's model 11.5 The Theory of Volatility Smiles 11.5.1 The smile effect in stock and index options 11.5.2 The smile effect for bond and currency options 11.6 Option Valuation with Information Costs: the Bellalah-Jacqudiat Model 11.6.1 The model 11.6.2 Empirical tests 11.7 Volatility Smiles: Empirical Evidence

Summary Points for Discussion Appendix l l.A: The Poisson Process CHAPTER

12: THE LATTICE

APPROACH AND THE BINOMEAL MODEL

Chapter Outline Introduction 12.1 Lattice Approaches l2.l.l A survey 12.1.2 The model for options on a spot asset without any payouts 12.1.3 The model for futures options 12.1.4 Model with dividends 12.1.5 Examples 12.1.6 The model for American options in the French market 12.2 The Lattice Approach for Interest Rate Derivative Assets 12.2.1 The Ho and Lee model for interest rates and bond prices 12.2.2 The Ho and Lee model for contingent claims 12.2.3 Deficiency in the Ho and Lee model 12.2.4 The Hull and White trinomial model Summary Pomts for Discussion -

242 242 243 244 245 247 248 249 250 250 250 251 251 252 256 256 257 257 259 259 259 260

260

-

261 263

264

'

CHAPTER

xi

13: NUMERICAL

METHODS FOR AMERICAN

OPTION

PRICING

281

Chapter Outline 241

Chapter Outline Introduction

CONTENTS

266 268 273 273 275 277 274 280 280

Introduction 13.1 Application to American Calls o Dividend Paying Stocks 13.1.1 The Schwartz model 13.1.2 The numerical solution 13.2 Application to American Puts on Dividend Paying Stocks 13.2.1 The Brennan and Schwartz model 13.2.2 The numerical solution 13.3 Application to Convertible Bonds 13.3.1 The specificities of convertible bonds 13.3.2 The valuation equation 13.3.3 The numerical solution 13.3.4 Simulations Summary Points for Discussion Appendix 13.A: The Algorithm for the American Call with Dividends Appendix 13.B: The Algorithm for the American Put with Dividends Appendix 13.C: The Algorithm for Convertible Call and Put Prices 13.C.l Initialization

CHAPTER

14: EXCHANGE,

FORWARD

START AND CHOOSER

OPTIONS Chapter Outline Introduction 14.1 Exchange Options 14.1.1 Identification and valuation 14.1.2 Applications 14.2 Options with Uncertain Exercise Prices 14.2.1 Analysis and valuation 14.3 Forward Start Options 14.4 Pay Later Options 14.5 Chooser Options 14.5.1 Simple chooser options 14.5.2 Complex chooser options

Summary Points for Discussion

CHAPTER

15: RAINBOWOPTIONS

Chapter Outline Introduction

281 281 282 282 282 284 284 284 285 285 286 287 288 290 290 290 291 293 293

297 297 297 298 298 300 301 301 304 305 307 307 308 310 310

313 313 313

CONTENTS

xii

15.1 Valuation of Rainbow Options 15.1.1 Analytic formulas 15.1.2 The discrete approach 15.2 Simulations M.; 15.2.1 Calls on the minimum 15.2.2 Calls on the maximum 15.2.3 Puts on the minimum Applications 15.3 15.3.1 Pricing currency bonds 15.3.2 Multi-currency bonds 15.3.3 Corporate option bonds 15.3.4 Spread options 15.3.5 Portfolio options 15.3.6 Dual-strike options 15.3.7 Delivery options and wildcard options

314

314 320 Y

4

Summary Points for Discussion

CHAPTER

325 326 326

327 327 327 328 33 I

333

Chapter Outline Introduction 16.1 Pricing valuation

context 16.1.2 Extendible calls 16.1.3 Extendible puts 16.2 Simple Writer Extendible Options 16.2.1 Simple writer extendible calls 16.2.2 Simple writer extendible puts 16.3 Applications 16.3.1 Extendible bonds 16.3.2 Extendible warrants

Summary Points for Discussion

CHAPTER 17: FOREIGN CURRENCY SECURITIES

324 325

×iii

17.2 Analysis and Valuation of Capped Options 17.2.1 The range forward contract 17.2.2 The collar

348 348

17.2.3 Indexed notes 17.3 Pricing Hybrid Foreign Currency Options 17.3.1 Tailoring hybrid foreign currency options: the Briysdrouhy approach 17.3.2 General pricing 17.4 Financing with Hybrid Securities: Peris 17.4.1 Specificities of the product 17.4.2 Cash-fiow identification 17,4.3 Valuation of perls

349 351

349

35 l 352

354 354 354 355

Summary

356

Points for Discussion

357

CHAPTER 18: BINARIES AND BARRIERS

359

331

16: EXTENDIBLE OPTIONS

16.1.1 The

323 323 324

ONTENTS

333 333 333 333 334 336 337 337 337

339 339 340 340 340

OPTIONS AND HYBRID

Chapter Outline Introduction 17.1 Equity-Linked Foreign Exchange Options and Quantos 17.1.1 The foreign equity option struck in foreign currency 17.1.2 The foreign equity option struck in domestic currency 17.1.3 Equity-linked foreign exchange call 17.1.4 The fixed exchange rate foreign equity options or quanto options

341 341 341 342 343 343

Chapter Outline Introduction 18,1 Analysis of Binaries and Barriers 18.1.1 Standard binary options 18.l.2 Complex binaries and range options 18.l.3 Barriers and structured barrier options 18.2 Valuation of Binary Barrier Options 18.2.1 Path-independent binary barrier options 18.2.2 Path-dependent binary barrier options 18.3 Valuation of Barrier Options 18.3.1 'In' barrier options 18.3.2 'Out' barrier options 18.3.3 Outside barrier options: an alternative approach 18.4 Analysis and Valuation of Standard and Exotic Options Using Binary Options 18.4.1 Example 1: Standard options 18.4.2 Example 2: Down-and-out call options 18,4.3 Example 3: Switch options 18.4.4 Example 4: Corridor options 18.4.5 Example 5: Knock-out range options 18.5 Continuous Strike and Continuous Barrier Options 18.5.1 Definitions and analysis 18.5.2 Valuation of the continuous strike option 18.5.3 Continuous strike range options 18.5.4 Soft barrier options 18.6 Static Hedging of Barrier Options 18.6.l Hedging at-the-money down-and-in cals 18.6.2 Hedging out-of-the-money down-and-in calls 18.6.3 Hedging in-the-money down-and-in calls

359 359

361 361 361 364

368 369 370 373 374 375 377 379 379 379 380 38 1 382 382

382 383

384 384 386 386 386

387

344

Summary

389

345

Points for Discussion

389

CONTENTS

xiv

CHAPTER

19: LOOKBACK

391

OPTIONS

391 391 392

Chapter Outline Introduction 19.1 Analysis of Lookback Options 19.1.1 Examplesofstandardlookbackoptions 19.1.2 The floating strike lookback 19.1.3 Fixed strike lookbacks 19.1.4 Lookback strategies 19.2 Analytic Formulas for Lookback Options 19.2.1 Standard lookback options 19.2.2 Options on extrema 19.2.3 Limited risk options 19.2.4 Partial lookback options 19.2.5 Simulations 19 3 Analytic Formula for an Exotic Tiratag Option:

394 395

395 396 397 398 399

tle BOMalah-PWgrat

Model 19.3.1 Analysis and valuation 19.3.2 Simulations and option characteristics 114 Analysis and Valuation of the Strike Bonus Option .

.

19.4.1 The strike option 19.4.2 The fractional bonus option 19.4.3 The hedging of strike bonus options Summary Points for Discussion

CHAPTER 20: ASIAN AND FLEXIBLE

About the Authors

392

393 394

ASIAN OPTIONS

409

Summary Points for Discussion Bibliography

425

Index

441

Introduction 20.1 The Average Price Options: Analysis and Valuation 20.1.1 Analysis 20.1.2 The valuation approaches 20.2 Analysis and Valuation of Basket Options 20.3 Analysis and Valuation of Flexible Asian Options 20.3.1 Flexible Asian options 20.3.2 Approximating flexible arithrnetic Asian options 20.3.3 Some properties

.

-

401 402 403 403 405 406 407 407 407

409 409 4l l 412 413 418 420 421 422 423 423 423

Chapter Outline

Professor Mondher Bellalah is Professor of Financial Economics at Université du Maine, France. He also holds teachmg and research positions at Université ParisDauphine, Université de Cergy-Pontoise and IHEC Tunis. He has been involved in activities at BNP. He currently acts as a financial engineering and market-making consultant for financial institutions. Associate Consultant at Philipponnat & Associés. Options, Futures and Exotic Derivatives is his third finance text. His articles have also appeared m the Financial Review and the Journal of Futures Markets. .

.

.

.

Dr Eric Briys is Director, International Fixed Income Research, at Lehman Brothers in London, UK. He was formerly Professor of Finance at Groupe HEC, one of Europe's leading business schools, where he was head of the Finance and Economics Department and Dean of the MBA programme. He is the Editor of the Review ofDerivatives Research and the former Editor of Finance, the journal of the French Finance Association. He has published widely in leading scientific journals such as the Journal of Finance, the Journal of Financial and Quantitative Analysis, and the American Economic Review. Dr Huu Minh Mai holds a PhD in Finance from Université Paris-Dauphine. He is currently a financial statistician and a research officer at MONEP, the French Options Markets in Paris. Prior to joining MONEP he held research positions at Université-Paris Dauphine and Associés en Finance. His articles have appeared m Fmance. .

.

.

.

Dr François de Varenne is Head of Financial and Economic Affairs at the French Federation of Insurance Companies. He is also part-time lecturer at Groupe HEC and ISFA Lyon. A graduate of Ecole Polytechnique, he is a co-founder and co-principal of Dixon Associates, a consulting firm specialising for in asset-liability management msurance companies. He has widely published in leading scientific journals such as the Journal of Risk and Insurance and the Journal of Financial Quantitative Analysis.

We would like to thank Ahnani Mohamed for his computational

assistance.

PREFACE

xy¡¡

'dissection'

of (very)exotic ones. Again, our main instruments and ends up with the has exhaustive possible. This effort is indeed more than been be to concern as as necessary. Derivatives are still viewed as some kind of evil, of permanent threat to the socalled economy'. Any financial collapse is usually attributed to derivatives in the place: indeed, first a scapegoat is most useful when the outside world does not understand exactly what it is made of. In ancient Greece the messenger was often killed simply because he would carry bad news. Derivatives are like this messenger. This reminds us of Baron' WorralL The a famous Battle of Britain ace, the late Air Vice-Marshal John Baron led a Hurricane fighter squadron throughout the fall of France and the Battle of Britain. The story2 has it that in the middle of a fierce fight he held the following exchange with his controller: 'real

'the

'24

Our colleague and dear friend Jamil Baz has his own unique way of describing modern Wall Street. He likes to compare Wall Street to the pharmaceutical industry. According to giants engaged in a fierce and him, investment banks are nothing but pharmaceutical This molecule will help molecule. towards the next pathbreaking endless competition producing new drugs (readfinancial instruments or strategies) capable of curing the pains and diseases (readrisk exposures) of Main Street (readcorporations, investors, individuals). This analogy indeed makes a lot of sense. The modern theory of financial argument to provide the required derivatives' pricing uses an or created priced together and by putting basic moleculos to achieve Derivatives are answers. the desired cash-flow pattern. In his foreword Oldrich Alfons Vasicek reminds us that derivatives pricing deals with the valuation of a security, called the derivative, relative to the value of another security, the underlying. If this pricing is not properly conducted, the derivative and the underlying security can be arbitraged against each other through the generate a positive gain on a position (the riskless asset) use of another that necessitates no investment and carries no risk To put it in a nutshell this text is about this fascinating process. It starts in 'Wall Street labs' where rocket scientists try to provide the solutions so badly needed by the agents actively involved in the growth of our economies. This is not just another text on options and futures. It goes without saying that there are already outstanding textbooks on the subject in the bookstore next door. It is less obvious, at least to the best ofour knowledge, that there are numerous textbooks that cover the A to Z of derivatives pricing. By A we mean the scientific and mathematical foundations. By Z we mean the study of the latest instruments insofar as it is possible to write (andpublish) as fast as Wall Street launches currently new exotjes! By the intermediate letters we mean the tools and techniques action'. available that we describe and analyze Three main blocks can be distinguished. The first and surely inescapable one deals with the economic foundations of the theory of derivatives (Modigliani-Miller propositions, arbitrage versus equilibrium, complete versus incomplete markets, notion of numeraire). It goes hand in hand with the second building block. This block develops the mathema_ tical toolkit that really leverages the previous concepts (stochasticcalculus, Itõ's lemma, Girsanov theorem, partial differential equations, numerical and simulation techniques). The third block combines the two previous blocks. It starts with the analysis of standard 'replication'

'assembly'

'to

'molecule'

.

'in

Controller: bombers with 20 plus more behind them.' Worrall: 'Got it.' plus more bombers and 20 fighters behind and above.' Controller: Warrall: 'All right,' Controller: 'Now 30 more bombers and further 100 plus fighters following.' Worrall: 'Stop. No more information please. You are frightening me terribly.' '20

Derivatives are in a sense very similar to Air Vice-Marshal Worrall's controller. That is why so many people are so frightened by them. We hold the opposite view and sometimes ask ourselves how people could make it without derivatives! In any case it is our hope that this textbook along with former and future ones will pave the way towards a better understandmg of the numerous benefits of financial markets. This book could not have been in your hands without accumulating a significant leveraged position (in many different currencies!). Indeed, we are indebted to many people for their invaluable help at different stages ofthe manuscript. Oldrich Alfons Vasicek has written a useful and insightful foreword that reallv helps to set the landscape. An old friend is the one who is always there to offer his advice and give you a hand when you need it. No doubt that Oldrich genuinely belongs to this most sought after category! Nassim Taleb, a best-selhng author of John Wiley & Sons and a truly remarkable ethnologist of financial markets, has done a superb job (beyondduty) in helping us improve what was m the begmnmg a very'rough draft. He has all our gratitude. This book is a joint effort between (former)academics and practitioners. In that last respect we owe special thanks to Lehman Brothers. Thanks to its friendly and highly professional atmosphere it has helped one ofus in his migration from an academic position into (hopefully!) a fully fledged investment banking role. This book is also the reflection of real life on a trading floor. It has benefited from the views, explanations and market insights of Alexandre Abadie, Kaushik Amin, Sandro Anchisi, Makram Azar, Riccardo Banchetti, Mark Benson, Robert Campbell, Glauco Cerri, Jos Corswarem, Benoît d'Angelin, Milena Dapcevic, Karim Derhalli, Dan Donovan, Adrian Fitzgibbon, Aògan Foley, Reza Ghodsi, François Girod, Lennart Hergel, Michael Hof, Roger Howgego, John Hunt, John McDonald, Benoit Migeot, Daniel Morley, Ken Nadel, Cyrille Paillard, Theo Phanos, Nicolas Pourcelet, David Prieul, Harsh Shah and Aubin Thomine-Desmazures. 'very

PREFACE

PREFACE

xviii

A book is like good wine. It has to age before it stands on bookstore shelves (nottoo long, hopefully). Even if they often do not realize it, a lot of different people contribute to make this aging period a very profitable one. Many thanks to Ramasastry Ambarish, Citibank; Michel Antakli, Morgan Stanley; Jo Assaraf, Discount Bank and Trust Company; Pierre Bernhard, Ecole Supérieure des Sciences Informatiques; Jean-Pierre Bobillot, Predica; Driss Ben Brahim, Goldman Sachs; Evert Carlsson, Skandia Investment Management; Jean Castellini (BARRA); Fred Célimène, Université des Antilles et de la Guyane; Jean-Louis Charles, AXA; Gilles Dupin, Groupe Monceau; Romain Durand, SCOR; Christian Ferry, Generali Finance; Claudio Giraldi, Istituto Nazionale delli Assicurazioni; Jean-Michel Hainard, Providentia; Denis Kessler, AXA; Ambroise Laurent, Paribas; François Martin, Assurances du Crédit Mutuel; Alessio Matteuzzi, Instituto Nazionale delli Assicurazioni; Philippe de Moustiers, Citibank; Gérard Pellieux, Caisse Nationale du Crédit Agricole; Michel Pelosoff, Union des Assurances Fédérales; Lester Seigel; Yves Simon, Université Paris-Dauphine; Denis Talay, Institut National de Recherche en Informatique et Automatique; Jean-Luc Vila, Convergence; and Patrick Werner, French Federation of Insurance Companies. Last but not least, Jamil Baz. Jamil loves the subway, especially the London Circle Line! This is a convenient way for exchanging ideas and turning a whole series of concepts upside down. Believe it or not, Jamil Baz and Eric Briys never missed either Liverpool Street station or High Street Kensington station! This book owes a lot to Jamil and his deep and sincere friendship. Jamil is indeed a rare combination: option theorist, option practitioner, math addict, philosopher and a fork'! We have also run into debt'. Many thanks to the Mathematics and Finance and Economics faculties and students of Groupe HEC, of Institut Supérieur de Finance et Assurance (ISFA), of Université des Antilles et de la Guyane (Martinique), of Institut National de Recherche en Informatique et Automatique (INRIA), of Université ParisDauphine, of Université de Cergy, of Université du Maine and of Université de Genève. There is another academic debt that we will never be able to repay. We have been bold enough to augment it. Robert C. Merton has kindly accepted! We are deeply honored by his words of endorsement. Paul A. Samuelson has described Robert C. Merton as a giant among the giants. If needed, our bibliography is a standing proof of it. Robert C. Merton has influenced the work and professional lives of numerous scholars and practitioners. We are very proud to be part of them. The professional staff at John Wiley & Sons is first class. Having heard our countless promises, we are sure they know exactly what default risk means, when an option is in or out of the money, what a forward position implies, what it means to be long or short. In any case, they can be sure that they taught us something very special indeed: trust. First, our deepest gratitude to Nick Wallwork who first decided that it was worth entering the forward contract. We hope that he did not leave London for Singapore because of us! Our editor, David Wilson, who took over from Nick, is now a close friend. We owe him a lot. He knows why! We are also grateful and proud of Louise Holden and Jennifer Mackenzie. They have done a terrific job even though we kept changing our minds. At the Oscar ceremony in Hollywood, movie directors, actors and actresses apologize for not naming people whose contribution was none the less instrumental. The same applies here. Our sincere apologies to those we have inadvertently omitted. These last few weeks have been a period where the four of us have gone hunting. More precisely typos, errors and inconsistencies hunting. We are afraid that in this matter zero 'solid

'academic

4

,

default does not any errors in the our manuscript. publisher permits

y¡

exist and we feel very sorry for that. Please let us know if you discover text or, on a more positive note, if you know about ways of improving Our only excuse is that this is the first edition. Hopefully and if our we will not have to invoke it again in the future!

Mondher Bellalah Eric Briys Huu Minh Mai François de Varenne

FOREWORD

Foreword

The theory of derivative asset pricing, or options pricing as it is more often referred to, is one of the great achievements of modern finance. It has become an essential tool in the valuation of derivatives. It is indispensable in risk management of derivative books. It allows valuation of corporate liabilities given the value of the firm assets, or valuation of corporate debt given the value of the firm equity. Besides its immense practical applicability, options pricing has also greatly advanced the theoretical understanding of financial markets and has led to many new insights. It fostered the development and application of new techniques, such as valuation in continuous time (whereasset prices are represented by Itö processes or other continuous time stochastic processes). It has led to the concepts of risk-neutral valuation and of the pricing probability measures. And it has engendered new areas of finance, such as models of the term structure of interest rates, What is options pricing? It deals with the valuation of one asset, or security, called the An asset is derivative, relative to the value of another asset or security, the underlying. called a derivative with respect to another if all of its payouts depend only on the value of the underlying. A simple example is a call option on a common stock. The payout on the call is determined solely by the value of the stock: it is the excess of the stock price over the strike price if the call is exercised, or zero if not. Another example is the value of a is the market value of the firm's assets (that corporate bond. In this case, the underlying is, the value of its ongoing business). It is obvious that the price of the derivative asset must be related to the price of the underlying one. Since the payout on a call option on a stock depends only on the stock price, the price of the call must be a function of the stock price. But which function'? Options pricing provides the answer. The function must be such that the return on the option (whichis given by the percentage increment of the function) in excess of the riskless return must be proportional to the return on the underlying asset in excess of the riskless return. If this were not the case, the option and the underlying asset could be arbitraged against each other (with the use of the riskless asset) to generate a positive gain on a position that necessitates no investment and carries no risk. This should not be possible in an efficient market. It turns out (and this is one of the achievements of the theory of options pricing) that function in relationship such a condition on the increment of the unknown to the increment of the underlying asset value, together with the specification of the option nrovisions, snecifies the function in full The ftmetion navouts nrovided bv its contractual

xx¡

can be sought out in either of two equivalent ways. One way is to determine the option value as the solution of a partial differential equation to which the price of any derivative asset must conform (this equation was initially derived by Black and Scholes, and independently, by Merton), subject to boundary conditions given by the form of the payout. This equation was originally solved by Black and Scholes for the European call and put, to yield their celebrated formula. The other way to get the option value has the tremendous appeal of bringing out the economic, rather than only mathematical, aspect of the options pricing theory, It is shown that the option is priced as the present value (discountedat the riskless rate) of the mathematical expectation of the payouts; provided, however (and this is a deep and subtle is calculated not with respect to the actual point of the theory), that the expectation probability distribution of the payouts, but rather with respect to an alternative probability distribution. This alternative distribution, called the pricing distribution, is the one the payouts would have in a risk-neutral economy. In such an economy, the expected rate of return on all assets is the riskless rate. The pricing probability distribution is thus obtained as if the underlying asset appreciated on average at the riskless rate, rather than at its own actual expected rate of return. The reason here is that the relationship of the option price to the price of the underlying does not depend on investors' attitudes toward risk, and therefore must be the same as in a risk-neutral world (and in a risk-neutral world, we know how assets are priced: as the expected present values of their payouts). Besides valuation, the theory yields powerful results for risk measurement and managenient. The hedging ratios, or deltas, for hedging the risk of derivatives positions can be calculated by differentiation, with respect to the price of the underlying asset, of the option value. These hedge ratios also provide a measure of elasticity, or exposure to the risk of the underlying security. By reducing the delta of a derivative portfolio (they combine linearly), the exposure to a given risk source is reduced. Other risk measures (gamma,theta, etc.), highly useful in managing a derivatives book, are likewise determined from the valuation formulas. By inverting the formulas, it is possible to obtain measures of stock volatility implicit in the market price of an option. Altogether, options pricmg is one of the most fascinating (not to mention useful m practical applications) areas of finance. Hundreds, if not thousands, of articles and tens of books have been written on the subject. Why, then, this book? One of the merits of this book is that it is self-contained. It was written for the sauvage intelligent: it does not necessitate any previous knowledge of the field. It can be read and understood by any quantitatively oriented person. One could well envision an investment bank or a brokerage firm hiring a PhD in physics, say, turning the book over to him and say: 'Read this, and develop for us a system for valuation and risk management of our derivatives portfolio'. It is both a textbook and a reference book. It covers the basics of the theory, as well as the techmques for valuation of many of the more exotic derivatives. It contains a detailed of knowledge of the field. What is more, however, it is written with a deep understanding the economics of finance. Not only how, but why of the options pricing; not only a compendium of existing results, but a methodology for deriving new ones. Congratulations on your selection, Oldrich Alfons Vasicek Managing Director The KMV Corporation, San Francisco, USA

Financial

CHAPTER This chapter

Markets, Innovation and Trading Activity

OUTLINE' is organized as follows:

1. Section 1.1 shows why and how managing financial risks may yield improved corporate returns. 2. Section 1.2 gives insights into the implementation of financial risk management. 3. Section 1.3 presents the main trading mechanisms in securities markets. 4. Section 1.4 illustrates the main trading strategies that are feasible with options.

INTRODUCTION Have you ever been in San Francisco at the auction house of Buttlerfield and Buttlerfield? No. Well, you should! The reason? In California, when a red wine is 10 years old, its owner can sell it legally without a retailer's license. It is still time to get there: the 1982 Bordeaux were something special. Their prices have been skyrocketing over the last ten years. Sell your 1982 Léoville-Las Cases now for a net hammer price of $700 a case and you will enjoy a nice 15% a year average return. Basically, it boils down to making a liquid asset really liquid! Assume now that you love wine so much that you are a shareholder of Léoville-Las Cases. Well, it sounds a lot more difficult to make your shares as liquid as your wine! Léoville-Las Cases is not quoted on the Paris Bourse. Many people will argue that you do not have to worry about it. After all, you own one of the safest assets in the world: a unique Bordeaux chateau producing a most prestigious wine! Well, things are not that simple and you will soon discover that your investment is both interest-rate and exchange-rate sensitive. How come? Wine has to age in cellars. As a consequence cash is tied up in the production process. The clientele base is primarily abroad and

his chapter

draws heavly on Briys and Crouhy 0993¾

OPTIONS,

2

FUTURES

AND EXOTIC

DERIVATIVES

FINANCIAL

most of the sales are dollar-denominated. Now, the question is: should Léovile-Las that is Cases hedge itself or should it just concentrate on its secular savoir-faire, Is there any value in managing financial risks should it just stick-to-the-knitting? instead of relying solely upon weather forecasts? Does it really improve the shareholders' situation? Does it help enhance the quality of wine? For a wine connoisseur all these questions may seem surprising and even spoil the genuine pleasure of winetasting! The authors of this book are nevertheless (not to say, obviously) wine lovers! They the deep purple color, the are convinced that there is much finance can do to unusual nose and blended flavor of a 1985 Cöte Rotie Cõte Brune. In other words, the authors truly believe that overlooking fmancial risks and their strategic implications and can even result in painful corporate always entails devastating consequences

MARKETS,

INNOVATION

Its supporters among Wall Street for long-term planning.

'hedge'

bankruptcies.

AND TRADING

analysts

ACTIVITY

say that the company

3

ís being penalized

To put it in a nutshell, the stock market is claimed to be too short-sighted to be trusted. But why is it so that modern finance theory has so far been so stubborn as not to recognize the obvious? The only answer is that its message has been misunderstood and deserves some clarification. Such a misunderstanding is all the more surprising since this message is so intuitive that it should appeal to everybody. First, it says that the benchmark is the share value: no surprise! As a manager you have got to care for your owners! You are in charge because they wanted you to be in charge. As long as you can show them that you care and do deliver, there is a fairly good chance that you will stay on board. Then, the message says that you cannot fool your fellow shareholders or outside analysts even by using the most inventive cosmetics. Well, again, a simple result: trust the law of supply and demand, the famous Invisible Hand. As they used to say in Chicago can fool some of the people some of the time, but you cannot fool all the people all of the time'. The market is a powerful machine for assessing and processing information. As a manager, you have to be humble enough to recognize that this machine is stronger than you. Last part of the message: the market will not pay at a premium for what it can undo. That is certainly the most controversial and vividly discussed part. Basically, it says that there is not much finance can do to enhance corporate share value. Nevertheless, there are virtues of financial decisions. For instance, corporate still a lot of believers in the thought financial leverage is by many to add value to the firm. Note that at the same time these people argue that in the current period of economic downturn German firms are better off than French firms because their leverage is much less. It is rather astonishing how Modigliani and Miller's simple proposition has been misinterpreted. It simply boils down to: you cannot change the size of the cake by the way you cut it. What matters is the ability of the firm to produce the cake (read:economic cash-flows). Afterwards, it does not take an expert to cut a cake (read:the market can undo it) but it takes a 'Three Star Michelin' baker (read:efficient, innovative and gifted human capital) to cook a tasty one! To many people and especially managers all this is too simple to be true. It contradicts casual daily observation: firms do care about finance. They hire CFOs, treasurers, etc. who spend days, not to say nights, balancing equity and debt, tuning corporate hedging sophisticated issuing warrants and convertibles. So, how come there is such a programs, gap between what theory says and what practice does? Well, the gap is artificial and it succumbs to a more careful analysis. It is true that firms do care about finance but not for the reasons that are most often quoted. For instance, some five years ago, it was commonplace to hear that the French corporation Thomson was making two-thirds of its profits in the financial area (throughits bank Altus). It was even suggested that financial investments with their appealing yields were cannibalizing Thomson's industrial efforts. Finance was even said to jeopardize the real economy. Such an analysis is obviously too superficial and too often confuses profits, earnings per share, cash-flows, stock prices, etc. There are only two ways whereby a firm can add value for its shareholders: either by mcreasing operating cash-flows or by lowering discount rates. The first solution is precisely what the corporation is all about. Based on its expertise, competitive edge, cash-flows. There patents, R&D expenses, etc., the firm has the ability to is not much the CFO can do inthat respect, except maybe teach the'cash-flow attitude' to 'you

DO FIRMS CARE ABOUT

I.!

FINANCE?

This section attempts to answer the following questions: .

• • • •

.

.

.

Why and how should one formulate efficient financial risk management policies? How do these policies impact in turn on company performance'? Why does proactive financial risk management add value to the firm'? How is this done? Are there good reasons for using financial markets for hedgmg purposes?

'real'

.

•

f.I.1

Finance

is a Fun Game

.

to Play, Hard to Win

Since the seminal paper by Modigliani and Miller (1958) much has been said about the objectives of the firm and the real impact of financial decisions on the firm's value. Modern finance theory contends that share value is the relevant benchmark for measur sinc$ ing performance it reflects both risk and the time value of money. An optimal by precisely focusing on those decisions allocation of corporate efforts is achievable value. Needless to say that this rather simple proposition which maximize shareholder analysts, CEOs, and policy_ has been (is still) subject to sharp criticism. Many shortmakers still consider this proposition to be a dangerous one: by over-emphasizing term effects (the price of the stock tomorrow) it penalizes long-term thinking which after all is the key to future prosperous growth. According to many critics (Hayes and Abernathy, 1980; Hayes and Garvin, 1982) such concepts as Net Present Value taught in most prestigious MBA courses are weird instruments which account for the decline of Corporate America. For instance. Lawrence M. Fischer wrote in the New York Times (May 20, 1991):

Ènancial

The more by Apple Computer to gain machines hui the resuhing is succeeding,

market strain

bv introducing lower-priced on profits has driven its stock down. share

\

'manufacture'

OPTIONS,

4

FUTURES

AND EXOTIC

DERIVATIVES

his colleagues. The second solution is precisely the battlefield of the CFO. But is it really that easy to modify the corporate discount rate, namely the cost of capital or the rate of things: first, return required by fund providers? A discount rate is basically made of two will have that chance fair Alan Greenspan, it is Unless you is a your name an interest rate. nevertheless slight exception to this: taxesThere is interest rates. a time lowering hard a After-tax rates are obviously lower than before-tax rates because the firm is subsidized by the government. Interest charges are deductible from taxable corporate income. Things this subsidy gives an obvious advantage to debt are even more complicated. Although they also equity, investors taxes: may prefer receiving dividends or capital gains pay over rather than interest payments. It depends on their tax brackets and the current fiscal which on regime. The CFO had better be smart to play around with these tax penalties time. changing over top of that are The second component is a risk premium. It is there to reward shareholders for carrying the risks of the business they own. Shareholders face two basic risks. The first the firm is involved in. Some businesses are more one is related to the type of business environment. volatile than others; they react more strongly to changes in the economic This kind of volatility cannot be changed in the short run: it takes major modifications and significant transaction costs to lower the business risk. The room for manoeuvre is noted that conglomerates of the sixties were assembled in narrow. In passing it should be would pay. 2 + 2 5 was the credo at the time. As diversification that corporate the hope time went by it soon became apparent that 2 + 2 was still equal to 4 in the best cases or add value. The market can just undo it even to three! Corporate diversification does not stocks. of In portfolio their famous 'In Search of Excellence', well-diversified by buying a should corporations that to the knitting'. The clear make and Waterman it Peters second risk is known as financial risk and arises mainly because of the use of debt leverage. We are back to the good old cake story. The situation seems to be hopeless for =

'stick

the poor CFO. Virginia, there is hope'. Indeed, the cake story As the Deep South song has it, holds true as long as some assumptions are met. For instance, every market player has to and know exactly what the cake is all about, its size, its composition (what Modigliani cake of the cherries should part There be perfect call on any information). no Miller (Modigliani and Miller initially assumed no taxation, hence no tax shield). The knife for cutting the cake should come for free (Modigliani and Miller assumed no transaction costs). The sliced cake (read levered) and the unsliced cake (read unlevered) should come risk for both the from the same baker (Modigliani and Miller assumed identical business levered and the unlevered firm). Some skeptical people will immediately argue that much is too much'. So many drastic assumptions cannot really account for the intricacies of the corporate environment. Assumptions should be taken the other way round. Now enhance that we know that in a so-called perfect world, corporate finance cannot result whether still holds the true when corporate share values, we should next ask result gives benchmark the another it To put way, are (re)introduced. indications to CFOs where to look so as to add their own part to the cake. At any rate, the imperfections, to take advantage of them, in other game for the CFO is to track those clientele. Brealey and Myers (1988) give a nice example of words to find the untapped attitude. In October 1970, ATT decided to issue Ma Bell savings bonds. such a proactive directly sold bonds to investors through the roughly two thousand ATT These were yield business offices. Up to now nothing really exotic. The clever idea was simply the 'yes,

'too

'imperfections'

FINANCIAL

i

, '

MARKETS,

INNOVATION

AND TRADING

ACTIVITY

5

offered. ATT savings bonds were copycats of US savings bonds because of ATT's high credit rating. Instead of offering a 5.5% yield like the US government, ATT offered for the same maturity 6.5%, namely a spread of 100 basis points. The issue was a success because it attracted an untapped clientele. Indeed, the rate on federal savings was locked in by law to protect savings institutions from a so-called destabilizing rate competition and to give people a less costly access to housing. As a result small investors snapped up ATT's offer. A good bargain for ATT: a regular issue on the bond market would have cost 8.6%! Nevertheless, ATT was forced to capitulate. The federal government required a cancellation of the total issue! Whatever the end, this story perfectly illustrates what we call hunt for the white elephant'. The CFO has to realize that he is not the only smart in the market. In capital markets, investors will try (and often be successful in) hunter there is no such thing as a patent in a stealing a brilliant idea from him. And remember financial market: that is why it is deemed to be efficient! 'the

1.1.2

Some

Commonly

Encountered

Pitfalls

In a recent Harvard Business Review paper Rappaport (1992)quotes a survey that had been conducted by Louis Harris and Associates in 1987 just one month before the market crash. The pollsters asked a sample of one thousand CEOs the following question: 'Is the current price of your company's stock an accurate indicator of its real value?' Fifty-eight percent answered no, and according to Rappaport most of them truly believed that the their shares. market was undervaluing the strategist guru of Harvard, supports the same view. According to Porter (1992a,b), him, the US economy is plagued by three major flaws: • • •

markets are only interested in short-term gains; Investors are impatient and they force managers to behave in a myopic manner; Managers are more interested in dealmaking than in production, service and long-term corporate strategy.

US financial

At the same time some companies like PepsiCo argue that shareholder value-based resources allocation (at all levels) ends up in truly competitive advantage. Their daily management practice does not seem to fit well with Porter's arguments. It is also true that many managers believe that the stock market does not fairly price their company's shares. They are deeply convinced that Earnings Per Share (EPS) is the real key to the game. In the Wall Street Journal (October 1, 1974) one could read that lot of executives apparently believe that if they can figure out a way to boost earnings, their stock prices will go up even if the higher earnings do not represent any underlying economic change'. and the manager is In other words, the market is only smart guy in zero upstairs' town'! The EPS or cosmetics mystique still remains very strong. But, how come managers can still reasonably advocate the foolishness of the market to justify their own behavior? According to Rappaport, the first obvious reason is that know more about their businesses than the market does and thus arrive at a different, often higher, value for their company's sharcs'. Managers strongly believe that even with fully disclosed information the market is still at a disadvantage when it comes to decipher the true and only corporate message. This belief has its roots in a second 'a

'a

'managers

'the

6

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

FINANCIAL

'short

termism vs. long termism'. The market is said not to reason which is usually called be able to decipher because it is too impatient. A lot of managers are convinced that the market does not take the long view and cares more about visible short-term results. This argument even leads some of them to sacrifice crucial investments to concentrate on more impressive short-term gains. General Electric and Westinghouse are good examples when they decided to leave the industrial robotics arena. Porter has recently argued that the reason for such mistakes lies in the widely dispersed ownership of American companies. According to him, Germany and Japan are much less exposed because of more concentrated ownership. Needless to say, this line of reasoning is rather dubious and not at all supported by empirical evidence. Nevertheless, a lot of managers continue to worship EPS although there is now ample evidence that the market knows how to read between the lines. To paraphrase Bernstein (1992),'American financial markets suffer from the ignorance and longstanding suspicion of people who function outside those markets. If something is wrong in the real economy, why not find a way to pin the blame on financial markets?'. McKinsey (1990)classifies the evidence against the hypothesis' into three classes: .

.

.

'scapegoat

•

Numerous studies show that accounting earnmgs are not very well correlated with share prices. The market is not reacting myopically to changes in EPS. As Rappaport appropriate, the market uses unexpected changes in earnings as a useful has it, 'when

•

•

proxy for reassessing a company's future cash flows.' Numerous studies show that earnings window dressing does not enhance share prices. Announced changes in accounting methods that impact on reported earnings but not on cash flows do not affect stock prices at all. For instance, switches from LIFO to FIFO (or the other way round) in inventory accounting do not affect corporate cash flows except for increased or decreased taxes. Numerous studies show that the market does indeed take the long-term view. A simple test of the time honzon of the stock market is to examine how much a company's current share price can be accounted for by its discounted expected dividends. Rappaport (1992)quotes a recent study by the Alcar Group which is quite revealingThe Alcar Group analyzed the stock prices of 30 Dow Jones Industrial companies. It found that between 80% and 90% of the stock prices were attributable to expected cash flows paid out in the form of dividends beyond five years'. Table 1.1 summarizes their results. 'typically

In the same vein, studies on R&D expenses reach the same conclusions. Studies by SEC (1985)and Randall Woolridge (1988) show that companies with the highest levels of spending on R&D tend to have the highest P/E ratios. This obviously contradicts the standard EPS view that the market working horizon does not go beyond its is king'! The market believes in one single god: cash fiow. There The truth is that is not much cosmetics can do. The irony is that even though more and more managers message' tend to recognize this, they still argue that they have to depart from the They simply forget that the market because of free-riders or aggressive competitors. analyzes not only the outputs of the decision to invest but also the damaging long-term of being able to sustain a real competitive consequences of not investing. The credibility advantage in the long run does not come for free. 'nose'.

'cash

'key

MARKETS,

I.I

TABLE

INNOVATION

The Stock

Market

AND TRADING

Takes

7

the Long View

Price of3 (os I Dec. 9 I)

Company

ACTIVITY

CumulativePresent Volue of Five Year Dividend Forecost

% o¶Current Share Pace . Attributable to E×pectations 8eyond Five Years

--

ALCOA Anled-Signal American Express

$

AT&T Bethlehem steel Boeing

14.00 47.75 43.88 69.00 80.25 I 14.50 46.63 48.25 60.88 76.50 28.88 53.50 89.00 70.75 38.00 166.50 95.25 68.63 80.25 93.88 37.88 61.25 20.25 54.25 18.00 26.50

caterpillar Chevron Coca-Cola

Disney Du Pont Eastman Kodak Exxon General Electric General Motors Goodyear

IBM international

Paper

McDonald's Merck 3M

J.P. Morgan Philip Morris Procter & Gamble

Sears, Roebuck Texaco

Union Carbide United Technologies Westinghouse Woolworth Source: Briys and Crouhy

1.1.3

Some

Good

64.38 43.88 20.50 39.13

$

6.22 3.56

3.80 5.92

2.15 5.! I 5.02 \3.59 4.72 3.l8 7.00 7.42 i \.36 8.77 6.06 l.79 19.73 6,65 l,6l I l.5l 12.67 7.96 10.05 9.45 7. I2 12.56 3.56 6.86 5.07 4.43

90.3% 91.9 81.5 84.9

84.7 89.3 88.6 80.3 94.I 97.2 85.0 84.6 8 i.3

88.5 79.0 96.7 77.8 90.6 95.8 93.1 86.7 88.4 87.5 89.9 8l 79.5 82.4 87.4 71.8 83.3 .2

(1993) Reasons

for Hedging

Before going into details of how a firm should hedge, the first question to be answered is whether a firm should hedge. As the reader may recall from the Introduction, the issue is not as straightforward as it might initially appear. Hedging is a financial decision which at first sight does not modify the size of operating cash flows. Moreover, even though wipe out financial risk, it does not pay a corporation hedging, asset/liability management to eliminate fmancial risk if shareholders can achieve the same result for the same or lower cost. The market does not value what it can undo. Nevertheless, as we have seen in our brief overview of Modigliani and Miller, some reasonable arguments can be put forward to legitimate proactive risk management.

OPTIONS,

8

FUTURES

AND EXOTIC

DERIVATIVES

1. Taxes again play a significant role. The following example based on Stulz and Smith why taxes may induce a firm to hedge. Assume that the before-tax corporate income may take only two values with equal probability, namely 200 and -100. The tax rate is 40% and there is no loss offset. In other words, when the firm experiences losses, it is not entitled to any tax credit. The expected after-tax corporate income is equal to:

(1985)shows

l /2(200 80) + 1/2(-100)

=

-

10

The expected income before tax is equal to: 1|2(200)

Namely, an after-tax

amount

equal

+ 1/2(-100)

=

50

MARKETS,

INNOVATION

AND TRADING

ACTlYITY

9

disastrous ventures in the high technology business filed for Chapter 11 protection. The Multigraphics Division of AM International had seven presidents in as many years. It is rather obvious that shaping a so-called corporate culture becomes rather difficult in such an environment. 5. Bankruptcy is usually a very costly procedure: legal fees, transaction costs, etc. Fund providers usually ask to be compensated for these extra costs through higher returns. 6. Loan covenants may sometimes be stringent enough to motivate a proactive risk financial risk may even reduce the corporation management. To some extent managing borrowmg costs.

to 30.

The firm is better off in the second case which is precisely the case whereby any variance in the profit function has been eliminated. The firm gets a clear incentive to undertake risk management policies. Note that in the case of a linear tax schedule (i.e. full loss offset) there is no room for hedging. Again, a market imperfection (non-linear tax schedule) helps justify a financial decision and improves the shareholder's position. 2. Shareholders may not know exactly the current risk exposure of the firm (asymmetric information) and even if they did they may not have easy access to risk control instruments. Levi (1990)for instance reports that a shareholder whose share exposure is $100 might face a bid-ask spread fifty times that of the company whose exposure is, say, over $1 million. In some cases, the shareholder may not have access to hedging instruments at all. 3. As Jensen and Meckling (1976)put it, a firm is just a nexus of contracts. A stable corporation may find it easier to negotiate its contracts with suppliers, clients, etc. No default could obviously be a strong selling argument. Reputation effects may also be very damaging. For instance, the value clients place on service agreements, warranties or technical assistance obviously depends on the firm's financial stability. This is especially goods, namely goods or services for which quality and care are true for credence important but for which quality and care are difficult to assess prior to consumptionWine, restaurants, hotels, air travel belong to this category. Frequent travelers knew exactly what it meant when it came to travel on Pan Am. Customers tend to think that product quality (read safety in the case of air travel!) is lowered when the firm is on shaky ground. A weak firm will have a hard time convincing its clients that it can really deliver. That is precisely why Lee Iacocca decided that Chrysler should not file for bankruptcy:

Our situation was unique. It was not like the cereal business. If Kellogg's was known to be going out of business, nobody would say: well, I won t buy their cornflakes today. What if I get stuck with a box of cereal and there is nobody around to service it? (Fortune, November 26, 1984) 4. Corporate employees suffer from poor asset diversification: slavery is prohibited and within the company. There is now ample evidence that more their human capital is risky firms command higher wages. A lower employment uncertainty, other things being equal, means lower risk premiums, namely lower wages. This in turn may increase operating cash flows. Moreover, financial distress may be conducive to a high turnover of Shapiro and Titman (1986) quote the example of AM International which after managers. 'stuck'

FINANCIAL

1.2

HOW TO IMPLEMENT RISK MANAGEMENT

STRATEGIC

FINANCIAL

exposure to risk on expected cash fiows jeopardizes the future of the corporation as the previous section has shown. As risk goes up, the corporation's cost of simply doing business rises and renders its survival precarious. Just the rumor of some prophecy) whereby clients, difficulties is conducive to a bandwagon effect (self-fulfilling suppliers, distributors, subcontractors and so on run away from the firm. Usually, firms which produce goods or services that require maintenance or repair, that are based on credence, that involve significant switching costs, whose production mainly depends upon intangible assets (managers,brand names, savoir-faire), are likely to suffer greatly from

Large corporate

risk exposure.

As a consequence, the first duty of the CFO is to determine the corporation's global risk profile. Some businesses are more exposed than others. Indeed, the very nature of the business the corporation runs is sometimes highly conducive to financial risks exposure. A few examples are in order. The introductory exampic of the vineyard business is the most obvious one. Indeed, the process of wine-aging exposes the producer to the volatility of interest rates. The airline industry is another industry whose financial risks. A famous example is the wellbusiness cycle carries numerous documented case of Lufthansa. In 1985, Lufthansa placed an order with Boeing for know that aircraft delivery takes place much later some $500 million. Airline companies on. In the case of Lufthansa the deadline was one year. At the time of the order the dollar was very strong vs. the Deutsche Mark and a lot of people believed that the dollar was on the verge of reversing its upward trend. After much debate, Lufthansa decided to hedge only half of the amount by buying dollar forward. The rest of the position was just left open. Ex post the story was that Lufthansa would have been off' not hedging at all. In other words critics were just recommending straight speculation! The truth is that Lufthansa should have first looked at its entire currency which is not exposure and then decided to cover the residual component hedged. Instead of running a careful investigation, Lufthansa just picked the least flexible instrument in the range. Airline companies are not only exposed because of the purchase of planes. They are also exposed because of the fierce competition on airfares. Deregulation which is now part of their daily business has forced them to ofTer not only competitive prices but also to these prices over a contractual time span. Needless to say, in the meantime the price of kerosene might have gone through the roof' Another example is the pharmaceutical industry. R&D is one of the crucial keys 'better

'naturally'

'freeze'

OPTIONS,

10

FUTURES

AND EXOTIC

in that business. For instance, Merck funds its US R&D through worldwide sales of which 40% are in foreign currencies. Any adverse change in currencies might endanger Merck's capacity to strengthen its R&D efforts. Some more subtle risk effects are also pervasive. Indeed, tackling the corporate risk issue requires taking a comprehensive view of the firm. The right approach has to encompass many dimensions such as investment, production, input-mix, pricing policies and financial decisions. These different items shape the structure of assets and liabilities, the stream of expected future cash flows, and obviously the risk profile of the firm. Risk is itself manifold and can be channelled to the firm through currencies, commodities and interest rates. The management of foreign exchange, commodity and interest rate risks has to be part of the global strategy of the firm. A comprehensive risk management program should incorporate expectations about changes in exchange rates and interest rates in all the above mentioned corporate decisions which affect the future cash flows, the financial structure, and the corporate risk profile. The question 'What is at risk?' must be treated stepwise. The identification procedure is based on a top-bottom approach rolling down from the strategic level to the financial level.

Strategic

FfNANClAL

DERIVATIVES

Level

The answers directly affect the investment, pricing, marketing and contracting policies of the firm. Foreign exchange swings may affect the company's competitive position. For example, whose costs are denominated in a depreciating currency will have a greater competitor a pricing flexibility and thus a potential competitive advantage: barriers to entry may become less effective and competition may affect profit margins. This is true even for domestic companies which buy and sell in their home currency. A purely domestic firm usually feels comfortable because it does not have the explicit burden of managing an interest rate or foreign currency exposure. This is the so-called 'I don't sell abroad, I don't have any exposure' syndrome. For instance, Nordic aluminum producers consume mainly local hydro-electricity and do not use imported oil. At first sight they do not have to worry much about the dollar's possible movements. Well, we are not so sure. A falling dollar may result in more competitive foreign producers. A falling dollar may even attract in Deutsche new market players. A firm need not pay bills in dollars or get revenues Marks to be exposed to currency risk. The very fact that its competitors are or could be thus exposed is enough to generate a strategic exposure. The message is then simple: are not exogenous, they are with your business'. The threat of functional substitutes should also be considered, whereby consumers switch from one product to another one. For example, a dollar appreciation against European currencies may alter the taste of Californians who will consider drinking French or Italian white wines instead of their tasty Chardonnays from the Nappa Valley. Foreign exchange and commodity movements may change the relative prices of raw materials and other key inputs of the industrial process, which may trigger technological changes. In most cases when these shocks are temporary, the best hedge could be a flexible oroduction process which allows for different input-mixes.

MARKEW dilillillillbaha11--IINiimliatagotNG ACTIVITY

¡¡

Economic Level "

A

The answers should come from a simultaneous analysis of asset and liability exposures, and in trying to locate natural hedges in the balance sheet. Changes in exchange rates, inflation, interest rates and commodity prices affect the company's cash flows, both: • •

directly, through interest rate payments, raw materials purchases or indirectly, for instance through the impact of higher financing costs for customers on the demand for the company's products; these gyrations may also induce a relative price effect which affects differently the costs and the revenues, putting the firm in a squeeze regarding its profitability.

Interest rate risk deserves special treatment since it cannot be dissociated from inflation risk. The main determinant of long-term yields is expectations of inflation. Therefore, for those companies whose long-term operating profits follow closely innation, higher inflation means higher revenues to offset the increase in financing cost. For such companies, fixed-rate debt increases the interest rate risk exposure since it will generate an unexpected loss when the rate of inflation (i.e. of interest and of revenue growth) unexpectedly slows down. On the contrary, it will provide a windfall gain to the company when interest rates and inflation increase. Thus, while fixed-rate debt seems to eliminate uncertainty about interest rate payments, it represents a bet on inflation. Floating rate debt, by contrast, offers a long-term natural hedge against inflation risk, although interest rate payments will fluctuate over time. It is the other way around for companies whose operating profits are quite insensitive to changes in interest rates, Finance Level At this stage the company is concerned only with the residual financial risks that, by any means, it wants to shift away from the balance sheet to the capital markets. This is a purely technical approach for which the company can hire the skills of fmancial managers recently trained in modern investment banking. Hedging tells you what type and how many forward contracts, futures contracts, swaps, options to use to transfer the undesired risks to speculators who stand ready to bear these risks for a reward (i.e. a positive expected profit). Financial engineering can be helpful in designing securities which bring together issuers and investors (read which complete the market) and allows firms to hedge at a lower cost than with traditional instruments.

'risks

Forward Rote Contracts They are the simplest and most basic hedging instruments. A forward contract is an agreement between two parties to set the price today for a transaction that will not be completed until a specified date in the future. For example, a forward contract for $1 million, to be delivered in 6 months, at a price of FF6.00 for a dollar. These terms oblige the seller of the contract, for example the bank. to transfer on the specified date, $1 million on the company's account, for the price set today. On the other hand, the buyer has no alternative than to accept delivery under the terms of the the contract at a later date, is to contract. The only possibility for the buyer to 'cancel'

12

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

FINANCIAL

MARKETS,

INNOVATION

AND TRADING

ACTIVITY

13

enter into a reverse forward contract, with the same bank or another institution, but at the expense of a loss (or a gain) since the new forward rate will be set at a new equilibrium level Forward rate contracts are flexible and allow for customized hedges since all the terms with the counterparty. However, each side of the contract bears the risk can be negotiated that the other side defaults on the future commitments. This is the reason why futures contracts are often preferred to forward contracts.

indexed Bonds When the operating profits of a corporation are exposed to the fluctuations of an index, as with a commodity price like oil, aluminum, or inflation - the exposure risk can be partially hedged by issuing bonds whose interest payments and/or principal repayment is linked to the index, in such a way that the effective cost of debt is movement reduced when there is an unfavorable in the price index, and is increased to the when the of the is favorable investors movement to the firm. benefit Some bond issues can be split into two parts: the bull and bear tranches, so that investors can choose only one side of the risk exposure. These bonds allow some investors

Like forward contracts, futures contracts Futures Contracts are used to lock in the company interest rate, exchange rate or commodity price. But futures markets are organized in such a way that the risk of default is (almost) completely eliminated. This is possible by trading futures contracts on an organized exchange with a clearinghouse which steps in between a buyer and a seller, each time a deal is struck in the pit. The clearinghouse adopts the position of the buyer to every seller, and of the seller to every buyer, i.e. the clearinghouse keeps a zero net position. This means that every trader in the and has strong expectations futures markets has obligations only to the clearinghouse, that the clearinghouse will maintain its side of the bargain as well. The credibility of the system is maintained through the requirements of margin and daily settlements. The main purpose of the margin is to provide a safeguard to ensure that traders will perform their obligations. It is usually set to the maximum loss a trader can experience in a normal trading day. Daily settlements or marking-to-market impose on each trader an obligation to realize any losses (or gains) on the day they occur. Since for each trader who loses (gains)there is another trader who gains (loses), this is a zero sum game. Marking-to-market just consists in a transfer of cash from one account to another. The elimination of default risk has a cost: contracts are standardized in order to bring liquidity to the market, there are only a few financial assets which are traded on futures markets and they do not necessarily correspond to the risk to be hedged. Therefore, there is no perfect hedge with futures contracts. The hedgers keep what is called a basis risk and a correlation risk which cannot be fully eliminated.

to take risky positions which are not directly available to them, or not allowed, on organized markets. Investors are ready to pay a premium for these opportunities which is translated into a reduced financing cost.

Swaps allow the exchange of one type of debt for another: fixed rate debt against rate debt, in the same currency or in a different currency. All combinations are possible. Like forward contracts, swaps are OTC instruments and subject to default risk. Swaps

floating

When a corporation has a low credit rating and must Warrants and Convertibles implement a large investment program to survive, it may well be too costly to issue standard debt, while raising equity might dilute considerably the current shareholders' position. Then, warrants (bondswith attached warrants) and convertibles become the only affordable financing instruments. When Faust struck his deal with the Devil, he ended up holding a naked position. Needless to say he would have been much better off reading our book and hedging his and financial very risky position. Obviously, the Devil would have been disappointed markets are certainly absent from Hell! Like Faust, any corporation strives for better returns. Unlike Faust, it knows it has to monitor the level of risk taken. Imagine a world without any financial instruments, any derivative assets, any financial engineering. Corporations would end up in a Faustian position with their fates gyrating at the same pace as currencies, interest rates or commodity prices. Their normal business course of action would be polluted by financial risks. Corporate performance would be affected by the dice thrown by the Devil. It would become hard distinguishing talented corporations because of a thick financial fog! Not many solutions are available to disentangle corporate performance from purely financial risks. Financial markets are the most natural candidates to fulfil this thorny task. They innovate every day to provide corporations with the most suitable risk hedging they also enable range. They not only serve as a most useful value barometer, entrepreneurs to demonstrate their true talents.

Options are more flexible than forwards and futures in the sense they provide Options the buyer with the protection needed, and leave him with the full benefits associated with a favorable development of the commodity price, interest rate or exchange rate. This nice feature has a price: the option premium. On the contrary, forward prices, futures prices and swap rates are set at a level such that the initial price of the contract is exactly zero. A cap is a series of interest rate options which allow a corporate borrower to put a ceiling on the borrowing cost of a floating rate debt.

Trading is done in several ways around the world and across financial markets and assets. The most common trading mechanisms are the continuous quote-driven system and the

There is no such thing as a free lunch in finance. The protection brought by Hybrids options is appreciated by corporate treasurers, but is very often considered too expensive. Hybrids (cf. Briys and Crouhy, 1988) are special options, or packages of options, whereby the upfront premium of the protective option is reduced by giving up part of the benefits derived from a favorabic movement in the market.

order-driven system. firm price In a quote-driven system, known also as a continuous dealer market. This is quotations can be obtained directly from market makers before order submission. the case, for example, in the NASDAQ the London International Stock Exchange. In or an order-driven system, orders transmitted by investors go through an auction process. This system can be continuous system or continuous or periodic. In the continuous

I.3

TRADING

MECHANISMS

IN SECURITIES

MARKETS

OPTIONS,

14

FUTURES

AND EXOTIC

DERIVATIVES

FINANCIAL

1.4

auction, orders submitted by estors can be executed immediately by dealers on the floor or against limit orders submitted by public investors or dealers. Since orders are executed upon arrival, the system is continuous. Yet, it operates as an auction since prices are determined multilaterally The Paris Bourse operates à la criée or as an open-outcry system and as an electronic system for options. The market is electronic for stocks. For more details, see, for example, Bellalah and Jacquillat (1995), The Swiss Option and Financial Futures Exchange (SOFFEX), the Toronto Stock Exchange's Computer Assisted Trading System (CATS), the Frankfurt Stock Exchange, and the Tokyo Stock Exchange's Computer Assisted Routing and Execution System (CORES) are also examples of continuous

of orders, no more market-on-close

l.4.I

compete The market giving their bid and ask quotes. The best quotes are displayed on the screen of the exchange. For more details, see Berkman (1992).

'strike

Options

and Synthetic

Long a synthetic

underlying

Short a synthetic. underlymg .

asset: long a call + short a put

asset:

=

(0,

1) +

(1, 0)

short a call + long a put

(-l,

-1)

=

(0,

'

l) +

(-1,

'

0)

.

Long a synthetic call: long the underlying

side.

book when on the floor

'maturity

Positions

(1, l)

1)

(0, Short a synthetic

i

=

call: short the underlying

(0, -l)

'open

with each other and with orders in the limit order

15

Options enable investors to customize cash-fiow patterns. We illustrate some of the most commonly used option strategies which apply to options on a spot asset, to options on a futures contract and in general to options with any particular pay-off. These strategies are illustrated with respect to the diagram pay-offs or the expected return and risk trade-off of standard options. The understanding of option strategies is based on the use of synthetic positions. Synthetic positions can be constructed by options on spot assets, options on futures contracts and their underlying assets, If we use the symbol 0 to denote a horizontal line, the symbol - I for the slope under 0 and the symbol 1 for the slope above 0, then it is possible to represent the diagram payoffs of a long call by (0, 1), a short call by (0, -1), a long put by (- 1, 0) and a short put by (1, 0). Adopting this notation for the basic option pay-offs, it is possible to construct all the synthetic positions as well as most elaborated diagram strategies using this representation. We give now the basic synthetic positions when the options have the same strike prices and maturity dates.

For more details, see Brock and Kleidon (1992) The EOE market combmes the features of a dealer market and an auction market. The outcry system' with competing market makers on the fioor. trading begins with an makers

ACTIVITY

CONTRACTS

'underlying

intermediary. However, in practice we observe a combination of the two systems. For example, on the Chicago Board Options Exchange, the NYSE, the European Options Exchange and others, the best quotes in the market can come from a dealer who is a market maker quantity or a specialist or from limit orders. A sell limit order can be defined by the with order and buy limit defined the limit, can be a to be sold at a stated price, called order limit is stated price. the quantity Hence, the a be bought at respect to to comparable to the quote of a market maker standing to sell at his ask quote and to buy at his bid quote . We give two examples of options markets: the NYSE and the European Options Amsterdam. (EOE) in Exchange The NYSE opens with a call market where the specialist matches buy and sell orders. Any resulting imbalance is made up from the inventory of the specialist. Then continuous double-sided auction trading is observed until the close. The usual procedure before the orders either by hand or by the close is that the specialist matches market-on-close automatic trading system, Opening Automated Report System (OARS). If there is an imbalance

OPTION

AND TRADING

'hors

The same trading mechanisms apply also to options markets where often a distinction is made between dealer markets and auction markets. In a dealer market, dealers provide liquidity by displaying quotations at which they are willing to trade. In auction markets, the orders emanating from the public are matched with other public orders without an

on the excess

TRADING

INNOVATION

This section illustrates the many possibilities offered by a specific set of derivative assets, namely options. It is a kind of d'oeuvre' to Chapters 5 and 6 where standard options are covered in a lot more depth. Some preliminary definitions are in order. We nrstdefine a call option. A call option is the right, not the obligation, to buy a specific security (the asset') at a specific price (the price') until a specific date (the date'). A put option is just the same, except it is a right to sell the underlying asset. When the right to buy ('call') or to sell ('put') can only be exercised at the maturity date, the option is said to be 'European'. When this right can be exercised at any time, the option is said to be 'American'.

auction systems. The second type of order-driven system, where orders are stored for execution at a single market clearing price, is known as a periodic auction. This is often used at the opening of many continuous markets such as the New York Stock Exchange (NYSE) and the Tokyo Stock Exchange. In practice, most trading mechanisms are complex hybrids of these systems. For example, at the open, the NYSE operates as an auction market and then switches to a dealer market. For more detail and an excellent analysis of the merits of each system, see, for example, Madhavan (1992).

orders are submitted

MARKETS,

=

asset + long a put (I, l) +

'

(-1, 0)

asset + short a put,

(-l, -- l)

+

(1, 0)

Long a synthetic put: short the underlying asset + long a call,

(-1,

0)

=

(-1,

-1) +

(0,

1)

OPTIONS,

16

FUTURES

FINANCIAL

ANDEKOfiCOGIU¥ATIVES

I.4.2

The Basic Option

=

INNOVATION

AND TRADING

ACTIVITY

=

(1, 1) + (0, -1)

Strategies

=

to trade

of options. The best known strategies in portfolio management involve combinations volatility They include vertical spreads, calendar spreads, diagonal spreads, ratio spreads, spreads and synthetic contracts. with the A vertical spread involves the purchase of an option and the sale of another produces the different strike When strategy and maturity a cash time pnce. to a same outtlow, we say that the investor is long the spread. When the strategy generates a cash intlow, the investor is said to be short the spread. The strategy can be implemented by calls or puts using different strike prices. option For example, when an at-the-money option is bought and an out-of-the-money is sold, the strategy is a vertical bull spread. The calendar spread strategy represents a position where the investor is long an option with a longer term and short an option with a short term for the same strike price. The diagonal spread involves the purchase of an option with a longer term and the sale of another with a short term where both options have different strike prices. The diagonal spread is bullish when the purchased option is at parity and the short option is out-of-thenao

Examples of l tility strategies are often based on the underlying asset volatility. corresponds combinations. A to the straddle volatility strategies include straddles and maturity A date. the and strike with the price call and same purchase of a same a put combination is a straddle for which the option exercise prices are out-of-the-money. when it When the strategy implies a cash outflow the investor is long the strategy, and the strategy. involves a cash inflow the investor is short of a A synthetic forward contract can be created by the purchase of a call and the sale synthetic long the forward the investor this is price. In with strike the case, same put that the contract. The main difference between forward contracts and option contracts is establish position. forward options nothing but to a investor pays a premium for

=

spread. The decision to trade an option with a given strike price depends on the trader's the market direction. If he thinks that the underlying asset price expectations regarding will end above 1040 at the maturity date, he can buy a near-the-money option with a strike price of 1000 at 34. If the underlying asset price is 1040 at maturity, the position shows a profit of 6 points, or (1040 1000) 34 6. If the trader estimates that the underlying asset price would not hit 1040 at the maturity date, he can sell the 1050 or 1100 calls. The sale of the 1050 call at 16.7 generates a gain of 16.7 if the underlying does not attain 1050 at the maturity date (since the call is worthless at that asseetLprice

Since there are at least three maturity dates and from three to sometimes 12 strike prices which option quoted on official option markets, the fundamental question is to determine

Trading the Underlying

a

-

Asset

If we put the underlying asset price on a horizontal line and the profit or loss on a vertical lme, the pay-off at maturity date to a long or a short position in the underlying spot asset is represented as shown m Figure 1.1. If the asset price rises or falls by one point, the profit or the loss will be of the same amount. Trading Calls When an investor buys a call with a strike price of 1000 at 34, the profit-loss relationship at the maturity date is represented as shown in Figure 1.2. When the underlying asset price is less than 1000, the call is worthless and the position shows a loss of 34. When the underlying asset price is 1034 (1000 + 34), the option is worth 34 and the position shows neither a profit nor a loss. This is the break-even point. Above 1034, the profit from the 1000 call is unlimited. On the other hand, under any circumstances the loss can not exceed 34 lf the investor is long the 950, 1000 and 1050 calls, the profit-loss relationship is represented by Figure 1.3. If the investor is short the 950, 1000 and 1050 calls, it is Profit

Consider, for example, an underlying asset with options traded on it. This asset is currently quoted at 990. Table 1.2 shows the call and put market prices for various strike TABLE

I.2

Call and Put Market

Prices for Different

Strike

i

Underlying asset

Prices

Strike Price, K

Call price, c Put price, p

17

10% and the volatility prices when the maturity date T 0.25 year, the interest rate r of the underlying asset is 20%. These prices correspond to the middle of the bid and ask

Short a synthetic put: long the underlying asset + short a call, (I, 0)

MARKETS,

850

900

950

I 000

I 050

I l 00

|39.2 2.7

96.4 8.4

60.43 2l

34 43

I6.7 73.7

109.8

4.

FIGURE

I.I

Long and Short

the Underlying

Asset

at 990

OPgligilig,4GTURES

AND EKORIG

iiafVATIVES

FtNANCIAL

Profit

i di

4AND

TRADING

ACTIVITY

19

Profit

E

=

1000

!

i

34

FIGURE 1.2

MARKETS,

i

1034

950

FIGURE I.4

Long a Call

Short

1000

Calls with D fierent

I

Underlying

105

asagt

Strike Prices

Profit Profit

950 950

i

1000

g

Long Calls with Different

1050

-

Underlying asset

Underlying asset

6

FIGURE Strike

i

t · .

(3) FIGURE 1.3

1000

105

Prices

I.5

Long Puts with Different Strike

Prices

Profit

represented by Figure 1.4. Note that the previous results are inverted, since the investor trades the same calls by taking the other side of the transaction. Trading Puts

Figure 1.5 represents the pay-off at the maturity date of a strategy which consists in buying the 950, 1000 and 1050 puts. Figure 1.6 corresponds to the sale of the 950, 1000 and 1050 put options. It is in some when the market way the inverse of the preceding one. Each position shows a limited gain risk unlimited when the market and falls. rises an It must be clear that the option seller assumes an unlimited risk, since he receives the option premium.

950

FIGURE

1.6

Short

s

1ooo

0

i

1050

Puts with Different Strike

Prices

Underlying asset

OPTIONS

20

1.4.3

Straddles

Trading

BITWRES AND EXOTIC

DERIVATIVES

FINANCIAL

INNOVATION

AND TRADING

ACTIMIT¥

11

Profit

Strangles

and

MARKETS,

When an investor trades a call and a put at the same time with the same strike price and maturity, this strategy represents a straddle. When the options have different strike prices, the strategy is a strangle. 77

Uf ±1y

Buying a Straddle

923

The straddle strategy can be implemented, for example, by paying 34 for the 1000 call and 43 for the 1000 put, or 77 (Figure 1.7). If the underlying asset is above 1000 at the maturity date, the put is worthless and the call is in-the-money. However, if the underlying asset is below 1000 at the maturity date, the call is worthless and the put is in-the-money.

or a Strangle

I.8

Short

g asset

107

(1)

FIGURE Selling a Straddle

1000

(2)

a Straddle

Profit

When the investor has strong expectations that the underlying asset price will lie in the interval (923, 1077) at the maturity date, he can sell the 1000 call and the 1000 put simultaneously. This is a straddle since the options have the same strike price. The profit this from strategy is limited to the option premium, 77, and the loss is unlimited when the market rises (Figure 1.8). When the options have different strike prices, the strategy is referred to as a strangle (Figure 1.9). If the investor does not expect great variations in the market until the maturity date, he can, for example, sell the 950 put for 21 and the 1050 call for 16.7. The profit from this position will be 37.7 when the underlying asset price is within the interval (950, 1050)· Note that the profit on this position, 37.7, is less than that on the previous one, 77, since reduction in risk must be accompanied by a decrease in the expected return a

1077.7

922.3

eso

1000

(1) FIGURE 1.4.4

I.9

Trading

Short

Underlying asset

1oso

(2) a Strangle

Spreads

The Can Spread Profit

0)

The strategy which consists in buying, for example, the 900 call for 96.4 and selling the 1000 call for 34 is known as a spread. If the underlying asset price is less than 900 at the options' maturity date, the options are worthless and the maximum loss is limited to the premium paid, 62.4 or (96.4 34) as shown in Figure 1.10.

(2)

923

1000

1077

e

.

Unde yingasset

If the underlying asset price is above 1000, the 1000 call is worth 100, i.e. (1000 900) and the maximum gain is 37.6 or (100 62.4), as also shown in Figure 1.10. with strike price. The investor sells the call spread when he is long the call a higher

77

The Put Spread

When the investor is long the 1050 put FIGURE

I.7

Long a Straddle

strategy is implemented

by paying

at 73.7 and short the 1000 put at 43, a put spread 30.7, or (73.7 43), as shown in Figure 1.11. The -

22

OPTIONS,

FUTURES

AMD EKOTIC DERIVATIVES

FINAg

INNOVATION

MARKETS,

AND TRADING

ACTIVITY

23

Profit

Profit

900

.. 950

37.6

g I

1000

Underlying assef

950

Underlying asset

i 1050

.

62.4

FIGURE 1.12 FIGURE

1.10

A Call Ratio

A Bullish Calt Spread

If the underlying asset price is 1050, the 1050 calls are worthless and the 950 call is worth 100, or (1050 950). The position is worth 89.67, or (100 10.33). Since the investor is long a call and short three calls with different strike prices, this position is similar to the sale of two units of the underlying asset. Hence, an increase in the underlying asset price by one point yields a loss of two points in the position.

Profit

16.3

Underlying asset 50

FIGURE

I.! I

A Bearish

Trading

Conversions

and Reversals

A strategy can be implemented by going long the 1000 call at 34 and short the 1000 put at 43. If the underlying asset price is above 1000 at the option's maturity date, the put is worthless and the call's value corresponds to the intrinsic value. The position will behave exactly as the value of the underlying asset. However, if the underlying asset price is below 1000 at the option's maturity date, the call is worthless and the put's value is its

Put Spread

investor is said to be long the put spread. This strategy is appropriate if the investor expects a falling market, If the underlying asset price is 1000 at the maturity date, the 1000 put is worthless and the 1050 put is worth 50. The maximum profit is 19.3, or (50 30.7). The investor can be short the put spread if he sells the 1050 put for 73.7 and buys the 1000 put for 43. This strategy implies an entry of cash equal to 30.7.

1.4.5

1.4.6

30 7

intrinsic value. The position will again asset, and is represented in Figure 1.13.

behave exactly as the value

Profit

Ratios

oo

A call ratio strategy (Figure 1.12) can be implemented by buying an option and selling more options with different strike prices. Similarly, a put ratio strategy can be implemented by selling an option and buying more options with different strike prices. The investor can construct the position by going long the 950 call at 60.43 and short three calls, strike price 1050 at 16.7. This call ratio implies a cash outlay equal to 10.33. If the underlying asset price is 950 at the option's maturity date, the 1050 calls are worthless and the loss on the position is equal to 10.33, or (60.43 3 × 16.7)

-

of the underlying

34

FIGURE

1.13

Long a Call, Short

a Put

Underlying asset

24

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

Buying the call and selling the put is a position equivalent to buying the underlying relationship asset. More generally, the following no-arbitrage (also called put-call parity, shown call, Chapter 5) between c, a put, p, the underlying asset price, S, and the as a m strike price, K, must hold: c- p where

=

S - Ke

"

(1)

for the riskless interest rate and Tis the option's maturity date. A conversion is a strategy based on the above relationship. It can be written as: short a call + long a put + long the underlying asset r stands

.

=

Areversal

short a synthetic underlying

corresponds

asset + long the underlying

simply to the reverse

long a call + short a put + short the underlying =

long a synthetic underlying

asset

conversion:

Trading

ACTIVITY

25

for the value

(6) premium

of the early exercise

for calls and puts

These relations account with different strike prices. If condition (5) were not satisfied then selling the American call and buying the European call (with a strike price K2), or buying the American call and selling the European call (witha strike price Ki), would allow an immediate profit. If the American call with a strike price K2 is not exercised before the maturity date, the position produces a zero cash flow at that date. If the call with a strike price K2 is exercised, the option with a strike price Ki can be exercised to generate a cash flow (K2 ¯ Ki) which will be invested until the maturity date T If the option with a strike price K2 is exercised before the maturity date at a date ti < T, the result at maturity is > K2

- Ki for options

(7)

traded on the Chicago the box strategy of the no-arbitrage years and found some violations when transaction profitable opportunities disappeared condition. However, the costs were taken into account. They concluded that the market is globally efficient.

(1989) tested

Board Options Exchange over

eight

SUMMARY a Box Spread

-

.

.

•

AND TRADING

P(K2) - p(K2) > P(Ki) - p(Ki)

Aimec and Ehud

with options on spot or options The box spread strategy can be implemented on futures. In the following discussion, c(K,), c(K2), p(Ki) and p(K2) denote respectively the Urices of calls and puts with strike prices K, and K2, with K2 > K,. Consider a portfolio correspondmg to the following strategy:

•

INNOVATION

(K2 - Ki) e' asset

If we substitute the underlying asset by a synthetic underlying asset in the conversion strategy for a different strike price, this eliminates the risks associated with the variations of the underlying asset price and gives a well-known strategy, the box spread. The box spread is simply a strategy equivalent to borrowing or lending money for a certain period.

1.4.7

MARKETS,

asset

+ short the underlying

asset

FINANCIAL

long a bullish call spread: buy c(Ki) and sell c(K2) long a bearish put spread: buy p(K2) and sell p(Ki).

This strategy is a box spread which costs

the practice of financial risk management are In this chapter, the main reasons underlying detailed The different tradmg mechanisms in securities markets are studied. In particular, a distinction is made between continuous mechanisms and periodic mechanisms. .

.

-

In particular, Fmally, the main trading strategies in options markets are illustrated. strategies involving calls and puts, straddles, strangles, conversions and reversals and the box spread are studied. These tradmg strategies can be used for most of the derivative implemented using options with any assets covered m this book, since they can be particular pay-off.

c(Ki) - c(K2) - p(Ki) + p(K2 The following non-arbitrage

condition must be satisfied:

POINTS

< (K2 c(Ki) - c(K2) - p(Ki) + p(K2) - Ki)e result At the maturity date, the of the strategy is always (K2 using - Ki). In fact, inequality (3),the pay-off of each option is

(3) •

the

•

max [Sr - Ki, 0] - max [Sr - K2, 0] - max [Ki (4) - Sr, 0] + max [K2 - S,, 0] This shows that the box is worth (K2 value is less than Ki) the maturity If date. its at Ki), then riskless arbitrage would be possible. the discounted value of (K2

Consider the following two relationships American options C and P:

between

European

options

e and

•

p and

• • •

• •

C(Ki)

- c(Ki)

<

C(K2)

- c(K2)

(5)

•

FOR DISCUSSION

policies? Why and how should one formulate efficient financial risk management How do these policies impact in turn on company performance? Why does proactive financial risk management add value to the firm? How is this done? Arc there good reasons for using financial markets for hedging purposes? financial risk Are there good reasons for using financial instruments and implementing management? Why are there so many new financial instruments'? What are the fundamental reasons behind the proliferation of financial assets? What drives the wave of financial innovation?

26

• • • •

OPTIONS,

FUTigilgS.

ID EXOTIC

DENVATIVES

What are the basic synthetic positions? How can one implement a conversion? How can one implement a reverse conversion? What is the main characteristic of a box spread?

The Dynamics of Asset Prices: Analysis and Applications

CHAPTER

OUTLINE

This chapter is organized as follows: time stochastic processes for the dynamics of asset 1. Section 2.1 introduces continuous prices. In particular, the Wiener process, the generalized Wiener process and the Itö process are presented and applied to stock prices. 2. In section 2.2, Itô's lemma is constructed and several of its applications are provided. 3. In section 2.3, we introduce the concepts of arbitrage, hedging and replication in connection with the application of Itô's lemma. This allows the derivation of the partial differential equation governing the prices of derivative assets. 4. In section 2.4, forward and backward equations are presented and an application is given. introduction to diffusion processes. 5. Appendix 2.A is a mathematical 6. Appendix 2.B gives the main properties of the conditional expectation operator. 7. Appendix 2.C reminds readers of the Taylor series formula.

INTRODUCTION

¯

Most fmancial models describing the dynamics of price changes, interest rate changes, exchange rate variations, bond price changes and derivative asset dynamics, among other things. present a term known as a Wiener process. This process is a particular type of a generaÍclass of stochastic processes known as Markov stochastic processes. A stochastic process can be defined either in a simple way, as throughout this chapter, sense, as in Appendix 2.A. or in a more mathematical in order to allow the Our presentation is at the same time intuitive and rigorous understanding of the necessary tools in continuous time finance. These tools are not as reader might think. complicated as an uninformed

OPTIONS,

28

FUTURES

AND EXOTIC

DERIVATIVES

Using the definition of a stochastic process enables us to define the standard Brownian motion and the Itô process. The Itô process allows the construction of stochastic integrals and the definition of Itö's theorem or what is commonly known as Itô's lemma. This lemma can be obtained using either Taylor series expansions or a more rigorous mathematical approach. In both cases, some applications of this lemma to the dynamics of asset and derivative asset prices and returns are provided. The introduction of the notions of arbitrage, replication and the hedging argument, which are basic concepts in finance, allows the derivation of a partial differential equation for the pricing of derivative assets. This equation first appeared in Black and Scholes (1973)and Merton (1973a).These authors introduced the arbitrage theory of contingent claim pricing and, using Itö's theorem, showed that a continuously revised hedge between asset is perfect. Since then, Itõ's theorem, the Black a contingent claim and its underlying and Scholes hedge portfolio and the concepts of arbitrage and replicating portfolios have been used by many researchers in continuous and discrete time finance. The basic equivalent results in the theory of option pricing in a discrete time setting were obtained by Cox, Ross and Rubinstein (1979) and Rendleman and Barter (1980). They showed that option values calculated with discrete time models converge to option values obtained by continuous time models. In other words, theoretical work on convergence shows that some discrete time processes converge to continuous time processes. For example, in the context of binomial models, an option can be perfectly hedged using the underlying asset, and Itô's theorem can be implemented when constructing the hedging portfolio for infinitesimal time intervals. Some important questions regarding the use of Itô's lemma and the perfect hedge can be studied. The main question is whether a continuously revised hedge is perfect over each revision interval or only when cumulating the hedging error to zero over a large number of revision intervals. In all cases, these basic arguments lead to a partial differential equation which must be satisfied by the prices of derivative assets. This can be derived using one of the two literature or simply definitions of Itö's lemma: the definition given in the mathematical the one obtained by an extension of the Taylor series.

2.1

CONTINUOUS TIME PROCESSES PRICE DYNAMICS

2.1.1

Asset

Price Dynamics

and the Wiener

THE DYNAMICS

OF ASSET PRICES

29

AW

=

(1)

e

where e is a random sample from a normal distribution having a zero mean and a unit standard deviation. short intervals of time, then the values of A W are If one takes two reasonably these properties, it is clear that A W also has a normal distribution independent. Usmg and a variance of At. with a zero mean, a standard deviation equal to Now, if one considers the change in Wover a longer time period (0, T), composed of N periods of length Ai, i.e. T NAt, then the change in W from W(0) to W(T), or W(T) - W(0), over this period of time is equal to the sum of the changes over shorter periods. Hence, one can write: =

W( T) - W(0)

(2)

E

=

Using the independence property, it follows from this last equation that the change W(T) - W(0) is normally distributed with a mean of 0, a variance of NAt T and a standard deviation equal to N. This is the basic Wiener process with a zero mean or drift rate and a unit variance rate. A mean or a drift rate of zero means that the future change is equal to the current change. A variance rate of 1 means that the change at time Tis 1 × T. =

Example Consider a variable W following a Wiener process, starting at W(0) 20 (in years). This variable will attain in one year a value which is normally distributed with a mean of 20 and a standard deviation of l. In two years, its value follows a similar type of distribution with a mean of 20 and a standard deviation of In n years, its value follows the same distribution with a mean of 20 and a variance of n. What happens if the interval At gets very small, i.e., tends to zero? When At gets close to zero, the equation analogous to equation (1) is =

,/i.

dW

FOR ASSET Figures 2.1-2.3

=

(3)

e

show the effect of progressively

reducing

At.

Process

The dynamics of asset prices are often represented as a function of a Wiener process or what is also known as Brownian motion. The Wiener process has some interesting properties and can be introduced with respect to a change in a variable Wover a small interval of time At, Wiener Process or Brownian

Motion

Let W denote a variable following a Wiener process and A W a change in its value over a small interval of time At. The relation between AW and At is given by the following equation:

FIGURE 2.1 At is Large

Wiener

Process

when

FIGURE 2.2 Wiener At is Small

when

Process i

(Il SIIIMII49UUUlllBAlliktiB@¾C

30

THE DYNAMICS

DalVATIVES

Z

6

2.1.2

Asset

3I

OF ASSET PRICES

Wiener

and the Generalized

Price Dynamics

Process

For a variable X, a generalized Wiener process can be expressed as dX=adt+bdW

(6)

where a and b are constants. This process shows the dynamics of the variable X in terms of time and d W. The first of X is a per term, a d t, called the deterministic term, means that the expected drift rate unit time. The second term, bd W, called the stochastic component, shows the variability or the noise added to the dynamics of X. This noise is given by b times the Wiener process. adt, or dX/dt a. This is equivalent to When the stochastic component is nil, dX =

=

X Xo + at. Hence, the value ofXat any time is given by its initial value Xo plus its drift multiplied by the length of the time period. Now, it is possible to write the equivalent of equation (6) using equation (1) for a longer At: =

FIGURE

2.3

Wiener

when At is Very Small

Process

AX

The Martingale Property

and the Brownian

Motion

The notion of martingale is useful in financial models, particularly when analyzing the A martingale concept of arbitrage. can be defmed as follows. Consider a probability space (Q, f, P) and a filtration (f,),o. An adapted family (M,),so of integrable random variables having a finite mean is a martingale when for all s < t we have E(M,

fs)

=

M,

(4)

where E(.) stands for the mathematical conditional expectation operator. Appendix 2,B gives the main properties of the conditional expectation operator. The notion of martingale asserts that the best approximation of M,, given all the available information fs, is Ms. In terms of fmancial markets, this means that the best way to predict future prices is to use the current prices. Hence, using current information is equivalent to using all the historical information, since only the most recent information matters. Using this definition, it must be clear that when (M,),,o is a martingale, then E(M,) E(Mo). The following result is advanced without proof. If (M,),,o is an f,-Brownian standard motion, then: =

=

aAt +

Hence, as before, since AX has a normal deviation is bv/At and its variance is b2AL

2.1.3

Asset

Price Dynamics

bEN

distribution, its mean is aAt,

and the Itô Process

Itô Process An Itô process for a variable Xcan be written dX

=

a(X, t)dt+

as follows: b(X, t)dW

•

•

(W,), is an f,-martingale W - t is an f,-martingale exp (a W, - jo2t)is an f,-martingale-

.,

X,

=

K, ds +

Xo +

=

exp

(9)

H, d W,

Jo

where K, and H, are stochastic processes adapted to f, for which the integral corresponding to the second-order moment is finite. It will be shown later that the second integral in the above expression is a martingale.

The first and second properties characterize the standard Brownian motion. The third property is useful when studying the dynamics of financial asset prices. In fact, as will be shown later, the price of a stock is often written as 9,

(8)

The dependence of both the expected drift rate and the variance rate on X and time t is the main difference from the generalized Wiener process. This process has been extensively used in the finance literature, especially for modeling stock price dynamics. We now give the mathematical definition of an Itõ process. Consider a probability motion. An Itö space (G, F, (§,),20,P) with a filtration and (M,),,o an f,-Brownian which for all te T: process is a process (M,)osta having its values in R for Jo

•

(7) its standard

(o W,

o2t)

(5)

The Dynamics

of Stock Prices

The dynamics of the stock price S are represented drift rate of uS and a variance rate of a2S2:

by the followmg Itô process

with

a

32

OPTIONS,

dS

=

FUTURIBAMBdlMOTdig

µSdt + aSdW

=

THE DYNAMICS

Example

Brownian motion, can be

µAt + acA dt

Consider now the Thomson stock price, having an expected return of 20% per volatility of 25% per annum. annum and a standard deviation or of three interval days, or 0.008 219 178 year, AS/S is normal, Over a time with

(11)

e is a random sample from a normal distribution with a zero mean and a unit standard deviation. When the variance of the stock price is zero, the rate of return expected drift in Sover At is where

dS=µSdt

or

dS/S=µdt

AS/S

/

~.

(0.00164, 0.0226)

,

(18)

(12)

So e"'. so that S When the variance rate is not nil and o2S2At is the variance of the actual change in S during At, the dynamics of the stock price are given by the expected instantaneous increase in S plus its instantaneous variance times the noise d W. This discrete time version says that the proportional return on the stock S, over a short period of time, is given by an expected return of µAt and a stochastic return of JeAdt. Hence, AS/S is normally distributed with a mean of µAt and a standard deviation of o Ài, or =

The Log-Normal

2.1.4

Property

Using the previous example, since the change in the underlying asset price between time variance of t and time t, is normally distributed, with a mean of (µ ja2)(t t) and a Ü2(l¿

-

f),

We

-

have:

N

/ [(µ-ja2XL-

ln(S,,)-ln(S,)~.

(19)

tL

or ,,)

/ (µAt,

33

OF ASSET PRICES

(10)

This process for stock prices, known also as the geometric written in a discrete time setting as

AS/S

SINVATIVES

~.

n

(13)

/

Xµ -ja2)(t, -

t) + In(S,), o

(20)

]

Hence, S,, has a log-normal distribution.

Example

Example

Consider the Peugeot stock price characterized by an expected return of 14% per annum and a standard deviation or volatility of 35% per annum. The initial stock price is 100 FE Using equation (13),the dynamics of this stock price are given

100, the expected return is 15% per annum and the volatility annum, the distribution of ln (Ss) in six months is

dS/S or for a small

=

0.14 dt + 0.35 dW

If S

=

In (S,,)

~.

ln (S,,)

dS S If the time interval A t is two given by =

=

weeks

0.14At +

0.35Ñ

46) +

0.35E

)

(16)

or

The price increase is a 0.538 and a volatility

=

random

100(0.005 384 + 0.068 639e) sample from a normal

× 0.09)(0.5),

0.3

]

(21)

/

~.

[4.657,0.212]

(22)

(15)

(or 0.038 46 year), then the price increase is

100(0.14(0.038

AS

[ln(100)+ (0.15 - 0.5

or

(14)

interval At:

AS

I

is 30% per

2.1.5

The Distribution

of the Rate of Return

compounded rate of return. What is the distribution of a? At a Let a be a continuously future date ti, the stock price can be written as

&

(17)

distribution with a mean of

S, e""

=

"

(23)

and

of 6.86

1 a

=

-

ti-t

In

So -S,

(24)

S ln -

~.

/

(µ -

(«2)(/

t)

(25) Af~

then a~.

variables

Now, if f depends on two equation (28)is

Using the log-normal property, i.e.

/

(µ

35

OF ASSET PRICES

THE DYNAMICS

PTIObtS,dilil$$llllllllAhlMBAIIIOTIC$lglIRIVAT1VES

34

8f ßt

8f Ax+ ßx

At+-

-

-

instead of x and y, the equation analogous to

x and t

82 f

l

2 Ox2

1 82f _2 8/2

Ax2

82 AXAl Ox8t

2

...

(30)

a2),

(26) Consider a derivative security, f(x, t), whose value depends on time and on the asset price x. Assuming that x folows the general Itõ process,

Example

dx

What is the distribution of the actual rate of return over two years for a stock having an expected annual return of 15% and a volatility of 30%? The distribution is normal with a mean of 10.5% (15% 0.09/2), and a standard deviation of0.3/v/ž, i.e. 21.21%.

=

t)dt + b(x, t)dW

a(x,

Ax

or

a(x,

=

t)At+

bev'Þ

(31)

In the limit, when Ax and At are close to zero, we can not ignore as before the term in Ax2 since it is equal to Ax2 b2e2At + terms of higher order in At. Hence, the term in At can not be neglected. 0 and a unit variance Since the term e is normally distributed with a zero mean E(e) 1 and E(s)2At 1, then E(e2) At. The variance of e2At is of order E(s2) E(e)2 At2 and consequently, as At approaches zero, e2At becomes certain and equals its expected value, At. In the limit, equation (30)becomes =

=

Itô's Lemma

2.2

and its Applications

Financial models are rarely described by a function that depends on a single variable. In general, a function which is itself a function of more than one variable is used. Itõ's lemma, which is the fundamental instrument in stochastic calculus, allows such functions to be differentiated. We first derive Itô's lemma with reference to simple results using Taylor series approximations. We then give a more rigorous definition of ftö's theorem. The formula for Taylor series is given in Appendix 2.C.

d

Intuitive

(g2

di

dx +

=

This is exactly Itô's lemma.

(32)

b(x, t)dW for dx, equation (32)

t)dt

a(x,

Substituting

b2 dt

gives

Of -ßx,

df= 2.2.1

=

=

=

-

Of 01

a+

-

1 +-

-

2

82f b Ox2

dt+

Of bdW -Ox

(33)

Form

f be a continuous and differentiable function of a variable x. If Ax is a small change in x, then using Taylor series, the resulting change in fis given by

Let

A If fdependson two variables

Af

~

Ax +

Ax2 8

Ax

Ax'+

x and y, then the Taylor series expansion

Ax2 +

Ay

(27)

...

Apply Itö's lemma to a function f(S, t) when the dynamics of stock prices are described by the following stocl astic equation

of Af is

Ay2 9

AxA y +

dW

...

(28) In the limit case, when Ax and Av are close to zero, equation df~

dx

dy

Example

(28) becomes

Equation d/

(29)

(34)

(33) gives Of US

'

uS

of

I

Ot

2

82 OS

o S2 dt+

Of -8/

USdW

(35)

oPTloNS,

36

FUTURES

AND EXOTIC

DERIVATIVES

1 ggè$$UIMI

BOF ASSET PRfCES

37

f'(X,)dX,=

Example Apply Itö's lemma to derive the process of First calculate the derivatives

f

=

In (S).

gesecondiategral

f'(X,)H

J'(X,)K,ds+

1

f"(X,)d(X,

X),

f"(X,)H

=

ds

f(Xo)

J'(X,)H,dW

J'(X,)K,ds+

f"(Xs)H

ds

(46)

More generally, if f(t, x) has first-order continuous partial derivatives in t and continuous second-order derivatives in x, then a2S2 dt

µS+

=

f(X,)=

(38)

Then fiom Itö's lemma one obtains df

(45)

ence, Itö's formula is

(37) 0

(

is given by

_

=

dW

USdW

(39) f(t,

X,)

f (0, Xo)

=

'(s,

+

X,) ds

f'(s,

X,) dX, +

f

",(s,

X,) d(X, X)

or o2]dt

df=[µ

odW

(47)

(40)

This last equation shows that f follows a generalized constant drift of (µ ja2)and a variance rate ofo2

Wiener process

with

Note that if we put K,

a

0, H,

=

-

1, in the Itõ process, i.e.

=

H,dW,

K,ds+

X,=Xe+ then it reduces to X,

Mathematical

2.2.2

=

W,, which

(48)

0

0

is simply the Brownian motion,

since Wo

=

0.

Form

The following theorem gives Itô's fortmda Example Theorem

In this example, Itö's formula is applied to the dynamics of the squared Brownian W2.

motion,

If(X,)os,«r

isanitô process,

i.e.

When

X,=Xo+ and

Tis a continuous

function (X,)=

K,ds+

with second-order

f(Xo)+

HsdW, continuous

J'(X,)dX

=

x2 and X,

(41) Since f'(x)

derivatives, then

"(X,)d(X,

X),

=

W =W =

(x2)'

=

0 and

2x,

=

W

(42) Since Wo

where

x)

f

(thecase

+

2W,dW

f"(x)

(x2)"

=

2W,dW,+

=

ds

W,

=

by definition =

H ds

=

0 and H,

=

l), we have

2d(X,X), 2 and d(X, X), 2ds=2

(49) =

ds, then

W dW,+t

W,dW

-

(50)

(51)

(43) Since the expected value of

The first integral is given by

K,

t, then W -t=2

d(X, X),

=

where

(W

ds is finite, then

(W

is martingale. - t) a

38

OPTIONS,

FUTURES

AND EXOTIC

TidiggsigWHCS

DERIVATIVES

39

OF ASSET PRICES

S,

Example

=

(59)

S,(µ dS + od W,)

So +

o

This process is given by

This application of Itõ's lemma concerns the calculation of an explicit solution to the process describing the dynamics of stock prices. Let us look for the solutions to (S,),,o of the following equation: S, which

=

So +

S,(µds

+ a d W,)

;

S,

=

As we will see later, this process is used in the derivation of the formula and is often referred to as the B-S process.

(52)

(60)

a d W,]

Otpek and Scholes

is often written in the form dS,

S,(µdt + a d W,)

=

Example

(53)

log (S,) where S, is the solution to the preceding equation. Since S, follows an Itô process with K, µS, and H, US, application of Itõ's formula log (S) gives to f(S) Let Y,

The Black and Scholes model for the valuation of European options uses two assets: a risky asset with a price S, at time t and a riskless asset B, at time t. The dynamics of the riskless asset or bond are given by the following ordinary differential equation:

=

=

=

=

'

In(S,)

dS

In(So)+

=

'

\

-1

a2Sids

(54)

dB,

Var(dS,)

=

(52)in (54)and

simplifying

=

=

a2S ds.

Replacing equation

gives

dS, Y,

Yo+

=

(«2)ds+a

(µ o

result

is straightforward

odW,

(55)

d W,

t,

=

o

=

W,

-

Wo with

Wo

=

0

o

(62)

µS,dt +US,dW,

U2t+adW,]

S,=Soexp[µt-

'

ds

=

where µ and a are constants and d W, is a standard Brownian motion. As we have shown, the solution to equation (62)is

since

'

(61)

1 r stands for the riskless interest rate. The bond's value at time 0, Bo that e". 8, so The dynamics of the risky asset or stock are given by the following stochastic differential equation:

=

=

rB,dt

=

where

1/S, f"(S) since f(S) -l/S2, and the term corresponding log(S), f'(S) of d(X, X), is the instantaneous variance of dS,, to the equivalence =

This

[(µ - jo')t +

Soexp

I

(56)

(63)

where Sois the initial asset price at time 0.

So we have a2)t+adW,

Y, =ln(S,)=ln(So)+(µApplying the exponential

to Y, gives the solution to equation S,

=

Se exp

[(µ

o')t

(57)

2.2.3

+ od W,]

(58)

This is the explicit formula for the underlying asset price when its dynamics are given by the stochastic differential equation (52)

The following theorem, equation (52) is unique.

which is stated without

proof,

Generalized

Itô's Formula

(52):

shows that the solution

to

Itô's formula can be generalized to the case where the Itö processes which are expressed in terms of standard Itö's formula or the vector This is the multidimensional is useful in deriving interest rate models and models several state variables. .,W(') where (W),,o Consider W,=(W,. are motions and W, is a p-dimensional Brownian motion. Itö'sformulaisintroducedwithrespecttonitöprocesses(XI,,...,X"):

X''

Theorem

=

K' ds +

X +

*

.

When

( W,),ao is a

every

i <

T,

Brownian motion, there is a unique Itô process (S,),,«

for which, for

f(.) depends on several independent Brownian motions. form of Itõ's lemma. The formula of derivative asset pricing with function

standard Brownian independent The mathematical expression of

HU

dX'

(64)

e

When the function f(.) has second-order partial derivatives in x and first-order partial derivatives in t, which are continuous in (x, t), then the generalized Itõ's lemma is

40

OPTIONS,

f(t,

X

,

...,

X")

f(0,

=

X

X")

..,

,

FUTURES

AND EXOTIC

(s, X

(s, X

...,

1

K ds

=

Hil dW

'compact'

In the financial literature, the notation is more mathematical literature. The term d(X', XI), corresponds neous covariance terms, or cov (dX;, dXI).

dc

than that used in the to the changes in the instanta-

AND THE

Let us denote by c(S, t) the option value at time t as a function of the underlying asset price S and time t. Assume that the underlying asset price follows a geometric Brownian motion:

dS/S where

=

U2

correspond respectively µ and the rate of return of the stock.

2.3.1

The Relationship

between

µ dt + od W(t)

(68)

to the instantaneous mean

Taylor

Series

and the variance

of

=

Oc dS + BS

-

Sc -St

1 dt + 2

dcuo + de(t)

(71)

a2S2 dt

dt

where de(t)

is

J'S2[dW2

=

2 \DS

(72)

82c BS2

Itô's Differential

and

the Replication

(69)

the last term appears because dS2 is of order dt The last term in equation (69) can be separated into its expected value and an term, as in Omberg (1991), in order to make the link between Taylor series (dc) and Itô's differential (dca), i.e.

Ûc

1

Portfolio

ScN

(74)

=

(ßc

=

l

(82c a2S2

(75)

r

where r is the nsk-free rate of return. In this spirit, the returns on the replicating

where

'error'

(73)

A replication argument is often used in financial theory. It means simply that in complete markets, the pay-off of an asset can be created or duplicated using some other assets. A combination of these assets gives a similar pay-off as that of the original asset. In general, the pay-off of any derivative asset can be created by an investment of a certain amount in the underlying risky asset and another amount in risk-free discount bonds. Also, the payoff of a derivative asset can be created using the discount bond, some options and the underlying asset. The portfolio which duplicates the pay-off of the asset is called the replicating portfolio. When using Itö's lemma, the term de(t) is often neglected and the equation for dcs is replicated by a portfolio of Q, units of the underlying asset and an amount of cash equivalent to Qcwith

Û,

(dS)2

- dt]

the error term.

and Itô's Differential

Using the Taylor series differential, it is possible to express the price change of the option over a small interval of time [t, t ÷ dt] as dc

=

dS +

=

de(t)

(67)

2.3.2

TAYLOR SERIES, ITÔ'S THEOREM REPLICATION ARGUMENT

(70)

(66)

"'ds

H'"H

=

«2S2dW2 + de(t)

dt +

or

dcas

d(X', X

2.3

dS +

=

(65)

with

dX

4I

OF ASSET PRICES

dc

X")ds

X )dX

...,

,

,

TSIggigimÞt1CS

DERIVATIVES

dOg

78c =

-

where the subscript R refers to the replicating

portfolio dly dS+

rQ dt

portfolio.

are

given by

(76)

42

2.3.3

FUTUREANOTICAIŒRIVATIVES

OPTIONS,

Itô's Differential

and the Arbitrage

THE DYNAMICS

2.3.4

Portfolio

43

OF ASSET PRICES

Why are Error Terms Neglected? 'rationale'

arguments, then the option value must be equal to the value of its replicating portfolio. The principle of arbitrage states simply that fmancial assets having identical characteristics must trade at the same price. If this principle is not respected, then selling the higher priced asset and buying the lower priced asset allows a risk-free profit. This principle is used to determine the fair price of a security or a derivative asset. In this context, we must have

If one

uses arbitrage

in ignoring the Now, we consider whether there is a mathematical or economic hedging error de(t) tends to self-financing and the revision portfolio each is if term error zero as the interval of time becomes extremely small. In the Black and Scholes theory, the error term de(t) is ignored because it is not correlated with the underlying asset price in the context of the Capital Asset Pricing Model. In that context, it is regarded as a diversifiable risk. This justification is referred to option pricing theory'. However, if one uses the Black and Scholes as the theory with respect to the implementation of Itö's lemma, then ignoring the error term 'equilibrium

c

=

c

=

QS

Q

Oc

(77) l

S+

/

OcN

l

/ 82c o'S2

(78)

N,

r

gives a pure arbitrage result. In the context of pure arbitrage results, Omberg and H2:

(1991)advanced

two explanations,

Hi

or

82c

l -

-2 OS2

U2S2 +

Oc rc - r

-OS

Hi: de(r)

Oc

S+

-

th

=

0

H2: de(t)

This equation is often referred to in financial economics as the Black and Scholes partial differential equation. Note that the value of the replicating portfolio is y

_

BS,

S+

Q

(80)

'

As we will see in another chapter, the original Black and Scholes hedge portfolio consists of positions in the option and its underlying asset, i.e·

Es

=

c -

-

S

=

(81)

Qc

where the subscript H refers to the hedged position. A hedged position or portfolio is a portfolio with no risk, whose return at equilibrium of Black and must be equal to the riskless rate of interest. This is the main contribution Scholes to the literature on derivative asset pricing A self-financing portfolio can be constructed by buying the option and selling the replicating portfolio or vice versa. This concept was introduced by Merton (1973a) when he proposed an arbitrage derivation of the formula for the pricing of derivative securities. The condition on the self-financing portfolio is

Ha

Oc =

c-

S-

Œ

=

0

(82)

the subscript A refers to the arbitrage portfolio. It is convenient to note that the omitted error term, de(t), may represent a replication error, a hedging error or an arbitrage error. In fact, when the revision of the portfolio is accomplished at [t, t + dt], and the term de(t) is nil, then the revision of the replicating portfolio is self-financing. When the term de(t) is positive, this means that additional cash must be put in the portfolio. When the term de(t) is negative, a withdrawal of cash from the portfolio is possible.

where

o(dt)

=

(83)

(79) =

O(dt) and

de(t) '

=

0 for

<>0

(84)

"

H, says that de(t) is o(dt), which is of a higher order than dt. Therefore, it is neglected with regard to the other terms in the Taylor series and Itö's differentials. In the limit, the hedging error disappears more rapidly than the risk-free return, since the latter is oforder dt and the hedging error is o(dt). In this context, the replication of the option value is perfect over a single revision interval. The smallness of the interval is a sufficient argument to justify neglecting the hedging error. H2 says that the error is of order dt. The true price dynamics are represented by the Taylor series differential rather than Itö's differential. In this case, cumulation of hedging errors over an infmite number of small intervals is necessary. The financial economics literature does not often say which of the two explanations is correct, except in some cases. For example, Merton (1973a)showed that the error term is O(dt) and neglected it by using the law of large numbers. Jarrow and Rudd (1983)argued that the error term is of order o(dt) and hence negligible. Leland (1994a) considered the term O(dt) and showed that the cumulative replication error disappears over an mfinite number of revision intervals. The mathematical literature does not support the first hypothesis since the proofs of Itö's theorem rely on a Taylor series differential. It shows that the integral of the error Omberg (1991) studied the behavior of the hedging error in a term vanishes. continuously revised hedge over one and several revision intervals. He showed that interval there are only two cases where the hedging error is zero over each revision for a diffusion process. The first case corresponds to the limit of the binomial option pricing model of Cox, Ross and Rubinstein (1979). The second case refers to a I over one revision stochastic revision strategy, which succeeds with probability of unproved that it succeeds over an infmite number interval, though it remains revision intervals. In other cases, the hedging error over one revision interval is of the For more details, see same order as the risk-free rate and can not be neglected.

Omberg (1991).

OPTIONS,

44

2.4

FUTURES

FORWARD AND BACKWARD EQUATIONS

AND EXOTIC

DERIVATIVES

THE DYNAMICS

45

KOLMOGOROV Example

When the asset price dynamics are described by the following Markov diffusionprocess: dS,=

OF ASSET PRICES

µ(S, i)dt+o(S,

t)dW,

When µ(S) µ and o(S) a, then Sis a Brownian motion with drift. It is possible verify that the solution equations to to (86) and (87) is =

(85)

=

f(S, t;

the probability density function for S at time t conditional on S, So, denoted by f(S, t; So, to), satisfies the partial differential equations of motion which are the backward and the forward Kolmogorov equations. The backward \(olmogorov equation is given by

xo, to)

=

.

/

_

(S+xo)

-µ(T-

to)

(90)

=

where.

I stands for the density of the standard normal distribution.

SUMMARY l 2

,

a-(So,

82f

to) BS

+ µ(So, to)

-

Qf DSo

+

of -

0

=

Oto

(86)

The forward Kolmogorov or Fokker- Planck equation is given by 1 2

97 BS2

g -OS [µ(S, t)f]+

[o2(S, t)f]-

,at

g

=0

(87)

-

These equations can be solved under the condition f(S, to, So, to) ös (S), i.e. at the initial time, S is equal to So and ös, is the Dirac measure at So. It is defined by =

ós

([a, b])

=

fl0

if Se é else

[a, b]

(88)

Since we are interested only in time-homogeneous a(S) and processes for which a sometimes of constraints obtain barrier absorbing imposed µ(S), types to two are an µ and a reflecting barrier when the drift is finite. An absorbing barrier means simply that value, this value will be conserved for all subsequent once the process attains a certain instants. A reflecting barrier means that when the process hits a certain level, it will return from the direction from which it comes. t) to take the value zero, then Si is an absorbing barrier from When we constrain f(S intuition of this result is that when S Si at ti, then S will conserve that above (+). The value for all the subsequent instants t after ti. When we constrain the term

Q

This chapter contains the basic and general material for the dynamics of financial assets and derivative asset prices in a continuous time framework. The presentation is made as simple as possible. The aim is to allow readers not familiar with these concepts to follow without difficulty the basic methods in continuous time finance. Each concept is proposed in two forms: an intuitive version and a more rigorous mathematicalversion. The Wiener and Itõ processes are used to model the dynamics of asset prices. Itö's lemma is proposed to differentiate a function of one and several stochastic variables. It is illustrated through several examples in different contexts. The Kolmogorov forward and backward equations are presented. These tools allow the pricing of standard options and more complex derivative assets. This chapter is necessary for the understanding of the basic techniques behind the theory of rational option pricing in a continuous time framework. However, it is not of all the formulas presented in this book. necessary for the use and application

=

=

POINTS FOR DISCUSSION • • •

,

=

2

[µ(S)f(S, t)]

OS2

Si, then Si is a reflecting barrier. to zero at S 0, then the value at which 11is convenient to note that when o(SI)

(89)

• • •

Define the Wiener process. What is a martingale? What is an Itô process? What is the intuitive form of Itö's lemma? What is a replication portfolio? What is an arbitrage portfolio?

APPENDIX 2.A: INTRODUCTION PROCESSES

TO OfFFUSION

=

=

a natural

a is nil represents

barrier.

When µ(Si) is nil, we have a natural absorbing barrier, and when µ(S, t) is strictly nonive (or negative), barrier from above (or below). we have a natural reflecting

The notion of a stochastic process variables. Using the notation

can be introduced

with respect

R: the space of all possible states o o-algebra defined over R; a class of partitions of R

f:

to the notion

of a vector of

OPTIONS,

46

FUTURES

AND EXOTIC

DERIVATIVES

X(w) is said to be a random variable when it is a measurable application from (G, f) to R X,,(w)] is a measurable application from A vector of random variables X(w) = [Xi(w), (R, f) into R". The notion of a vector of random variables is similar to that of n ordinary variables defined on the same probability space A stochastic process is the extension of the notion of a vector of variables when the number of elements becomes infinite. It is a family of random variables, {X,(w)}, 1 E T, when the index varies

THE DYNAMICS

OF ASSET PRICES

APPENDIX

2.C: TAYLOR SERIES

47

...,

in a finite or an infinite group. When w we, X(wo, t) is a function of t called a realization t to, X(w, to) is a vectorofvariables A stochastic process will be denoted by X(t). =

or a path of the process.

When

=

If a function x gives

f

has derivatives in the region (x, x + h), then the development of this fugattissiaround

/(x Ifthe function

fand

+ h)

f(x) + J'(x)h + jf

=

=

f(x) + J'(x)h + l

stochastic process having its values m a space E with a mhe f is a family of A continuous-time random variables (X,),, defined on the probability space (9, f, P) takmg tes values m (E, f)

Definition

+-

n!

A stochastic process X(t) for which the changes in its values over successive intervals are random, The process has no derivative and its independent and homogeneous is said to have no process is the most regular path can be represented by a continuous curve. The Wiener-Levy process among those whose changes are independent and homogeneous. 'memory'.

A filtration(§,),20is an increasing family of sub-tribes of f in the probability

space (G, f, P) and let the definition of the conditional expectation.

F be

space (R,

f,

finance when deriving models for

The following theorem allows

f

Theorem For any integrable random variable X, there is a unique associated all elements in 11: E(X1

)

=

E( Y1

random variable

•

If X is El-measurable, then E(X

•

E[E(X/N)]

• • • • •

=

B}

=

X

X

For any random variable Z, measurable

with respect to

B,

E(XX + µY/N) AE(X/N)+ µE(Y/N) If X > 0 then E(X/N) > 0 if C is a sub-tribe of9, then E[E(X/F)/€] E(X €) E(X). If Xis independent of N, then E(X/El) =

=

=

E[ZE(X

Y such that for

(91)

)

Yis known as the conditional expectation of X given II, or E(X/N). The conditional expectation operator obeys the following properties:

)]

=

E(ZX)

+

(n -

1)!

f""

.

(92) we

"(x)h"¯

(93)

i

d'fÏx, ?)

2

Dy2

k2 +

82f(x, y) h Ox

by

1 82 (x, y) 8 (x, y) k+h 2 dy Ox-

Ox8y

hk

D2f(x, y) - Oy

Similar expressions can be derived for functions of three or more variables.

P).

EXPECTATION

a sub-tribe of

...

in Taylor series expansions

D (x, y) h dx

l

Consider a probability

+ - f'"'(x)h" n!

f,"'(x*)h"

n!

Definition

+ (f"(x)h2

where x* is in the region (x, x + h). A function of two variables x and y is represented

f(x+h,y+k)=/(x,y)+

2.B: THE CONDITIONAL

...

up to order n exist in the same region, then usingTaylerseries

its derivatives

Definition

APPENDIX

+

have (x + h)

An example of stochastic processes often used in continuous-time the valuation of financial assets is the Brownian motion.

"(x)h2

(94)

Applications to Asset and Derivative Asset Pricing in Complete Markets

CHAPTER

OUTLINE

This chapter is organized as follows: 1. In section 3.1, we give some definitions and characterize complete markets. 2. In section 3.2, derivative assets are priced with respect to the partial differential approach. Both methods are applied to the equation method and the martingale valuation of stock options. 3. In section 3.3, we introduce numerical analysis and simulation techniques. 4. Appendix 3.A presents the change in probability and the Girsanov theorem. 5. Appendix 3.B gives in great detail the resolution of the partial differential equation under the appropriate condition for a European call option. 6. Appendix 3.C gives two approximations of the cumulative normal distribution function. 7. Appendix 3.D introduces finite difference methods with respect to the heat transfer equation. 8. Appendix 3.E lists an algorithm for a European call. 9. Appendix 3.F states Leibniz's rule.

INTRODUCTION The pricing of derivative assets is usually based upon two methods which use the same basic arguments. The first method involves the resolution of a partial differential equation under the appropriate boundary conditions corresponding to the derivative asset's pay-offs. This is often referred to as the B-S method. The second method initiated by Harrison and Kreps uses the martingale method, and Pliska (1981), where the current price of any derivative asset is a nd (1979) Harrison

OPTIONS,

50

FUTURES

AND EXOTIC

DERIVATIVES

by its discounted expected future pay-offs under the appropriate probability probability. Both methods measure. The probability is often referred to as the risk-neutral pricing call options. the of for illustrated in detail European are Unfortunately, for most problems in financial economics, and in particular for the pricing of American options, there is often no closed-form solution and option prices

given

Therefore, financial economists often resort to numerical must be approximated. niques. A brief presentation of these methods is given in the present chapter.

3.1

CHARACTERIZATIONS

OF COMPLETE

APPLICATIONS

TO ASSET PRICING

IN COMPLETE

MARKETS

5i

If we consider a contingent claim specified by its final pay-off h which is of the form (S, K)* for a European call and h (K - S,)* for a European put, then an attainable strategy ¢ simulates or duplicates the option price when its pay-off at the maturity date T is equal to h, or:

h

=

=

-

V,(¢)

tech-

=

h

(1)

The sequence of the random derivative asset prices between the initial time 0 and the under the uníque probability P*. Hence, we have option's maturity date T is a martingale Vo(¢)

MARKETS

=

E*[Vr(¢)]

(2)

E*(h/Sr)

(3)

and In financial markets, there are two classes of financial assets: securities and derivative assets. Securities correspond fundamentally to common stocks and bonds. Derivative assets are contingent claims characterized by their intermediate and final pay-offs. There are several definitions of complete markets. The idea of complete markets was first proposed by Arrow and Debreu (Arrow, 1953, 1970; Debreu, 1954). They defined a complete market with respect to state securities or state contingent claims. A state security or an Arrow- Debreu security is a security which pays off one dollar if and only if a given state of nature occurs. A state of nature or a possible state in the economy is said to be an insurable state when it is possible to construct a portfolio of assets which has a non-zero return in that state. In this economy, the price of each traded asset at the beginning of the period is ps. where every state is insurable, a price vector can be completely For an economy determined with unique state prices. This implies the absence of arbitrage opportunities. A contingent claim is attainable if there is a strategy that gives the same value at the derivative asset maturity date as the contingent claim terminal pay-off. Hence, a complete market can be defined as a market in which all the contingent claims are attainable, i.e. all the contingent claims are obtained by implementing a replication or a duplication strategy. In this sense, they are redundant. A complete market can be defined with respect to the concept of a viable financial market. A viable financial market is a market where there is no profitable riskless arbitrage opportunities The absence of riskless arbitrage opportunity. means that a with a zero cost must have a zero which initial implemented the time is at strategy terminal value. It is important to note that there is a relationship between the notion of arbitrage and the martingale property of securities prices. The latter means simply that the best estimation of the future price is derived from the latest information. Hence, when historical security price data are used to predict the future price, only the most recent information matters, i.e. the last price. This concept also defines that of an efficient markcL

ln mathematical terms, a financial market is viable if and only if there is a probabihty P* which is equivalent to a probability P, under which the discounted asset prices have the martingale property. This is so under the theorems of the change in probability and, in particular, the Girsanov theorem: see Appendix 3.A for more detail. A viable financial market is complete if and only if there is a probability P* equivalent to the probability P under which the martingale property is satisfied by security prices. It P* is unique. can be shown that this probability

'°(¢)

=

If at the initial time a derivative asset is sold at its expected price, E*(h/Sr), then an investor following a replication strategy can obtain the exact pay-off h at time T.

3.2 3.2.1

PRICING

DERIVATIVE ASSETS

The Problem

Since each financial asset is specified by its intermediate and terminal pay-offs, option pricing consists in finding the fair price at the initial time when the derivative asset is bought or sold. There is a unique approach for the pricing of derivative assets. The value of each option is given by its expected terminal and intermediate pay-offs discounted to the present. However, there are two methods for the pricing of options. The first was initiated by Black and Scholes (1973).The second is the martingale approach due to Harrison and Kreps (1979) and Harrison and Pliska (1981) The first method, known as the B-S method, is based on the resolution of the following partial differential equation: (a2S2 -

ß2c Sc + rS + OS2 BS

-

Oc -

-

-8t

rc

=

0

(4)

under the appropriate boundary conditions. The second is based on the use of martingale techniques. In either the first or the second approach, the boundary conditions are the same. These conditions refer to the appropriate pay-off conditions corresponding to the value of European and American contingent claims. For a European call, the final pay-off is given by c

=

max

[0, Sr

- K]

max

[0, K

- Sr]

(5)

For a European put, the final pay-off is p

=

For an American call, the following additional differential equation:

condition

(6) must

be satisfied by the partial

Eg

OPTIONS,

C

>

max

FUTURES

[0, S, -

AND EM

1AII libERIVATIVES

K]

(7)

The difference from the condition for a European call is that the American call holder can exercise his option at each instant. This condition indicates that at each instant, the American call value must be at least equal to or greater than the intrinsic value, which corresponds to the value of a European call at maturity. For an American put, the following additional condition must be satisfied since the put holder can exercise his option at each instant: P

>

max

- S,]

[0, K

(8)

This condition shows that at each instant, the American put value must be at least equal to the intrinsic value. All these conditions apply in the absence of dividends. When there are distributions to the underlying asset, there are in general no explicit solutions to these problems and numerical methods are often used. First, we illustrate the partial differential equation method for the valuation of European call options. Second, we illustrate the use of the martingale approach for the pricing of European calls. The reader can verify that the price of the call is the same under both methods, Third, since it is difficult to get closed-form solutions for American options with and without distributions to the underlying asset, financial economists often use numerical methods. Therefore, we develop the main principles of these techniques in the last section of this chapter

APPLICATIONS

MARKETS

IN COMPLETE

TO ASSET PRICING

53

where N(.) is the cumulative normal distribution function. A detailed resolution of this system is given in Appendix 3.B. The approximation the cumulative normal distribution function is provided in Appendix 3.C.

3.2.3

The Martingale

of

Method

The B-S pricing of options requires first the knowledge of the probability under which the In the B-S context, there is an equivalent probability to P asset price S, is a martingale. under which the discounted expected stock price, S*

is a martingale. In fact, if one we have

e "S,

=

uses the stochastic

dS*'

differential equation for theateck price,

"'S, di + e

-re

=

(15)

"

dS,

(16)

or dS*

S*[(µ

=

-

'

r)dt

+ od W,]

(17)

r)/a]t

(18)

If we put the change in variables W*

W, +

=

[(µ -

then 3.2.2

The Partial

Differential

Equation

Method

4

dS,

Consider

the search for the solution to the following partial differential equation, under the boundary condition corresponding to the call's pay-off at the maturity date:

82c

(a2S2

c(S, T) Using an appropriate

=

change of variables, =

c

(Oc

rS (ßc

OS2

SN(di)

max

-

[0, S, -

rc

=

0

K]

(9)

- Ke

"'

"N(d2)

ln(S/K)

d2

+

=

ln (S/K) + =

(r

í¤2X T

(1l)

- t)

T (r - ¼¤2)(

=

Si

t)

=

-

S* exp [a W* -

jo

2

t]

(20)

.

with

di

(19)

Using the Girsanov theorem, 8, (µ r) o, there is a probability P* equivalent to P, under which ( W*),,,, is a standard Brownian motion. For more detail, see Appendix 3.A. Hence, under the probability P*, we deduce from equation (19) that S is a martingale and

(10)

the European call's value is given by

S,e a d W,e

=

.

In the B-S model, each derivative asset is defined with respect to its terminal pay-off, is positive and F,-measurable under the probability P*. Hence, at each instant of expected time, the option price is given by its conditional terminal value under the same probability P*:

which

V,

(12)

=

E*[e

'

"h/F,]

with

h

(S, - K)

=

=

f(S,)

The option price at time t can be expressed as a function of time and the underlying price, In fact:

(13)

V,

=

E*[e

"^

"h|F,]

(21) asset

(22)

and N(d)

=

-

exp

-

dx

(14)

V, Smce for all t,

=

E*[c-ar

"f(S,e"T

"exp

{o(W*

-

W*) -jo2(T

-

t)}] F,]

(23)

S,

Soexp [oW

=

-

ja

2

LICATIONS

Alli$$$2ENVATIVES

FUTURESA

oPTIONS,

s4

t]

(24)

TO ASSET PRICING

IN COMPLETE

( r-

exp{-r(T-t)}xexp

H(t,x)=E

o

MARKETS

55

2

(T-t)+o(W,-W*)-K

and at the option's maturity date Sr

Soexp[o W*

=

Ü2T]

(25) =E*

the value of =exp[o(W*-

W*)-la2(T

'

S,

-

2

t)]

(26)

where

r

=

oYn--r

xexp

-Ke"

2

(34)

T -- i. This last formula is equivalent to equation(28)

by (z, K) Now, using the

where

f(x)es

laced

.

Since the random stock price S, is F,-measurable and (W* - W P*, it can be shown that

) is independent

of Fr

under

V,

H(t, S,)

=

notation

(27)

In

di

with

H(t,

x)

=

E*

'"

Under the probability P*, (W Y When (W, - W*) =

exp {-r(T

-

t)}

-

W*)

(T - t)

-

W*) follows Gaussian law N(0, a and Y follows N(0, 1), then:

-

=

H(t, x)

o(W*

"exp

xe'"

"f

e

xexp

H(t, x)

dy

exp

=

=

exp

dy

E

\

xexp

\

r

de

(T - t)+ o(W,

2/

K)

E[(Z-

(35)

a

K - W,) -

(36)

]= E(((Z- K)]1z,«]

the condition Z E,.[G(

e

)Y)

1

1

(30)

since =

=

(37)

where

When Y follows N(0, 1), under P*, then

H(t, x)

d2

Using the following lemma:

(29)

Es-[h(Y)]

2)r

o

weconsader again the equation

)

(T - t) + oy

r -

(28)

(r

=

(31)

xexp

>

K is equivalent

[a Y

ifZ>

K

(38)

__

,,

to

a2r]

K

>

[oY

or

-

þr24>

M

X

where

Hence G(

)Y

=

f

i)Y

o(v

xexp

r -

(T

-

t)

(32)

l exp

(

y')

(33)

the call's pay-off function, h

=

(S, - K)*, then using equation

(28):

,

(40)

,

or }'

under P*. If we replace

'

Y

and Y follows N(0, 1) with a density

>

-d

or Y + d2

Using this remark, H(t, x) can be written as

>

0

(41)

OPTIONS,

FUTURES

AND EXO1WCOERIVATIVES

APPLICATIONS

1Y+dp0

3.3.1

«2

H(t, x)

=

=

E

oY

xexp

('

=

x exp

xexp

/

- Kexp(-rr)

-r U2

a Y

r

-JY

-r

- K exp

(42)

1

(-rr)

exp

(-jy') dy

exp(-jy2)dy

- Kexp(-rr)

(43)

TO ASSET PRICING

MAIWETS

IN COMPLETE

to Finite Difference Methods

Introduction

The non-existence of analytic solutions to some types of partial differential equation leads to the choice of an appropriate numerical scheme. This allows the approximation of to the real solution. For an introduction to finite difference methods and their applications the heat transfer equation, see Appendix 3.D. Consider, for example, the discretization of the B-S partial differential equation

(44) 472S2

82c

+ rS

Oc

Oc -

This integral can be divided into two integrals: d2

2

x exp

YO -

-a

2

r

exp

O

,

(-jy2) dy

da

+

exp(-¡y2)dy

- Kexp(-rr)-

(45)

57

rc

=

(49)

0

asset price S and on The discretization of a function c(S, t) depending on an underlying time t requires the division of the asset price (the state variable) and the time period until the maturity date T (the time variable) into a finite number of equally spaced intervals of length AS and At. For example, the time to maturity, T - t, can be divided into M subintervals of length At with At (T - t)/M. This gives M + 1 time points from 0 to T. The underlying commodity price can be divided into M sub-intervals of length AS with AS So N where So is the highest underlying asset price. This gives N + l stock prices 1) for 0 to Sm. This formulation can be represented on a diagram with (N + 1)(M =

=

The second integral is equal to -Ke "N(d2). If we use the change in variables z y + a can be written as =

,

or y

=

z - a

,

then the li

integral

pomts. k as the step with respect to the time variable, or the time step, We will refer to At respect to the state variable, or the state step. the with and AS h as step =

Jx

xexp

a2

-a(z-an)O--r

1

=

exp(-(z-of)2)dz

(46)

Developing this quantity and simplifying gives exactly xN(di). Finally, the sum of the two integrals gives the following formula for a B-S call option: H(t, x)

=

xN(di)

- Ke "N(d2)

(47)

with

N(d)

=

exp

-

dx

The implicit Difference

Scheme

The option price c(S, t) is represented in the diagram by a point (i, j) where i refers to the space index and j refers to the time state variable. Each interior point c(i, j) can be approximated at point (i, j) by either a forward difference 1, j)

c(i,

j)]

(50)

l c(i - 1, j)] -h [c(i,j) -

(51)

[c(i +

=

(48)

-

or a backward difference

Oc OS

3.3

NUMERICAL ANALYSIS TECHNIQUES

AND SIMULATION

or an average of the two approximations

Oc There are many problems in financial economics which do not have closed-form solutions analysis and or analytical approximations. However, it is always possible to use numerical simulation techniques to approximate derivative asset prices. Numerical methods are often based on the discretization of the state and time variables. Simulation methods rely on simulating random variables. The price of any financial asset can be simulated when it can be expressed in the form of an expected value of a random variable. The best known method is the Monte-Carlo technique which allows the calculation of an option price. This method is only used when there is no other method to value the contingent claim

1

+ g [c(i

¯

1, j)

-

c(i

-

(52)

1, j)]

The term 82c/BS2 in the partial differential equation can be apptoximated point by 42 =

-

[c(i +

¯ hy [c(i +

c(i, j)] l, j) - -

[c(i, j) -

2c(i, j)] 1, j) + c(i - 1, j) -

c(i

-

et the

(i, j)

1, j)]

(53)

OPTIONS,

58

AND EXOTIC

FUTURES

The partial derivative with respect to time, ßc/8t, can be approximated bv the forward difference:

l

oc

=

-

[c(i, j +

at the

(i, j)

1) - c(i, j)]

PRICING

ßc 8t

po

(54)

IN COMPLETE

=

k

[c(i,j+

If we replace these derivatives by their values get the following system:

in the B-S partial differential

these partial derivatives with their values 1 to N - l and j 0 to M - l: we get, for i

If we replace equation.

DERIVATIVES

c(i, j)

a*c(i

=

1, j+

-

MARKETS

59

1) - c(i, j)]

(62)

in the B-S partial differential equation,

1) + b c(i, j+

1)+ c*c(i+

1, j+

1)

we

(63)

=

=

reti.j)=jo2i2h

[c(i+l, j)+c(i-1,

h

with

j)]

j) -2c(i, 1

1 +rih-[c(i+l,j)-c(i-l,j)]+-[c(i,j+1)+ 2h

a

(55) (i,j)

<

'

b

l

rk

[-(rik

I

rk

[l

1

rk

[jrik+

+)o2i2k]

-
(64)

This system can be rewritten as

j + 1)

c(i,

=

arc(i

-

1, j) + b c(i, j) + c,c(i + 1, j)

c*

(56)

with

This method gives =

a,

b,

(rik-ja2i2k

=

=

c,

1 + a2:2k + rk

-(rik- jo2

(57)

c(i, M)

=

max

[0, ih

K]

-

for i

=

=

max 1,

[0, Sr

...,

K), is approxi-

-

N

(58)

M - I gives N - I simultaneous equations: c(i,fo M - 1) + c,c(i + l, M - 1) a.c(i - 1. M - 1) + b,c(i, Equation (57) with j

)a2i2k]

between three different values of the option at time 1), c(i+ 1, j+ 1), and one value of the deriva-

t+ (j+ l)k, i.e. c(i - 1, j+ 1), c(i, j+ tive asset at time t + jk, or c(i, j).

2k

for the European call, c(S, T)

The boundary condition mated by

a relationship

=

3.3.2

Application

to European

The Partial Differential

Calls on Non-Dividend

Paying

Stocks

Equation

=

Consider the discretization of the B-S partial differential equation

=

=

1,

N - 1

...,

(59)

c(N - 1, unknowns: c(1, M 1), c(2, M - 1), which can be solved for N - I M - 1) where the c(i, M) are known and given by equation (58). Solving this system gives the option prices at time 0 as a function of different levels of the underlying c(M - 1, 0). asset, c(1, 0), c(2, 0), This method gives a relationship between three different values of the option at time l, j), c(i. j). c(i + 1, j), and one value of the derivative asset at time r jk. i.e. c(i ik. c(i, l j + l). ( r -- j or

2S

rS

- rc

=

0

(65)

...,

under the conditions c(S, T)

=

max

[0, S,

K]

(66)

...,

-

lim se

-

Scheme

The Explicit Difference

Oc 82c BS2

=---[c(i

l h2

l

[c(i +

c(i l j) - 1, j + 1)] ,

l,j+l)+c(i-l,j+l)-2c(i,j+l)]

BS

This last condition shows that for sufficiently option behaves like the underlying asset.

The Numerical

The scheme is simpler than the preceding one. The partial derivatives in the partial respectively by differential equation are approximated

=

[

1

(67)

high values .of the underlying

asset, the

Solution

Using finite difference methods, the partial derivative with respect mated by either a forward difference

to S can be approxi-

(60) (61)

L J or a backward difference

[c(S + h, T) - c(S, T)]

(68)

Oc

l

T) -ßS (S, At points

(i, j), the opt ton

=

-

h

AND EdígDTIC DERIVATIVES

FUTURES

OPTIONS,

40

[c(S, T)

BS

h

[c(i + 1, j)

c(i, j)]

-

b,

j) h [c(i, -

A better approximation of c(i, j) at point asset price and divides by two: l

oc

(70)

(i, j)

1, j)]

-

two steps h a the underlying

considers

c(i - 1, j)]

l

Oc(S + h, T)

Oc(S, T)

Ã

OS

OS

1, j)

[c(i +

The term 82c/OS2 can be approximated

1

=

The partial derivative with point by

=

at

k [c(i,

j) -

c(i,

lo2i2h2

1

Simulation

Monte-Carlo

gives

(74) 4

.

can be approximated

at the

(i, j)

e

(75)

j - 1)]

j) -

'

[c(i + 1, j) + c(i - 1, j) - 2c(i, j)]

[c(i + 1, j) - c(i - 1, j)] +

.

.

k and

gathermg

the terms

c(i,

[c(i,j) -

.

Multiplymg by c(i, ) - 1) gives

in

c(i

j - 1)] .

.

-

.

1, j), c(i, j),

rc(i, c(i

j)

j - 1)

=

a,c(i

2

c(n

=

ih - K 0

underlying

-

l, j)

=

=

max

[0, S, -

K], is approxi-

if ih a K else

asset, the condition h

for j

=

0,

...,

on the optienkpartial m

(80)

Option prices are calculated using the expected value of the terminal pay-off. Since this value is unknown, it can be simulated. Boyle (1976, 1986, 1988) proposed a simulation methodology using the distribution of asset values at the option's expiration. This distribution is determined by the process which generates future movements in the asset price. If the process is specified, then it is possible to simulate its values. Each time a simulation is done, the computer generates a terminal value of the financial asset. Repeating the simulation procedure 1000 times gives the distribution of terminal asset values. This distribution allows one to extract the expected asset terminal value. The simulation problem of the price of a financial asset can be introduced as follows. Given a random variable with a law µdx, if we draw n times on a computer Xi, Xa, X,, with sufficiently large n so the X,, follows the same law µdx, and the sequence (X,,),, is a sequence of independent random variables, then by the law of large numbers the derivative asset price F can be expressed as follows:

-

l, j) + b,c(i, j) + cic(i + 1, j)

No

1, j) and

(77)

N

(Xs)

=

f(x)µ(dx)

(81)

The Monte-Carlo method can be implemented on a computer in the following way. We which correspond assume that we can construct a sequence of numbers (Us),,, to a uniform sequence of independent random variables on the interval 1]. Then, we look [0, p) for a function to which corresponds, F(ui, u u p), such that the law of the (ui, random variable up) is the unknown law of µ(dx). The sequence of random F(ui, independent variables with (X,) ...,

...,

with

2

Method

(76)

.

c(i,

(78)

_la2i2k

Methods

lim + rih

Irik

...,

=

=

=

c(n,

3.3.3

Hence, the option price can be determined at each instant j as a function of its value at instant (j- 1). If we replace these partial derivatives with their values in the B-S equation, we get, for i 0 to M - 1: 1 to N - l and j 0

1 + a2i2k + rk

l) linear equations with (n + 1) unknowns u(i, j). The resulting system gives (n Since the option value at the maturity date is given, it is possible to generate at each time step all the option prices by inverting the resulting linear system. A detailed algorithm is given in Appendix 3.E.

3)

2c(i, j)] 1, j) + c(i - 1, j) -

respect to time, ßc/8t,

-c

4

(72)

.

-[c(i+

41

by

Replacing the partial derivatives of Oc OS in this last equation

82c

0)

For sufficiently high values of the derivative is approximated by

(71)

-

=

82c

c(i

MARKETS

The boundary condition for the European call, c(S, T) wilatedby u(i,

=

=

Ci ¯

or a backward difference BS

IN COMPLETE

a,=(rik-ja2i2k

by a fdrward difference

partial derivative is approximated =

TO ASOU¶PRICING

(69) Ÿ

c(S - h, T)]

-

APPLICATADNS

..

.,

OPTIONS,

62

(X,,)

F[U,,,_a

=

FUTURES

U,,,]

...,

,

AGRESIIIRTICDERIVATIVES

APPLICATIONS

(82)

follows the law µ. To generate the sequence (U,,),,,, with a computer, we use the random function which gives a pseudo-random number between 0 and 1 or an integer in a fixed mterval. The function random or Rand(.) is often avadable on computers. It is convenient to note that the main drawback of the Monte-Carlo methods is that they are cumbersome in the valuation of American options and are most often used for the valuation of European-style options.

A random

Gaussian Variables

. • • • •

•

X is a Gaussian variable with a zero mean and a unit variance

variable

MARKETS

63

How can we define a complete market? What do we mean by option pricing? What are the different methods of option pricing? What is the partial differential approach? What is the martingale approach? What is the Girsanov theorem? What is the principle of the Monte-Carlo method? What are the difTerent finite difference schemes? How can we simulate a Gaussian variable? What are the differences between an explicit and an implicitalmstericat acisease?

. .

•

of

IN COMPLETE

POINTS FOR DISCUSSION

.

Simulation

TO ASSET PRICING

when

its density function is given by n(x)

=

exp

- 2 /

(83)

The law of X is known as the standard normal distribution When a random variable Y m + aX is of the Gaussian type, it follows a normal It is often denoted distribution with parameter m as the mean and o2 as the variance. N(m, a2). If there are two random uniform independent variables (Ui, U2) on the interval [0, 1], then cos(2;rL/2)¾ follows a normal distribution with zero mean and unit

APPENDIX 3.A: THE CHANGE THE GIRSANOV THEOREM 3.A.!

The Equivalent

IN PROBASKSTY Atigt

Probability

=

variance.

a Gaussian law with mean m and variance a, we just use the m + o g. For example, the following instruction may be used ín in Turbo Pascal to generate the Gaussian law:

In order to simulate change in variables: z a program written Gaussian

:=

comúnuous with considerthe probability space (G, f, P) A probability Q is said to be smictly =0. P if for all A in f, P(A) = 0 implies that Q(A) The foHowing respect to the probability theorem gives what is well known as the equivalent probability.

Theorem

=

m+

sig*cos(2

pi

sqrt(-2*log(Random))

Random)

.

.

A probability Q is said to be strictly conunuous with respect to a probability P if and onlv if there exists a random variable Z taking on positive values on (Q, §) such that for all A in f we have

(84)

Q(A)

=

z(w)dP(w)

(85)

where Z is the density of Q with respect to P and is often denoted by dQ/dP The probabilities and Q are equivalent when each probability is continuous with respect to the other.

P

SUMMARY .

.

This chapter contains the basic material for the priemg of derivative assets in a continuous time framework. The presentation is made as simple as possible in order to enable umnformed readers to understand these derivations First, we present in detail the search for an analytic formula for a European call option within the partial differential equation method. Second, we illustrate in detail the martingale method for the derivation of a European call formula. Third, we apply fmite difference methods for the valuation of European call options. of the basic techniques of option While this chapter is necessary for the understanding pricing theory, it is not necessary for the use and applications of all the formulas presented in this book.

3.A.2

The Girsanov

Theorem

Consider the probability space (Q, F, (f,)os,sy, PT If (8,)os,«r is an adapted process such that 8 ds
j

L,

=

es d W, -

exp

8 ds

where dW, is standard Brownian motion, then under the probability a respect to P, the process (W*)oster defined by '

is a standard Brownian motion.

+

J

8, ds

(86) P

with the density Lr with

(87)

OPTIONS,

64

AND EXOTIC

FUTURES

APPLICATIONS

DERIVATIVES

TO ASSET PRICING

OF THE PARTIAL

APPENDIX 3.B: RESOLUTION DIFFERENTIAL EQUATION

IN COMPLETE

-rf(t)y(ui,

MARKETS

af

u2) +

-81 y

u2)

i,

65

0

=

(96)

We seek a solution to the differential equation

82c

a2S2

rS

Oc

Oc

-

-

rf(t) - rc

=

0

(97)

=

(88) Hence

under the boundary condition

f(t) c(S, t*)

=

max

[0, S*

K]

-

(89)

=

(90)

f(t)y(ui, u2)

2

a2 and simplifying

Of St

2)

(ui,

Oc

f(t)

=

+

f(t)

By Bui

Oy

Ou

Bu, 81

(!) Sv

By Bu2

Bu2 8t

f(t)

(99)

the differential equation to identify the other terms, we 6nd

and rewnting

82y

(gu

S2

¼J'Ë'

+rS

gu -

py, -

(100)

is equal to 0 if a2

Bu

(92)

-

-

1)

by

gy --

(o'S

=

that the term

where f(t) and y(ui, u2) are the unknown functions that must be determined, and we begin by deriving the first option's partial derivative with respect to time and its first and second partial derivatives with respect to the underlying asset.

Sc St

(98)

Using the notation

The search for the solution needs an appropriate change in variables in order to transform the partial differential equation into the heat transfer equation in its simplest form, We postulate a solution of the form

c(S, t)

'

e"'

=

-- ja2S2

Du

rS

BS2

OS

(101)

Ot

Assuming that

this condition + 2f(t)

if u2(t)

(93)

(94) is reduced

Substituting these partial derivatives in the B-S differential equation, we obtain =

(o2S2

(t)

2

-

=

then u

-a

(t)

Bu

DS

OS

t*)

(t

,

(103)

=

0

(104)

Using the expression for a2, we have By Bu

a

(105)

oS

8 DS2 rf(t)y(u

rSf(t)

-a

+rS

Du Bu

=

to

la2S2

¯

2

( l 02)

is rewritten as

The B-S equation

0

0

=

=

or u

)

+ b(t)

In

(106)

Replacing in (104)givese2S2 y(u

Now, equafion

(94)can

,

u

)

(94)

(t)

2\O2u /

Then, one can search for the functions T, ui(S, t) and u2(S, t) with some additional assumptions:

rS

=

0

(107)

or

be solved if one uses the heat transfer equation for the function y: Ou2

b'(t)

-

ao

(95) Since b'(t) is given by

b'(t) +

rn

=

\a

0

(108)

OPTIONS,

66

b'(t)

FUTURES

AND EglIþTIC DERIVATIVES

laan=

IN COMPLETE

K

(1

r

r

TO ASSET PRICING

APPUCATIONS

MARKETS

+qÃs)-

exp(u,

67

1 exp

dq

-

(122)

Using equations (I l 1) and (122) we have h(t)=

- r

-

(t-t*)

(110)

In

=

_

(t-t*)

r

-

a(t

t

)

(123) S

a,(t)=n

In K

+

in

a2 --r

(t-t

2

(ll1)

)

-2 -

r

(t

-

r

t

-

_

Also, as

c(S, t

)

=

(t)y(ui,

u2)

(112)

In

-

and

(113) the mateity

=

y[ui(S, t*), u2(S, t*)]

=

In

y

Uo(e)exp

=

-

d

(I14)

0

,

+ qa

(124)

y(u, t)

(115)

de

-In

=

r

(t -

t*) +qa

(125)

these values in the integra solution of the heat equation,

and substitute

Recgl that a solution to the heat transfer equation is of the form y(ui, u2)

¡*)

, _

qa

If we denote the lastteem by

conétion is rewritten as c(S, t')

t*)

(t -

=

in

exp

(t -

- r

/*)

we have - 1 exp

+ qa

-

dq

(126)

which can also be written as

with

Uo(s)

=

(116)

lim y(e, u2) --o

y(u, t)

=

'

Ke

q2

(' exp

) exp

(qu

2

At the option's maturity date, the option value c(S, t*) must satisfy the condition

c(S, t*)

=

y(k, 0)

K exp

=

- l

if k

>

0

×

else

0

exp

(t -

- r

lf we make the change in variables p

=

-

t

dq

-

)

,

exp

-

dq

(127)

q then

with y(u, t)

k

in

=

exp

=

r

(118)

(t -

t*)

exp

dq

exp

'3N(d - Ke'"

(128) or

.

Usmg the fact that

(119)

ln

and letting p

hence S Applying this condition,

a2(t

_

equation

=

K exp

=

(120)

- t)

(129)

the integral can also be written as

q + o

)dq

(-qa

exp

=

exp

-

dq

(l15) becomes

Making a change in variables, equation (l21) becomes

=

exp

(t - t)

exp

(t*

-

1)

exp

(q +

exp

(

) dq

a

p')dp

(130)

)

FUTURES

AND EXOTIC

t*) SN(di)

'UN(d2) - Ke'"

OPTIONS,

=

=

e'"

'Dexp

a2(t

t*)

(t -

r 2 -

exp

-

DERIVATIVES

SN(di) - Ke"' 'UN(d2)

(132)

APPLICATIONS

TO ASSET PRICING

APPENDIX ANALYSIS

3.D: INTRODUCTION

3.D.I

The Heat Transfer

IN COMPLETE

MARKETS

69

TO NUMERICAL

Equation

with In

d'

K/

=

(t*

r

\ 2

Using an appropriate change in variables, most equations of the parabolic type in financial economics can be transformed into the heat transfer equation. We restrict the analysis to the

t)

-

(133)

a

d2 whichäthefordwikforàEuropean

(134)

=

calL

equations

with one state variable,

u(0, t)

APPENDIX 3.C: APPROXIMATION OF THE CUMULATIVE NORMAL DISTRIBUTION we have to calculate the cumulative N(d)

("

I =

/

exp

,

x

u(x, 0) standard normal distribution:

x2\

dx

- 2

(135)

We give two methods for the approximation of the cumulative normal distribution which has a precision of order 10 3, is as follows If x The first approximation, coefficients

the formula

=

a2

=

a3

=

0.000 344

a4

=

0.019 527

The second approximation,

~

l -

(1 +

aix

0.115 194

which has a precision of order 10

=

82u/8x2

=

u(L, t)

=

uo(x)

Condition (139) is the heat transfer certain value L.

equation

Some

Numerical

(140)

for 0 ExsL

(141)

for which the state variable x lies between 0 and a

(x, t)

+ a4x")-4

(136) ,

then with

where each point M(j, n) is determined by its position

Suppose that the discrete option values u(j, n) are known for all the space indexes j at each instant of time 0, 1, n - 1, n. We will look for the values u(j, n + 1) and we limit the analysis to the schemes with two steps, i.e. u(j, n + 1) are calculated using only the uU, n) The heat transfer equation is discretized in points MU, n) The time derivative, Bu/St, is approximated by the finite decentered difference: .

as follows. If x>

(139)

for t > 0

Schemes

Uh,nk).

is

for 0 ExsL

Condition (140) is a boundary or a limit condition which indicates that the option is worthless asset is zero. Condition (141) is a limit or a boundary condition which indicates the option's value at the

Consider a grid of points in the plane

2,

0

=

is to ftad the value

when the underlying

3.D.2

+ a2x2 + a3x

instant of time, the

maturity date.

is

N(x)

at each

0, then with

0.196854

a,

t. Hence,

At the option's maturity date, time 0, the value of u(x, t) is given and the probkm of u(x, t) which satisfies the following system: Bu/St

When pricing derivative securities,

x, and a time variable,

derivative asset price can be written as u(x, t). The simplest form of the heat equation is

.

.,

"'+

p

bi b2 b3

=

=

=

=

«U, n +

0.231 6419

c

=

1.78 1 477 937

"(.j,

N(x)

1-

n + l) - 2u(

n) + u(j -

1, n)

by the finite centered

o(h2)

=

(143)

1.330274429

This gives what is often known as the explicit scheme:

1 (1 + px)

u(j, (l/vi¯T)cxp(-x2/2)(bic

(142)

which is exact to the first order. The derivative with respect to the space variable, 82u/8x2, is approximated difference:

-0.356563782

the formula is ~

o(k) '

=

b,

«U, n)

0.319381530

/>4 -L821255978 =

1) -

+ b2c2 + b3c3 + b4

4

5

(137)

n + 1)

=

u(j,

n) +

k

-[u(j+ h2

This scheme is accurate to the first order, if k

=

1, n)

o(h).

-

2u(j, n)

u(j

-

1, n)]

(144)

70

FUTURES

OPTIONS,

AND EXOTIC

DERIVATIVES

It is also possible to apply (143) at the level (n + 1) rather than n. If we replace (142) and by

(143)

APPLICATIONS

TO AiglET

PRigglilBdillCOMPLETE

g (n) u (n, j) 1 down to 1 For i = n -u(i)u(i+1, u(i, j)g(i) end for End(forj=1tom)

MARKETS

71

=

-

Du

utj, n + 1) - utj, n) k n+1)-2u(j,

u(j+1,

L

n+1)+utj-1,

uso

+ o(k)

n+1)

we get the totally implicit numerical

scheme

u(j, n + 1)

-[u(j+

u(j, n) +

k

We can take the average of the numerical Nicholson numerical scheme: u(j, n + 1)

u(j, n) +

=

-2u(j,

1)

UU,n)

=

(144) and

(146)

1, n + 1)1

uU -

(147)

(147) to obtain the well-known

1, n) - 2u(j, n) + u(j-

1, n) + u(j+

(148)

k + - {(I -

0)[uU+

1, n) - 2"(1, n) +

<

8 e 1:

1, n)]

uU-

+0[u(j+1,n+l)-2u(j,n+1)+u(j-l,n+l)]}

(149)

l, the scheme is of When 8 0, this scheme reduces to the explicit numerical scheme. When 6 the implicit type. When 6 = we get the Crank-Nicholson numerical scheme. A numerical scheme is convergent if the solution of the discretized problem tends to the solution of the problem with partial derivatives when h and k approach zero, in practice, for a sufnciently small h and k, we approach the real solution and the problem is solved numerically. =

=

i,

APPENDIX j

3.E: AN ALGORITHM

FOR A EUROPEAN

1 to M 0, 1+rk+o2k -1 2rk - 1 2o2k c (1) d(1) =u(1, 1-1) For

a(1) b(1)

=

=

=

=

Fori=2ton-1

a(i) b(i)

=1

2o2i2k

2rik-1

=1+a2i2k+rk

2U2i2k c (i) = -1 2rik - 1 =u(ij d(i) 1) end(fori=2ton-1) b (n) 1, c (n) a (n) = -1,

=

This is the procedure

=

0, d (n)

=

h

for the matrix inversion:

Fori=1ton

u(i) g(i)

end

=c(i)

=

(for

(b(i) (d(i) i=

-u(i-1)a(i))

-g(i-1)a(i)) 1ton)

(b(i)

-u(i-1)a(i))

CALL

RULE FOR INTEGRAL

Consider the following integral: I(x)

=

F(x, t)dt

(150)

Ifthe function Fand its derivative are continuous in i in the interval [A, Blamixtilkesanits in the interval [a, b], then the derivative of this mtegral is given by

1, n 4 I)

scheme. More generally, it is possible to write, for 0

APPENDIX 3.F: LEIBNIZ'S DIFFERENTIATION

Crank-

n + 1) + u(j - 1, n + 1)1

which is also an implicit numerical

uU, n +

1, n + 1) - 2u(j, n + 1) +

schemes

[u(j+

2h

©(6

+

h

=

(145)

J;

j)

=

dt + F(x, B)B'(x)

-

F(x, A)A'(x)

values

(151)

If A and or B are infinite, then this rule is applied only if the absolute value of thedetivative of F with respect to t is less than a certain value C(t) for all x in [a, b] and t in [A, B). Also, the indefinite integral f C(t) dt must be convergent in the interval [A, BJ.

4 Asset Pricing in Complete Markets: Changing Numeraire and Time

CHAPTER OUTLINE This chapter is organized as follows: 1. Section 4.1 describes the building assumptions. We consider a very general stochastic economy where transactions are possible at any moment. Interest rates are assumed to be governed by a Gaussian uncertainty. 2. Section 4.2 considers valuation of a European call option in the Black-Scholes economy. numeraire and time change. These two transformations are used to value a European call option in a stochastic interest rate environment. An economic interpretation is also given. 4. Appendix 4.A illustrates the use of the previous transformations of on the valuation barrier options, where the barrier is defined stochastically.

3. Section 4.3 introduces the concepts of change of

NTRODUCTION The main contribution of modern finance to the subject of risk is clear: risk is quantified and its remuneration on financial markets specified. As shown in the previous chapter, nsk evaluation has benefited considerably from application to finance of the contingent claims theory since Arrow (1953)and the introduction of the continuous time modeling theory since Merton (1973a).These methods assume that there are as many tradable assets on the markets as there are states of nature. In other words, markets are supposed to be complete. The use of stochastic processes has immensely enriched the dynamic financial asset valuation theory. Prices are presented as the solution of a partial differential equation, hereafter PDE, or as the conditional expectation of functionals of stochastic processes. These representations produce explicit formulas in the simplest models. In other cases, numerical methods are necessary.

74

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

Historically, as shown in Chapter 3, Black and Scholes (1973) and Merton (1973a) determined the price of a European option by establishing the PDE governing the dynamics of the security. The basic idea is to create a portfolio made up of an option, its underlying asset and a risk-free asset. At each instant and for a given combination of each asset, the total risk can be eliminated. The resulting portfolio becomes locally insensitive asset. In a way, this portfolio duplicates a Treasury to price changes of the underlying bond. Arbitrage is supposed to give this portfolio instantaneous return which is equal to the interest rate on the Treasury bond, the risk-free rate. The probabilistic method, also known as the martingale method, was developed by Cox and Ross (1976), Harrison and Kreps (1979) and Harrison and Pliska (1981). If the markets are complete, there is a unique probability measure, known as the probability, under which the discounted price of any fmancial asset is a martingale. Thus, 'risk-neutral'

valuation of any asset amounts to calculating, under the risk-neutral probability, the conditional expectation of discounted future cash fiows at a risk-free interest rate However, The PDE and martingale methods obviously produce the same valuation. these theories were developed independently from each other. Application of the Feynman-Kac formula, as shown for instance by Duffie (1992b), makes it possible to switch from price as a solution of a PDE to a probability representation. The main objective of this chapter is to introduce two mathematical instruments, namely the change of numeraire and the time change, both very effective in solving valuation problems. The valuation process used throughout this chapter is based on the martingale approach. However, in order to go beyond purely mathematical and technical aspects, as shown by de Varenne (1997), partial differential equations offeran economic interpretation of the proposed transformations. .

.

COMPLETE

=

(A.2) Complete

Financial Markets

without Financial markets are deemed to be complete. Trading takes place continuously, exists absence of arbitrage, there friction, transaction costs or taxation. In the a unique probability measure Q, equivalent to the historical probability P, under which the

This probability Q is Q-martingale.

in the term structure of interest rates, interest rates are To capture the uncertainty assumed to be normally distributed and to follow a Gaussian process. This framework has been used, for instance, by Merton (1973a), Vasicek (1977),El Karoui and Rochet (1989),Jamshidian (1991), and Heath et al. (1992).In such a framework, the short-term riskless interest rate r(t) at time t follows a Gaussian diffusion process and the volatility structure is a deterministic function. The only drawback of such a representation is that it should be noted that for negative interest rates are not precluded. Nevertheless, reasonable values of the parameters, this event has quite a low probability of occurrence.

The process followed by r(t) is governed under the risk-neutral stochastic differential equation dr(i)

a(t)[b(t)

=

r(t)]dt

-

probability

Q by

+ a(t)dW:(t)

the

(1)

for some deterministic functions a(t), b(t) and a(t). o(t) is the instantaneous standard deviation of r(t). According to (1), the dynamics of return of the default-free zero-coupon bond P(t, T) maturing at time Tis given at time t under Q by dP(t,

T)

r(t)dt

=

-os(t,

T)dW,(t)

(2)

T) is a deterministic function defined by T)=o(t)

og(t,

a(s)ds

exp

du

(3)

We let B(t) represent the value as of time t of a portfolio which, at date t 0, has invested one dollar, continuously reinvested at the prevailing spot interest rate r(t). In other words, the B(t) fund defines a capitalization factor over time. It is given by =

Economy

There are two sources of uncertainty across the economy, represented by two independent standard Brownian motions, (Wi(t), W2(t), TE [0. T]), on a probability space (R, F, P). The information flow reaching the economic players is represented by the filtration filtration generated by the augmented (F,, tE [0, T]). (F,, I E [0, T]) represents (Wi(t), W:(t)) with F Fr. The filtration is augmented to include all P-null events in F that right and complete. These two sources of risk spring from the F, is continuous so stochastic character of the interest rates and a risky traded asset.

is a

75

(A.3)Gaussian Interest Rate Uncertainty

ASSUMPTIONS

(A.f) Stochastic

AND TIME

'risk-neutral'

.

We consider a very general economy where transactions are continuous on a givea áme period [0, T]. This economy is characterized by the following four assumptions.

NUMERAIRE

CHANGING

continuously discounted price of any security probability. referred to as the

where o,(t,

4.1

MARKETS:

B(t) (A.4) Dynamics

=

exp

r(u)du

(4)

of the Risky Asset .

.

Let us assume that there is a risky asset A, in the economy, price dynamics under the risk-neutral probability Q:

characterized

by the following

.

d A,

=

r(t)dt

+ aa[p d Wi(t) +

-

d W2(t)]

(5)

where as denotes the instantaneous standard deviation of the asset return. The coefficient between the risky asset and the stochastic interest p, pE [0, 1], introduces a correlation rates. W2(t) can be viewed as a standard Brownian motion, independent of W,(t) capturing the asset risk other than the interest rate risk. the price of any Given the definition of B(t) and the absence of arbitrage opportunities, We define a European asset, relative to the capitalization factor B(t), is a Q-martingale.

AND EXOTIC

FUTURES

OPTIONS,

76

DERIVATIVES

If it gives contingent asset as a non-negative L2 random variable, Fr-measurable. right to receive at time / T the final stochastic pay-off h( T), where h( T) is Fe measurable, its time t 0 value Vo can be stated by the following conditional expectation probability Q: under the risk-neutral

COMPLETE

MARKETS:

CHANGING

d

=

NUMERAIRE

In(Ao/K)+

(r +

=

AND TIME

2)T

a

d2

=

77

4

=

Vo where

EU[

EU

=

operator under the probability

expectation

] is the conditional

(6)

ß(I)

Q. with

respect

to the filtration E

4.2

VALUATION ECONOMY

and N( ) is the cumulative standard normal distribution. in equation (8): Let us make successively the following three changes of variables " e'« n, C, Ce'" and r The first two changes of variable are a change A, t. a f, of numeraire. Indeed, f is the forward contract on the underlying asset A for delivery at time T defined by the cost-of-carry formula between the spot price A and the forward price f In the same way, C, denotes the price of a forward call option. The last transformation is a time change. Substituting the change of underlying asset f in equation (8) gives =

=

=

BC

IN THE BLACK-SCHOLES with

The Black-Scholes (1973) economy is based on two simplifications. It assumes that interest rates are constant and equal to r. The price of the risky asset A,, as defined by equation (5),is thus given under Q by dA --' A,

=

r

dt +

«»

d W:(t)

(7)

Let us consider a European call option C on the underlying asset A, with an exercise price K, a maturity Tand a spot asset price of Ao at time t 0. In their seminal papers, Black and Scholes (1973)and Merton (1973a)show by an arbitrage argument that, at each moment t, the call option C must satisfy the following PDE: =

BC + St

(alA

,

82C BC -BA2 + rA -DA - rC

0

=

the boundary condition

rc

0

=

(12)

K, 0}. Equation (12) will be recogC(f, 0) max nized as the Black (1976)pricing equation for a call option on the forward contract f Comparison of equations (8) and (12)shows the impact of the change in underlying asset. Spot price A is replaced by forward price f The coefficient o a is not affected because A and f have the same volatility. This result holds, of course, only because interest rates are =

{f -

non-stochastic.

However, the term (rAßC ßA) in (8) has disappeared in equation (12). This is not surprising. Indeed, forward contracts cost nothing to initiate and to adjust the position in the replicating portfolio Vwhich consists of an investment in (BC ßf) forward contracts and an investment in amount C at the risk-free interest rate r at any time t before T Because positions in forward contracts require no investment, the dynamics of this portfolio V can be stated on one hand by

(8)

:

dV

the boundary condition C(A, 0) w max {A - K, 0}. This equation results from the creation of a replicating portfolio V consisting of (dC BA) risky assets and an investment m the amount (C - ABC/OA) at the risk-free rate r. In other words, this is the combination of the risky asset with the riskless asset that duplicates the option C at any time t until maturity T Indeed, the dynamics of this portfolio Vare given on one hand by ith

82C

(alf2

Crdt+-df

=

BC

(l3)

=

.

.

and, on the other hand, by ltö's lemma: dV

BC

=

-+)olf2

82C

OC

dt

df

'

'

(14)

which gives the

dV

C- A BA

=

rdt

(9)

dA + BA

and, on the other hand, by Itõ's lemma: dV

OC =

=

,

+)o-A

82C -

DC

dt+DA

Taking the difference between equation (10) and equation solution to (8) is thus given by the well-known formula C(Aa, T, as, where

r)

dynamics (12) of C. As noticed by Black (1976), the price of a call option on the spot asset A and the price of a call option on the equivalent forward contract f have the same value. 0, If C, is defined by C, Ce" this change of variable in (12) induces the following expression:

=

AoN(di) - Kc

(9) '"

(10)

dA gives the POE

N(d2)

82C,

DC, -+)o2]

=

0

(15)

(8). The

(11)

with the boundary forward contract,

C,(f, 0) K, 0}. Since the option C, is itself a max { portfolio V, is just no initral investment. Thus the replicating (OC,/8f) forward contracts f, and its dynamics are given by

condition it

requires

made of a position in comparing, on one hand,

=

78

OPTIONS,

d V,

FUTURES

AblD EXOTIC

DERIVATIVES

d

=

(16

and,on the other hand, by Itö's lemma:

coMPLETE

MARKETS:

79

NUMERAIRE AND TIME

CHANGING

the price of any financial asset is a martingale as regards the domestic capitalization equation (6),the price of any fmancial asset with factor B(t), ln other words, as stated by numeraire However, Geman, El Karoui and Rochet B(t) is respect to the a Q-martingale. quantities than investment fund B(t) may be chosen as a other that shown have (1994) numeraire and that a given problem, there is an optimal numeraire'. It is, of course, of the calculations involved. complexity optimal for the generally defined by a process X(t), almost surely positive for numeraire is A which does not pay any dividend. Geman, El tE [0, T]. It is assumed that X is an asset Karoui and Rochet (1994) have shown that this gives a probability measure (F, defined derivative with respect to Q: by its Radon-Nikodym 'for

dV,

df

dt +

=

(

Because the entrance cost of the forward option C is zero, the term rC in equation (12) has disappeared in equation (15). Finaly, let us transform the previous equation (15) one last time. If r is defined b oit, with r* 0 T, then equation (15) is obviously simplified into r =

=

ß2C,

SC,

=

0

with the boundary condition C,(f, 0) K, 0}. Equation (18) defines the price max of a forward option on a forward contract whose volatility is equal to l but whose maturity is r'. The last change of variable generates a new maturity for the forward contract and therefore for the option. The solution to equation (18)is thus given by =

C(

o,

r*)

=

{ -

C(fo, r*, 1, 0)

=

foN(di) - KN(d2)

(r

+ al/2)T

forrnula and the time change

X(0)

h(T)l E

X(0) E

=

NUMERAIRE

This section introduces the methodology pricing of a call option under stochastic martingale approach, changing numeraire computational point of view. However, to ah vaena ongneconhem cDcEonee odology is st

4.3.1

The Change

h(T)

x

Q (21)

(22) 1

I

P(0, T) B(T)

'digests' .

Combining equation

(22)and

equation

(21) gives

AND TIME for changing both numeraire and time. The interest rates is taken as an illustration. In a and time may be quite powerful from a go beyond the pure mathematical techniques' made to show that these transformations also

of Numeraire

We can demonstrate the impact of this specific change of numeraire by means of a European call option C, on the underlying asset A as defined by equation (5), with a strike price K and a maturity T The only difference from the Black-Scholes (1973) risky economy in the previous section is that interest rates are henceforth stochastic. The of this option the price Formula notably by correlated to them. C (6) defines asset A is the following expectation

under

the probability

C.,

=

Q: (24)

1

E

where the indicator function le for E is the real-valued Ic(w)

As seen in assumption (A.2), the completeness of the financial markets means that there is a unique probability measure Q, equivalent to the historical probability P, under which

(23)

[h(T)]

P(0, T) E

=

B(T)

CHANGING

X is a

P(T, T) B(0)

dQ"

EU

4.3

B(T)

in such a way that the relative price of any asset compared with numeraire martingale. The conditional expectation (6) can thus be rewritten as

A comparison of the pure Black-Scholes (1973) equation as given by (8) and the last dy namics (18) shows the final impact of the above-mentioned transformations. Everything goes in equation (18) as if we were in a Black-Scholes economy with an interest rate r of0, a standard deviation a of I and a maturity Tof r*. But this is a pure cosmetic the forward contracts, captures the argument. Indeed, the choice of new numeraires, through the cost-oficarry mterest rate uncertainty the volatility of the forward contract.

(20)

=

dQ

For example, let us take as a new numeraire no longer B(t) but the riskless zero-coupon bond P(t, T) with maturity T As we shall see, what makes the choice of this numeraire P(t, T) especially judicious is the fact that interest rates are stochastic. The numeraire derivative makes it possible to define a new probability Q" by giving its Radon-Nikodym with respect to Q:

because ln(Ao/K)

i

(18)

=

=

1

Vw é E

0

otherwise

random

variable defined by

ONS,FUTURESAND

80

Equivalently

IC DERIVATIVES

EX

Q":

under

Ca where

K

A,

P(0, T)-Ey

=

\P(T,

-1

T)

(25)

a

T))

P(T,

obviously P(T, T) l Analytically, the objective of the change of numeraire is clear m formula (25). It makes it possible to avoid. the stochastic character of interest rates. In order to specify the let us examine the impact on the economic interpretation of this transformation, correspondmg PDE With stochastic interest rates, the price C., of such an option at time / is usually calculated by considering C as a function of t, r and A. This of course gives an equation with partial derivatives relative to these three variables. Another less frequent approach is this the price C of the European call take of and In C P A. function case, to t, as a option must satisfy the followine PDE: =

-

.

.

.

.

BC, 8t

.

.

,02C, 82C, o A -+)o2,,P2--posoeAP BA2 OP2

82C,

,

ßABP

+rA-

DC DA

rP

BC DP

rC,=0

COMPLETE

MARKETS:

CHANGING

boundary condition C (A, 0) max {A - K, 0}. Because C, is first-degree homogeneous in A and P, namely

SI

.

.

'forward-neutral'

.

.

.

dQ -dQ"

A, =

P(0, T)

(29)

-

Ao P(T,

T)

=

C4 expression

AND TIME

under the martingale approach amounts to a change of R The change of numeraire variable in the PDE method. Equation (28)has the same structure as equation (l 5) in the P(t, T) through the cost-ofBlack-Scholes (1973)economy. The change of numeraire formula the stochastic character of As Fis a forward price, captures interest rates. carry it sums up itself the random character of interest rates. The only point distinguishing this economy from a standard Black-Scholes (1973) economy is the fact that the volatility a,(t, T) of the underlying asset is not constant but deterministic. A second transformation is necessary to recover the exact assumptions of the foregoing section. The intuition behind this particular change of numeraire was already found in Merton (1973a)and Bick (1988), Indeed, Merton (1973a) normalized the price of the underlyina asset in his PDE by means of a change of variable A,/KP(t, T). This new probability Q", often referred to as the probability, was examined m depth and used by Jamshidian (1989) and El Karoui and Geman (1993). Equation (25) can still be simplified by a new change of numeraire. Indeed, let us consider the underlying asset A itself as a numeraire. It defines a new probability measure Q*such that

(26) with the

NUMERAIRE

(26) is simplified 8t

-

A-

C SA

+ P

Substituting this change of probability

OC

C,

DA2

+ jo2,P2

,

-

KP(0, T)E

1

(25) gins >K

(30)

SP

into

a A2

laars

AoE

=

in the first term of equation

po go,AP BABP OP2 -

=

0

(27)

Taking P(t, T) as a new numeraire amounts to changing the variables in equation (27). Let us assume that F is the value of the underlying asset A relative to the new numeraire A,/P(t, T). We also define the value of a European call option C, by P, with F, C,/P. The economic interpretation of F and C, is simple thanks to the cost-ofC, formula. F is the price of the forward contract on the underlying asset A at time t, carry for delivery at time T, while C, is the price of the forward contract on the corresponding European option. With these changes of variables, equation (27)can be easily rewritten as

Moreover, after these two changes of numeraire, the relative price of any financial asset compared with P(t, T) (respectivelyA,) is a Q"-martingale (respectivelya Q"-martingale). It can therefore be expressed as stochastic integrals based on two standard Brownian motions (W"(t), W,"(t), t é [0, T]) under Q" (respectively (W (t), W (t), tE [0, T]) under Q').

=

=

DC,

ay(t,

T)2F2-

82C, =

(28)

0

4.3.2

The Time

Change

The change of numeraire is a useful instrument which can be used to express, under an appropriate probability measure, the relative price of any security as a martingale. However, in many financial problems, some distribution functions are only known in the method case of the standard Brownian motion. In these cases, the time-change for martingales is essential to obtain the necessary explicit formulas. Let us consider M P-martingale, satisfying the {M,, 0 st < oc a continuous condition lim, (M), almost surely under P, and defined by the following x, stochastic differential equation: =

T) with Cy(F, 0) max {F - K, 0} and ay(t, T)2 o , + os(t, T)2 + 2po,,ap(t, It is interesting to note that the factor or(t, T) represents the instantaneous volatility of P. Simply put, it is the asset F, in other words, of the asset A relative to the numeraire volatility Of course, interest rates environment. of the forward contract F in a stochastic 0 0 and p has disappeared, as in the specific case where interest rates uncertainty which leads to os(t, T) a Equation (28) gives the economic interpretation of this particular change of numeraire =

=

=

=

=

dM,

=

xi(t)

d Wi(t) +

x2(t) d W2(l)

(31)

=

where xi and x2 are deterministic functions. Under the previous assumptions, it can be shown that there exists a unique one-dimensional standard Brownian motion B such that

AND EXOTIC

FUTURES

OPTIONS,

82

COMPLETE

DERWATIVES

CHANGING

MARKETS:

(32)

x

(u)du.

C

AoE

=

numeraire,

A'

in

=

P(t, T)

/

=

Jo

1

(33)

T) and P(t, T)/A am respectively

,

(34) T)2 + 2paso,(u,

+ os(u,

[a

price C, is now a

0

(41)

=

P(0, T)·C,(Fo)

=

P(0, T) C

A ,

r*, 1, 0

(42)

where C is the standard Black-Scholes (1973)solution as defined by (11). The change of numeraire absorbs the stochastic character of the interest rates. This the volatility of the forward contract. It was found intuitively in time change Merton (1973a).It defines a new economy in which the only source of risk is a standard one-dimensional Brownian motion and in which the volatility of the asset is equal to 1. In such an economy, it is necessary to for a time equal to r* in order to achieve the of risk borne the initial economy until T same degree as m

.

/

(28). Option

=

Ca

At

\

,

T)EU

A,/P(t,

But, as mentioned in the previous paragraph, Q"and Q"martingales which satisfy In

=

- KP(0,

1

=

SF2

83

0) max {F - K, 0}. The instantaneous volatility ay(t, T) of the forward contract F is fully absorbed by r. Thus, this environment is extremely similar to the Black- Scholes (1973)economy and equation (18).Equation (41)only defmes the price of a forward contract on a European call option with strike price K, maturity r* on an underlying asset with a volatility of 1. Finally, the solution of(24) as given by the martingale approach by expression (40)can also be stated as the solution of the PDE (8):

of the European call optionavith stoch the price C, as of time i 0 has

Let us take again the previous example

02G, -

Or with C,(F,

interest rates. After two changes of defmed by equation (30):

2

/

=

AND TIME

T)2 du in equation BC,

2

(M),

where

By,

=

L'

y(u, change of variable r(t) a function of Fand r which satisfies =

M,

x,(u) d W,(u)

=

NUMERAIRE

'digests'

T)]du

'wait'

futro¢ucing the time change r defined by r(t)

[oi

=

T)2 +2posoc(u,

+ or(u,

T)]du

os(u,

=

J0

-

T)2 du

(35)

0

SUMMARY

with r

r*

=

T(T)

[oi

=

T)2 + 2poso,(u,

+ o,(u,

T)] du

(36)

o

from (32) there exist two standard one-dimensional respectively under Q" and Q" such that A,

Ao =

-exp

=

A,

\

Ao

l

B

exp

Brownian

r(t)

motions

ß" and 84

\

(37) (38)

B "" r(t) 2

O

Combining these expressions in equation (33) gives C,

=

This last expression

C

=

AoN

AoEU

KP(0,

1

(39) is straightforward

In Ao KP(0, T) + r* 2

This time change

is interpreted

T)EU

(39)

1

to compute and gives KP(0, T)N

by comparison

POINTS FOR DISCUSSION •

r*/2 ln Ao KP(0, T) -

with the PDE method.

The purpose of this chapter is to introduce two mathematical instruments, the change of and the time change, in the process of valuing complex financial securities. They offer undeniable interest for analysis since they greatly simplify certain problems. Apart from their purely mathematical interest, each of these two transformations has an economic significance. The resulting probability changes reveal new economies, even if virtual, where certain numeraire) of risk absorbed of adjusted are (timechange). sources or (change In such a framework, valuation of exotic options can be simplified. The example of a down-and-out call option with a stochastic knock-out barrier, under Gaussian interest rates uncertainty, is taken. Briys and de Varenne (1997) use the same approach to value a risky corporate bond with the possibility of early default and deviations from the absolute priority rule (see also Chapter 9 of this book).

numeraire

•

(40)

Let us make the

•

What are the underlying assumptions of the Black-Scholes economy? What is a numeraire? G ive some examples of numeraires.

•

What is the risk-neutral

•

What is the forward-neutral probability?

probability?

84

• • •

FUTURES

OPTIONS,

AND EXOTIC

DERIVATIVES

Can any asset be chosen as a numeraire? What is a time change? How can a time change be interpreted?

COMPLETE

.

*

NUMERAIRE

MARKETS: CHANGlNG

AND TIME

85

Early Expiration Under this second scenario, when the underlying asset A, reaches the barrier a(t) for the first time, the owner of the barrier option receives an amount equal to ,)

R(T,,,)=a-a(T

if Te,,
APPENDIX

4.A: APPLICATION

TO BARRIER

OPTIONSI

In this appendix, we demonstrate the effectiveness of the change of numeraire method and the timechange method in valuing exotic options. The main interpretation of this example is that it shows the power of these techniques in a complex valuation context. Barrier options are options whose final value is path-dependent on the underlying asset during the entire term of the contract. Indeed, the value of such an option is determined not only by the price of the underlying asset at maturity, but also by the fact that this same asset, during the life of the option, reaches a predetermined level, the barrier. Like standard European options, the strike price and maturity of the contract are known in advance. As noted by Rubinstein and Reiner (1991a),barrier options are intermediate options between standard European options and American options. Because their value depends on the path of the underlying asset, barrier options have a definite American character. However, in contrast with American options, the exercise frontier is exogenous and specified in advance, which makes their valuation less complicated. In this section, we will attempt to value a down-and-out call option Coo on the underlying asset A, with a strike price K and a maturity T Down-and-out options are European options which expire at an early date if the underlying asset reaches the predetermined barrier before maturity. In the case of early expiration, we assume that the owner of the down-and-out option Coo receives a rebate. The barrier a(t) is assumed to be stochastic. As soon as the value of the underlying asset A, falls below the barrier a(t), the option contract becomes unexercisable but gives an immediate rebate R(t). Ofcourse, once the rebate R is received, the option is worthless at maturity T More specifically, the barrier a(t) and the rebate R(t) are defined as follows:

R(t)

To better understand the picture of the various maturity Tby the following expression:
=

(43)

whereFdenotesapositiveconstantsuchthatAo>FP(0, T)and0saml. Without loss of generality, we derive the time t 0 value of this down-and-out option. To clarify the valuation procedure, we first look at the cash flows which the owner of the contract is entitled to under the various scenarios. He has indeed a claim on the following potential cash flows =

Con

Coo

=

AoE

=

inf

{t a

0, A,

=

«F-1,

E

=

But, given the definitions stated as

[1,

a(t)}

where

,ar.x,ax

Coo -- AoE

(38) of the Brownian

1,

x,

aF-Ira

This appendix

is directly related to Chapter 18.

(48)

,,]

K-Ir

B" and &

er

,

equation

(49) (49) can be

,,¡,,472

2)

K-1,

-

,,<e

(50)

.

where a^(t) ln [Ao/FP(0, T)] + t 2 and a"(t) = In [FP(0, T)/Ao] + t|2. These two conditional expectations in equation (50) are now straightforward to compute, Indeed, as shown for instance by Harrison (1985), according to the reflection principle, we have the Brownian motion on a following results for the first passage time of a standard one-dimensional straight line: =

FP(0,

EU"

[1,

[1

]

=

y,,,

,,

A"

N(-d4)

y,.

2]

+

=

FP(0, N(d

2)

where InAo/KP(0,

T) + r* 2

(45)

InAo/FP(0,

T)+r*/2

InAoK

a

r -

motions

(44)

1, is the indicator function for E.

x

T)E°'[aF-1re

In that case, the option expires at maturity Tand its pay-offs are equal to (Ar-K)-Ir

(47)

,,

+ (Ar - k)-17

,

x] + P(0,

,,

(37) and

E

Tra

K)-lroar

r + (Ar -

at

This complex expression (48) can be evaluated by using the two previous techniques, namely the numeraires A, and change of numeraire and the time change. We use the two above-mentioned P(t, T) and the time change r(t) defined by equation (35).Under the new probabilities Q and Q", expression (48) can be rewritten as

No Early Expiration Under this first scenario, the value A, of the underlying asset has always remained above the default T). Let Ta denote the first passage time of the process A through the knockbarrier a(t) (0 Grs out barrier a:

the pay-offs

cash flows, one can summarize

=

+ P(0, T)E

a-a(t)

T)

The price as of time t 0 of the down-and-out call option Coo is given by taking the conditional expectation of its Onal pay-offs under the risk-neutral probability Q, as indicated by expression (6):

F P(t, T)

a(t)=

TL,

""

=

Equation (50) can finally be stated as

F2P(0,

T)+r*

2

T)

T)

N(-d3) A"

-

FP(0

T)

(52) N -d

53

86

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

*

.

Coo

=

AoN(d ) KP(0, T)N(d2) + aAoN(-d3) AoK

+-N(-ds) F

It is interesting

to rewrite equation

C

=

+ aFP(0,

T)N(-d4)

(54) - FP(0, T)N(-ds)

(54) as follows:

KPa Ca(Aa, K) -

Ao F -F'K -

+a(AoN(-d3)+

FP(0,

T)N(-da)

(55)

•

ical

where C, and P denote respectively two standard European call and put options. The first term in corresponds to the possibility (55) is the price of a standard European call option. The second term of early cancellation of the option contract before maturity. It is characteristic of the down-and-out of early expiration, feature of the option. The last term is the value of the rebate received in case

European

Option n r ricin g Mod els •

CHAPTER

•

•

OUTLINE

This chapter is organized as follows 1. Section 5.1 gives an overview of the option pricing theory in the pre-Black-Scholes (B-S hereafter) period. 2. Section 5.2 presents the main results in the pathbreaking work of B-S for the pricing of derivative assets when the underlying asset is traded on a spot market. It also proposes the different applications of the model to several derivative assets. 3. Section 5.3 reviews the main results in Black's (1976) model for the pricing of derivative assets when the underlying asset is traded on a forward or a futures market. Some applications of the model are also given. 4. Section 5.4 develops the main results in Garman and Kohlhagen's (1983) model for the pricing of currency options. 5. Section 5.5 presents the main results in the models of Merton (1973a) and BaroneAdesi and Whaley (1987)for the pricing of European commodity and commodity futures options. Some applications are also proposed.

NTRODUCTION Numerous researchers have

worked on building a theory of rational option pricing and a general theory of contingent claims valuation. The story began in 1900, when the French mathematician, Louis Bachelier, obtained an option pricing formula. His model was based on the assumption that stock prices follow a Brownian motion. Since then, numerous studies on option valuation have blossomed. The proposed formulas involve one or more arbitrary parameters. They were develoned hv Snrenkle tioAlì Ronem (19641 Thorn Avres (1963ì Samnekan (1965)

88

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVE

The Black and Scholes (1973) formulation, hereafter B-S, solved a problem which had occupied economists for at least three-quarters of a century. This formulation represented a significant breakthrough in attacking the option pricing problem. In fact, the B-S theory is attractive since it delivers a closed-form solution to the pricing of .European options. Assuming that the option is a function of a single source of uncertainty, namely the underlying asset price, and using a portfolio which combines options and the underlying asset, Black and Scholes constructed a riskless hedge which allowed them to derive an analytical formula. This model provides a no-arbitrage value for European options on shares. It is a function of the share price S, the strike price K, the time to maturity T, the risk-free interest rate r and the volatility of the stock price, J. This model involves only observable variables to the exception of volatility and it has become the benchmark for traders and market makers. It also contributed to the rapid growth of the options markets by making a brand new pricing technology available to market players At about the same time, the necessary conditions to be satisfied by any rational option pricing theory were laid out in Merton's (1973a) theorems. The post-B-S period has seen of many financial economists to the many theoretical developments. The contributions extensions and generalizations of B-S type models have enriched our understanding of derivative assets and their seemingly endless applications. It is helpful, as in Srnithson (1991), to consider the B-S model within a family tree of option pricing models. This allows the identification of three major tribes within the family of option pricing models: analytical models, analytical approximations and numerical models. Each analytical tribe can be divided into three distinct lineages: precursors to the B-S model, extensions of the B-S model and generalizations of the B-S model. This chapter presents in detail the basic theory of rational option pricing of European options and its applications along the B-S lines and its extensions by Black (1976) for options on futures, by Garman and Kohlhagen (1983) for options on currencies, and indirectly by Merton (1973a) and Barone-Adesi and Whaley (1987) for European commodity and futures options. The questions of dividends, stochastic interest rates and stochastic volatilities are len to other chapters since the main concern in this chapter is with analytical models under the Black-Scholes (1973) assumptions

5.1 5.1.1

PRECURSORS MODEL Bachelier

OF THE BLACK AND SCHOLES

Formula

M

ANALY

c(S, T)

=

SN \ «

MODELS

KN

-

S- K o

89

+

ap/Ïn

K

S

where: Sis the underlying common stock price, K is the option's strike price, Tis the option's time to maturity, o is the instantaneous standard deviation of return, N(.) is the cumulative normal density function, n(.) is the density function of the normal distribution. As pointed out by Merton (1973a)and Smith (1976),this formulation allows for both negative security and option prices and does not account for the time value of money. 100 and various volatilities and strike prices. Table 5.1 lists call values for S the option pricing problem by assuming that the Sprenkle (1961) reformulated dynamics of stock prices are log-normally distributed. By introducing a drift in the random walk, he ruled out negative security prices and allowed for risk aversion. =

TABLE 5.1

Call Values:

Bachelier

S

=

100 Time to Moturity

Volatility

Striking Price

0./

0.3

0.5

0.7

0.9

l.I

10%

90 100 i 10

0.00 0.01 0.00

0.00 0.02 0.00

0.00 0.03 0.00

0.00 0.03 0.00

0.00 0.04 0.00

0.00 0.04 0.00

25%

90 \00 I lo

0.00 0.03 0.00

0.00 0.05 0.00

0.00 0.0/ 0.00

10.00 0.08 0.00

10.00 0.09

|0.00 0.10 0.00

50%

90 100 l io

0.00 0.06 0.00

io.oo 0.11 0.00

10.00 0.14 0.00

\0.00 0.i7 0.00

lo.00 0.\9 0.00

5.l.2

o.oo

10.00 0.2| 0.00

sprenkieFormula

Sprenkle (1961) derived the following formula: c(S, T)

The story begins in 1900 with a doctoral dissertation at the Sorbonne University in Paris, France, in which Louis Bachelier gave an analytical valuation formula for options. Using an arithmetic Brownian motion for the dynamics of share prices and a norrnal distribution for share returns, he obtained the following formula for the valuation of a European call option on a non-dividend paying stock:

PRICING

OPTION

=

eer SN(di) -

(1 -A)KN(d2)

(2)

with

di

1 =

"

5

In

S 'K

(p+ (a

T

(3)

OPTIONS,

90

DERIVATIVES

AND EXOTIC

FUTURES

and d2 di av/T, where p is the average rate of growth of the share price and A corresponds to the degree of risk aversion. to the average rate of As can be seen from this formula, the parameters correspondmg. growth of the share price and the degree of risk aversion must be estimated. This reduces considerably the interest and use of this formula. Sprenkle (1961)tried to estimate the values of these parameters, but was unable to do so. Table 5.2 lists call values for the Sprenkle formula. =

-

TABLE

5.2

Call Values:

Sprenkle

S

=

ANALYTICAL

EUROPEAN

TABLE 5.3

Boness

Volatility

Striking Price

10%

25%

50%

A

0.I

0.5

/.0

0./

0.5

I.0

90 100 I 10

56.45 33.04

6f.63 45.43 15.59

68.76

l90.08 124.83

0.73

!88.08 \52.32 54.75

188.16 \69.29 109.17

90 \00 \ \0

53.l6

53.98 40.33 26.45

60.69 49.86

38.80

176.4! 110.10 29.41

154.84 122.68 83.76

153.ls 132.43 107.13

5 i,02 42.58 34.71

59.44 52.73 46.42

f46.00 105.10 61.78

t 28.53

I 28.32 117.89 106.87

0.19 30.70 7.9| 46.79 31.78 17.98

90 100 I 10

56.78 34.70

I I I.91 94.39

91

100

=

Price

0./

0.3

0.5

0.7

09

/.I

10%

90 100 I 10

10.90 l.8l 0.00

l2.67 3.95 0.32

14.42 5.85 l.14

16.l3 7.67 2.25

17.8! 9.44

3.53

19.44 I l.l7 4.92

25%

90 \00 I 10

Il.f4 3.66 0.59

13.75 6.98 2,90

!6.09 9.58 5.12

18.23 i 1.87 7.22

20.22 13.97 9.20

22.09 15.95 I l.IO

50%

90 100 I10

12.92 6.78 3.06

1774 12.27 8.20

21.44 16.26 12.16

24.57 19.61 15.53

27.35 22.56 18.53

29.88 25.23 2\.27

volatility

2.0

=

Call Values: S

MODELS

-

100

0.5

=

PRICING

Time to Moturity

Time to Maturity A

OPTION

growth of the call's value, he proposed the following formula: c(S, T)

=

"

Se

N(di) - Ke

av

N(d2)

(6)

T

(7)

-

with d

In

=

a

a

K

2)

and d2 di a d. Note that all the proposed formulas show one or more arbitrary parameters, depending on the investors' preferences towards risk or the rate of return on the stock. Table 5.4 shows Samuelson call values for S 100. =

-

=

5.1.3

Boness

Formula

(1964) presented the following option pricing formula accounting of money through the discounting of the terminal stock price using rate of return to the stock: Boness value

c(S, T)

=

SN(d

)-

Ke

"'

for the time the expected

TABLE 5.4

and d2

5.1.4

=

di

--as/Ï.

Samuelson

In

a

(p+ ja

(5)

T

Table 5.3 shows the Boness call values for S

100 Time to Maturity

Volatility =

=

(4)

N(d2)

with di

Samuelson Call Values: S

=

from the stock. Samuelson (1965) allowed the option to have a different of the and Defining p as the average rate of growth of the share price a as average rate level of risk

2.5%

a

Price

0. |

0.5

/.0

90 I00 I 10

9.48 0.88 0.00

7.57 0.29 0.00

0.00 0.00

25%

90 \00 110

9.82 2.84 0.37

10.46 5.I I 2.07

10.38 5.63 2.59

50%

90 100 !!0

11.83 6.00 2.60

16.77 12.20 8.72

19.68 15.68 12.44

%

100.

Formula

=

=

5%

striking 0. |

0.5

/.0

2. I4 0.00

2.30 0.00

10.26 3.\ \ 0.44

]2.33 6.60 3.09

13.97 8.74 5.11

12.19 6.26 2.75

!8.32 13.55 9.87

22.65 18.41 14.93

9.95 I I9 .

o.oo

OPTIONS,

92

FUTURES

AND EXOTIC

DERIVATIVES

In another paper, Samuelson and Merton (1969)proposed a theory of option valuation by treating the option price as a function of the stock price. They advanced the theory by realizing that the discount rate must be determined in part by the requirement that hold all the amounts of stocks and the option. Their final formula depends on investors investor. the utility function assumed for a Black and Scholes (1973) used a formula derived by Thorp and Kassouf (1967) for This formula determines the ratio of shares of stocks needed to pricing warrants. position. This position is constructed by buying an asset and selling hedged construct a situation, the another. However, Thorp and Kassouf did not realize that in a no-arbitrage expected return on a hedged position must be the return on the riskless asset. This breakthrough was due to Black and Scholes as we will see in the derivation of their 'typical'

ANALYTICAL

5.2.1

BLACK AND SCHOLES

of equity in a hedged position, 1 options, is S

(B-S) MODEL

•

•

•

containing

a stock purchase

and a sale of

c(S, t)

BS Over a short interval At, the change in this position is Ac(S, t)

'ideal

•

93

ßc(S, t)

The Model

Under the following assumptions, the value of the option will depend only on the price of the underlying asset S, time t, and other variables assumed to be constant. conditions' as expressed by B-S, are the following: These assumptions, or •

6

MODELS

'market

[8c(S, t)/8S]

5.2

PRICING

OPTION

risk', B-S expressed the formed, where portfolio weights are chosen to eliminate expected return on the option in terms of the option price function and its partial derivatives. In fact, following B-S, it is possible to create a hedged position consisting of a sale of [8c(S, t)/OS] ' options against one share of stock long. If the stock price changes by a small amount AS, the option changes by an amount [8c(S, t)/BS]AS. Hence, the change in value in the long position (the stock) is approximately offset by the change in [8c(S, t)/8S] I options. This hedge can be maintained continuously so that the return on the hedged position becomes completely independent of the changes in the underlying asset value, i.e. the return on the hedged position becomes certain. The value

model.

EUROPEAN

The option is European. The interest rate over the lifetime of the option is known. The underlying asset follows a random walk with a variance rate proportional to the square root of the stock price. It pays no dividends or other distributions. There are no transaction costs, and short selling is allowed, i.e. an investor can sell a security that he does not own. and the standard form of the capital market model Trading takes place continuously holds at each instant.

(10)

Oc(S, t)

L BS where Ac(S,

t), is given by c(S + AS, t At) t). - c(S, Using stochastic calculus for Ac(S, t) gives

;

l 82c(S, t) Sc(S, t) a2S2At+ AS+- OS ßS2 2

Ac(S,t)=

Oc(S, t)

St

At

The change in the value of equity in the hedged position is found by substituting

from equation (11) into

equation

00 Ac(S, t)

(10):

'observable'

The main attractiveness of the B-S model is that their formula is a function of variables and that the model can be extended to the pricing of any type of option. The assumption concerning the underlying asset states that its dynamics are governed equation: the following by

dS S

The Model

return

ß2c(S, t) OS2

on the common

stock

and o

is

the

Since the return to the equity in the hedged position is certain, it must be equal to rAt where r stands for the short-term interest rate. Hence, the hange in the equity must be equ \ to the value of the equity times FA1, or 82c(S, t)

'short'

(12)

BS

for Call Options

namely the Assuming that the option price is a ftmetion of a single source of uncertainty, time intervals, At, a hedged stock price and time to maturity, c(S, t), and that over portfolio consisting of the option, the underlying asset and a riskless security can be

Oc(S, t)

Oc(S, t)

(8)

=adt+odW

a is the instantaneous expected instantaneous standard deviation of return.

where

jo2S

-

-

Sc(S, t) =

Oc(S, t) L

BS

J

S

- --Oc(S, t) L DS

r Al

(13)

FUTURES

OPTIONS,

AND EXOTIC

and d2

terms yield the B-S partial differential equa-

Dropping the time factor and rearranging tion:

EUROPEAN

ANALYTICAL

DERIVATIVES

=

di

-

avi,

OPTION

ß2c(S, t)

-rc(S,t)+

OS2

-

(14) A

This partial differential equation must be solved under the boundary condition expressing the call value at maturity date: c(S, t*) K is the option's strike price. For the European put, the equation

=

K]

[0, Se -

max

(15)

where

must be solved under the following maturity date

condition: /*)

p(S,

=

[0, K

max

- S,4]

(16) (15), B-S make the

To solve this differential equation, under the boundary condition following substitution: c(S, t)

=

2

e"'

y

r

/S'

(r - go2)ln

-

(r jo'Xt -

-

2(t

th

;

-

-

t*) -

i

the differential equation

(18)

y(u, 0)

The solution

Churchill

to equation

l

the solution

(18) is

K

=

Substituting from withT=t*-t:

Paray Reasonsay

The put-call parity relationship that is distribution-free. is a very important relationship It can be derived as follows. Consider a portfolio A which comprises a call option with a maturity date t* and a discount bond that pays K dollars at the option's maturity date. Consider also a portfolio

max

u<0

inax

else

to the heat transfer

qvis)Ü2

(u +

exp

-

a

r

(20) into (17)gives c(S, T)

=

equation

given in

- Ke

'

N(d2)

dq

(20)

(21)

with

di

i =

In

/SN

+

(r + («2)7

[0, K

-

S,-] + S,-

c, - p,

the following solution for the European call pnce SN(di)

=

max [K, S,.]

(24)

(22)

=

max [K, S,.]

(25)

Since both portfolios have the same value at maturity, they must have the same initial value at time t, otherwise arbitrage would be profitable. Therefore, the following put-call relationship must hold:

2

I exp

K] + K

[0, S,- -

The value of portfolio B at maturity is

(1963): y(u, s)

see Chapter 3). It is important to note that the option value is independent of its underlying asset expected return. This may sound rather strange. One intuitive way to account for it is to say the expected return on the stock is already embedded into the stock price itself. It is also worth noting that the option price rises when the asset price, the time to maturity, the interest rate and the variance increase. The partial derivative [Sc(S, t)/OS] which is equal to N(di) gives the ratio of the underlying asset to options in the hedged position. It refers also to what is known as the option's delta. Since S/c(S, t)[Sc(S, t)/OS) is always greater than 1, the option is more volatile than its underlying asset. The value of the put option can be obtained from that of the call option using the put call parity relationship.

=

K exp

(23)

B, with a put option and one share. The value of portfolio A at maturity is

equation is the famous heat transfer equation of physics The boundary condition is rewritten as

This differential

if

dx

2

-oo

(fora detailed resolution,

rue Fut-con

becomes

0

&

--

') 2

(17) Using this substitution,

s

X2

exp

-

95

normal density function given by

d

1 =

.4

Sc(S, t) Oc(S, t) +rS-=0 BS Ot

MODELS

where N(.) is the cumulative N(d)

¼o2S2

PRICING

=

S, - Ke

'"

"

(26)

Ifthis relationship

does not hold, then arbitrage would be profitable. In fact, suppose, for example, that ""

p, > S, - Ke (27) At time t, the investor can construct a portfolio by buying the put and the underlying asset and selling the calL This strategy yields a result equal to c, p, - S,. If this amount is positive, it can be invested at the riskless rate until the maturity date t*, otherwise it can be borrowed at the same rate for the same period. At the option maturity date, the option will be in-the-money or out-of-the-money according to the position of the underlying asset S,, with respect to the strike price K. c,

-

-

OPTIÒNS,

96

FUTURES

AND EXOTIC

DERIVATIVES

ANALYTICAL

If S,. > K, the call is worth its intrinsic value. Since the investor sold the call, he is assigned on that call. He will receive the strike price, deliver the stock and closes his position in the cash account. The put is worthless. Hence, the position is worth "[c,

K + c"

-

p, - S,]>0

"

K + e'

[c, -

p, - S,] > 0

97

di

=

2.8017 implying N(di)

=

0.997

d2

=

2.7268 implying N(d2)

=

0.996

the formula for the put price is written as p

(29)

N(-d2)

14.629(1 - 0.996) 18(1 - 0.997) -

=

14.629(0.004)

=

- 18(0.003)

=

0.0045

-

(30)

.

We give here five examples to illustrate determination of call and put prices.

MODELS

Example 3

+ Ke-,r

-SN(-di)

=

PRICING

Using the same data, what is the put price? Since the values of di, d2, N(di) and N(d2) are respectively

(28)

In both cases, the investor makes a profit without initial cash outlay. This is a riskless arbitrage which must not exist in efficient markets. Therefore, the above put-call parity relationship must hold. Using this relationship, the European put option value is given in the B-S model by

p(S, T)

OPTION

Example 2

lf S,, < K, the put is worth its intrinsic value. Since the investor is long the put, he exercises his option. He receives upon exercise the strike price, dehvers the stock and closes his position in the cash account. The call is worthless. Hence, the position is worth '

EUROPEAN

the application

of the B-S model

for the

Using the call price,

what

is the put price when the put-call

parity relationship

applies?

The put price is given by p=c-S+Ke-,r 3.3659 - 18 + 15 e-0.l(0.25)

=

=

Example

I

When the underlying asset S 18, the strike price K 15, the short-term interest rate r 10%, the maturity date T 15%, 0.25 and the volatility a the call price is calculated as follows. First, we compute the discounted value of the strike price: =

=

=

=

*

Ke Second, the values

di

=

15 e

2"

0.0045

This price is exactly the same as that given by the application in the B-S model.

of tle put formula

=

14.6296

=

Example

4

of di and d2 are calculated:

ln(18/l5)+[0.1

(0.l5)2(0.5)]0.25 0.21013

=

0.075

0.15

2.8017

the underlying asset S=18, the strike price When the volatility a=0.15, 0.1 and the time to maturity Tis 6 months, we have 15, the interest rate r K =

=

di

d2

=

0.15B

d -

=

2.7267

C

l8N(2.8017)

- 15 e

d2

C

18(0.997)

=

2.1373

implymg

=

0.9876

=

0.9844

.

N(d2)

.

The "

2"

N(2.7267)

Using the approximation of the cumulative normal 2.8017 and 2.7267, the call price is 3.3659, or =

2.2433 implying N(di) .

Replacing these values in the formula gives =

=

- 14.6296(0.996)

distribution

c- p

call =

price S - Ke

is '

3.7461

and

the

3.3659

put

is

0.0084.

:

c

at the points

-

p

S - Ke =

price

This result shows that the put-call

=

*

3.73 =

3.73

parity relationship

is satisfied.

Also,

since

98

FUTURES

OPTIONS,

Example

AND EXOTIC

DERIVATIVES

•

sivis

PRICING

MODELS

99

5

In this example (Table 5.5), call and put prices are simulated using the B-S formula. Figures 5.1 and 5.2 show the same results graphically for a volatility a of 25% and for a range of striking prices K from 80 to 120.

20 18 16

TABLE 5.5 10% r

and

Black

Scholes

European

Option

Values:

S

=

100,

-

-

-

14

-

=

12

Time to Maturity

i

O.

10

-

-

.0

Call Volatility

8

Put

6

Striking Price

0. /

0.5

90 \00 i10

(0.90 L8 1 0.00

14.42 5.85 \.\4

90 100 110

11.14 3.66 0.59

90 100 1 IO

12.92 6.78 3.06

-

-

0.8 0.7 0.6 0.5 0,4 T: Time to matut®y 0.20.3

.0

.0

0. /

0.5

f

4

!8.63 10.31 4.22

0.00 0.82 8.9|

0.03 0.97 5.78

0.07 0.79 3.75

2-

16.09 9.58 5.\2

21.16 14.97 10.16

0.24 2.66 9.50

1.70 4.70 9.76

2.60 5.46 9.69

K: Striking price

2\.44 16.26 12.16

28.64

2.02 5.79 I l.97

7.05 i I.39 16.79

10.08 14.41

FIGURE 5.2 Black and Scholes 25% 10%, « r

!

_

'

(0%

25%

50%

23.93 19.93

·

0

_

60 84 6 100104108112 ¾6120

=

Put Values

0.1

Function

of (K, T): S

=

100,

=

19.46

5.2.2

Applications

Valuation and the Role of Eguity Options 25 _ 20 15

io

_

-

1.0 0.9 0.70.8

-

5

-

O

-

80 84 06 92

0.6 0.5 0,4 T: Timeto maturity 0.20.3 96 100 104108

K: Striking price

FlGURE 5. I Black and 25% 10%, « r =

=

Scholes

0.1 112 116 120

Call Values

Function

of(dt. T): S

=

100,

Broadly speaking, there are four groups of equity options: traded options, over-thecounter options, equity warrants and covered warrants. Traded options are standardized contracts which are listed on options exchanges. These options are not protected against dividend and their strike prices and maturity dates are set by the exchange. Over-the-counter options are tailor-made to the investor's needs and are usually written by investment banks. Equity warrants are long-term options and are often traded in securities markets rather than in option markets. When these options are exercised, new shares are issued by the company. Covered warrants are over-the-counter long-term options issued by securities houses. All these equity options can be valued using the B-S model. However, the following specificities of these instruments require some adjustments to the B-S model. First, these options are frequently traded on an asset which distributes dividends and they are quite often of the American type, namely they can be exercised before maturity. Second, the assumed diffusion process may not represent reality since equity prices may jump downward or upward in response to bad or good news. In addition, the issue of new shares upon exercise triggers the question of dilution. Third, it is more dif6cult to justify a constant volatility for the underlying asset when

OPTIONS,

100

FUTURES

AND EXOTIC

DERIVATIYES

ANALYTICAL

the option maturity is long. The same argument applies for the riskless interest rate. In this chapter, we restrict our analysis to the assumptions of the B-S model, which will be relaxed afterwards when studying the extensions and generalizations of the model. Many strategies can be implemented with equity derivatives. These strategies are obviously not specific to equities. They also apply to options on other types of underlying Buying or assets. Equity options can be used in several ways in portfolio management. selling options involves the payment or the receipt of the option premium at the initial time when the transaction is done. Smce the option pay-off is asymmetric, this gives rise to an asymmetric distribution of returns. Hence, options can be used in portfolio manage ment to structure the distribution of expected returns The best known strategies in portfolio management involve combinations of options. They include vertical spreads, calendar spreads, diagonal spreads, ratio spreads, volatility spreads and synthetic contracts (see Chapter l for more details) The main difference between futures contracts and option contracts is that the investor pays a premium for options and nothing to establish a futures position Calls and puts are bought or sold in anticipation of future cash flows, for defensive purposes or speculative reasons. The investor must choose the appropriate options to be bought or sold. Therefore, the question of the management of an option position is as important as the question of option valuation and strategies. We deal with the question of monitoring options with respect to their sensitivities in the next and managing chapter Valuation

and the Role of Index Options

Stock index options and futures markets have experienced remarkable growth rates. Stock index options are of the European or the American type and often involve cash settlement procedure upon exercise. Stock index options are traded on the major indices around the world. Options on the spot index are cash-settled and there is no physical delivery of the underlying index, i.e. of a weighted average of prices of the stocks that constitutes the

EUROPEAN

OPTION

PRICitiG

MODELS

10 i

with di

.

l a

and d2

=

di

--«

.

Se-dT 2)

ln

=

(r + (o

- K

.

index

The formula for a European

¡>(S, T)

=

Ke

*N(-d2)

(32)

put is

- Se

Table 5.6 lists a selection of B-S index option call Figure 5.3 shows call results graphically for a volatility

T

"

N(-di)

and

(33)

put values of 20%.

for S

=

100,

while

Arbitrogebetween Index Options and Futures It is convenient to note that the same strategies for stock options can also be used in portfolio management with index options. Also, these options can be used in asset allocation and portfolio insurance. Since these financial instruments are based on the same underlying index, their prices must be should instantaneously interrelated. If this is not the case, the relative mispricing disappear given the variety of cross-market strategies. These strategies include arbitrage between index options and index futures, and between index futures and the stocks comprising the index. Many researchers have studied these arbitrages which imply that significant deviations from prices dictated by the relevant market interrelationships should disappear. It is of the no-arbitrage often found that there are some violations bounds. However, when taking into account transaction costs, the futures price lies between the no-arbitrage bounds. Even if all tests and published studies are in favor of market efficiency and integrated markets, it is often reported that relative mispricing does exist between index options and futures contracts. For more details, see, for example, Evnine and Rudd (1985),Brennan and Schwartz (1990)and Lee and Nayar (1993)among others.

index. There are several weighting schemes. The most commonly used is the market capitalization scheme where each equity price is weighted by the market capitalization of the firm, i.e. the number of shares times the share price. Two alternative methods are sometimes used: equal weighting and price weighting. The than do last two methods assign greater relative weight to smaller company constituents capitalization-weighted indices Index options are also sold in the OTC market as over-the-counter warrants. In this case, they refer to long-term options on the spot index. Because they are traded on OTC markets, they are subject to credit risk.

TABLE 5.6

Adjustment of the 8-S Model for Index Options When the option underlying index is constructed to pay continuous dividends, the index price is adjusted by the discounted value of the continuous dividend yield. The appropriately adjusted B-S version when the continuous dividend yield is d corresponds to the following formula for a European index call:

c(S, T)

=

Se '"N(di)

- Ke

N(d2)

(31)

Index Option

Values:

S

=

100, r

=

10%, d

=

4%

Time to Moturity Put

call

.

.

Black and Scholes

Votatility

Striking Price

0./

0.5

l.0

\00 110

10.50 1.56 0.00

12.47 4.32 0.60

25%

90 100 I 10

10.76 3.43 0.53

50%

90 100 i lo

12.60 6.56 2.93

10%

90

0.I

0.5

I.0

14.78 6.96 2.1\

0.00 0.97 9.31

0.06 l.43 7.22

0.l4 1.37 5.56

14.44 8.30 4.26

17.92 12.23 7.98

0.27 2.84 9.83

2.03 5.4l 10.88

3.27 6.63 I l.43

l9.99 \5.03 I I.\3

25.72 21.29 17.57

2.1 I 5.96 12.24

7.58 I2.\3 17.74

I LO7 15.69 21.02

OPTIONS,

102

FUTURES

ANALYTICAL

AND EXOTICDERIYATIVES

EUROPEAN

OPTION

PRICING

c(B, T)

=

MODELS

BN(di) - Ke

103 "

N(d2)

(34)

with 25-

di

1 =

o

20

and d2

10

a 0

In

-

=

di - o#,

B -K

+

(r 4

la2)

T

(35)

where:

B is the bond's price, K is the option's strike price, Tis the option's time to maturity, o is the instantaneous standard deviation of the bond price, r is the spot rate on a risk-free investment with a maturity date T

-

_

Using the put-call

d Dividend

parity relationship,

the put's value is given by

-

84

es

p(B, T)

92 96 1001041081121161200%

=

-

BN(-di)

+ Ke

N(-d2)

(36)

However, other European models, which are extensions or generalizations of the B-S model, such as Merton's (1973a)model are more appropriate for the pricing of these

K: Striking price

options. FIGURE 5.3 Black and Scholes 20% 10%, « r

Call Values

Index

Fune iam

of

(K, d): S

=

100,

=

=

On the other hand, despite the controversy about index arbitrage and program trading, these financial instruments are beneficial to stock portfolio managers and institutional investors. Before the emergence of these contracts, market participants could not hedge and control the market risk of their portfolios. Even if there is some evidence that trading in index futures increases cash-market volatility, arbitrage activities via program trading may cause prices to adjust more rapidly to new information. This helps to keep the movements of index futures prices and the value stock index more synchronous. The deviations of futures prices from their and substitutability between spot including various considerations imperfect results from futures markets, the speed with which mformation is incorporated in prices in the different markets, and market imperfections including transaction costs and regulatory

Interest rate options are often used in the management of interest rate risk in the same way as equity options. A direct implication is that option strategies for equity options apply directly to interest rate options. The most common and specific strategies based on short-term interest rate options are caps and floors. These strategies place either a cap on the future level of interest rates on a floating instrument or a floor on the interest rate receivable on deposits. A cap is an option strategy which protects from a rise in interest rates and allows a profit when interest rates are falling. A floor is an option strategy which protects from a decrease in interest rates at the time when the deposit rate is reset. When an investor buys a cap (floor) and sells the floor (cap), this strategy is known as the collar,

'fair'

.

constraints,

Valuation

.

.

among other thmgs. and the Role of Short-Term

Options

on Long-Term

rates. short-term

The B-S model is sometimes used to price counon bonds. In this context, the call's value is given by

European

.

The price of any financial asset is given by the present value of its expected cash ows. The first step m determimng the bond's price is to determine its cash flows, i.e. the periodic coupon interest payments until the rnaturity date and the par value at maturity.

Bonds

markets. Short-term options on long-term bonds are often traded on over-the-counter These options may be of the European or the American type. There is a traded option on the Chicago Board Options Exchange which is based upon the yield to maturity on a portfolio of bonds. The yield to maturity is driven by changes in the term structure of mterest

Valuation and the Role of Bond Options

options on zero-

Since the bond price is given by the present value of the cash flows, its price is given by adding all the discounted future payments at the appropriate interest rates. Some bonds do not make any periodic coupon payments and the interest due to the bond holder is given by the difference between the maturity value and the purchase price. This class of bonds is referred to as zero-coupon bonds. It should be emphasized that there are several types of bonds: bonds with call provisions,

putable bonds, convertible

bonds, bonds with warrants attached, exchangeable bonds, etc. A bond with a call provision gives the right to the issuer to call the issue before the

104

OPTIONS,

specified redemption

FUTURES

AND EXOTIC

DERIVATIVES

ANALYTICAL

date. The call price is different from par and is specified at the bond

•

EUROPEAN

Nine-month

PRICING

OPTION

interest

MODELS

105

8.5%.

rate:

issue.

A bond with a put provision gives its holder the right to put the bond back to the issuer at a fixed pnce. It is a putable bond A convertible bond entitles its holder to convert the bond into a certain number of units of the equity of the issuing firm or into other bonds. This number is often called the rate of conversion which is specified when the bond is issued. A bond with an attached warrant is simply a package comprising the bond and a It allows the holder to purchase the equity of the issuing firm. Most of these warrant. bonds are eurobonds issued in international capital markets. An exchangeable bond is similar to a convertible bond, with the exception that it gives its holder the right to exchange the bonds for the equity of another company. The call or the put provision in these bonds can be valued using the B-S model. However, the model is not appropriate if there are many call or put dates and if the embedded options in the bonds are of the American type. We now give an application of the model to options on zero-coupon bonds and options on coupon-paying bonds under the B-S assumptions. Zero-Coupon Bonds When there are no coupon payments, follows for bond options: =

c

P=

BN(di) - Ke

-BN(-di)

Ke

"

The option has a strike price equal to 100 FF and its maturity The present value of coupon payments is 9.59 FF, or *

5e

+ 5e

""

""

4.9 + 4.69

=

=

date is inene yeat

9.59 FF

Applying the B-S formula gives

96 - 9.59 86.41 FF [In (86.41/100) + 0.1 + 0.0032] 0.080 B

di

=

da c

=

=

=

di - 0.08

=

86.4l N(-0.5358)

=

l.25 FF

× l

=

+ 100e

=

-0.5358

-0.6158 "

N(-0.6158)

'incoherence'

with this model since it assumes It is convenient to note the and time the stochastic rates interest bond price. constant at same a

the B-S model is applied as

N(d2) "N(-d2)

(37)

5.3

90

5.3.1

BLACK MODEL The Model

with

di

=

-

o and d2

=

di - a

v'†

ln

K

+

(r +

lo 2)

T

(39)

Using some assumptions similar to those used in deriving the original B-S option options and formula, Black (1976)presented a model for the pricing of commodity forward contracts.

VT. The Model

Coupon-Poying Bonds If the bond is a coupon-paying bond, then the present value of all coupons due during the option's life must be subtracted from the bond's price.

Example

for Forward,

Futures and Option

Contracts

In this model, the spot price S(t) of an asset or a commodity is the price at which an investor can buy or sell it for an immediate delivery at current time, t. This price may rise steadily, fall or fluctuate randomly. The futures price F(t, t*) of a commodity can be defined as the price at which an investor agrees to buy or sell at a given time in the future, t*, without putting up any t*, the futures price is equal to the spot price. money immediately. When t A forward contract is a contract to buy or sell at a price that stays fixed until the maturity date, whereas the futures contract is settled every day and rewritten at the new futures price. Following Black, let v be the value of the forward contract, u the value of the futures contract and c the value of an option contract. Each of these contracts is a function of the futures price F(t, t*) as wel as other variables. So, we can write at instant i the values of =

Consider a European call for which the following characteristics: • • • • • •

the underlying

Bond's price: 96 FF One-year interest rate: 10% Time to maturity: 10 years Volatility of the bond's price: 8% Coupon payments: 5 FF in three months Three-month interest rate: 8%

asset is a coupon bon&with

and nine months

these contracts respectively as v(F, t), u(F, t) and c(F, t). The value of the forward contract depends also on the price of the underlying at time /* and can be written v(F, t, K, t*).

asset, K,

106

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

ANALYTICAL

It is important to distinguish between the price and the value of the contract. The futures price is the price at which a forward contract presents a zero current value. It is written

0

MODELS

107

(42)and (44),the

v(F, t*, K, t*)

F - K

=

value of a commodity

option

=

F - K*

if F else

f

0

>

K'

(42)

-

K*N(d2)]

(46)

di

l =

F -K*

In

-

a

e

4

+

(jÛ2

(47) .

Ö.

It is convenient to note that the commodity option's value is the value the of an option on a security paying a continuous dividend. The rate of same as s, is substituted in the distribution is equal to the stock price times the interest rate. If Fe original formula derived by B-S, the result is exactly the above formula. In the same context, the formula for the put option is given by and d2

d, - o

=

.

.

p(F,

T)

+ K* N(-d2

e "'[-FN(-di)

=

(48)

The value of the put option can be obtained directly from the put call parity.

The Put-Call

Parity Relationship

The put-call parity relationship

for futures options is

contracts and comrnodity options, Black assumes that:

In order to value commodity

p - c

The futures price changes are distributed log-normally with a constant variance o2 All the parameters of the Capital Asset Pricing Model are constant through time. There are no transaction costs and no taxes.

Under these assumptions, it is possible to create a riskless hedge by taking a long position in the option and a short position in the futures contract. Let [8c(F, t) BF] be the weight affected to the short position in the futures contract, which is the derivative of c(F, t) with respect to E The change in the hedged position may be written as Oc(F, t) - [8c(F,

(43)

t)/8F]OF

Using the fact that the return to a hedged portfolio must be equal to the risk-free rate and expanding ßc(F, t) gives the following partial differential equation:

at

t)

- rc(F,

t) - 2la

F

82c(F,

t)

- F)

(49)

nows

interest

(44)

¿)µ2

e "(K

=

This relationship can be explained as follows. Consider a portfolio where the investor is long a future contract, long a put on the future contract and short a call with the same time to maturity and strike price. Note that the combination of the call.and the put is equivalent to a short synthetic future. At expiration, the pay-off is given by the difference between the option's strike price and the current futures price. Hence, the current value of that portfolio must be equal to the present value of this difference, Since these options are European, they have the same cash as options on the spot asset. This is because at the maturity date the futures price is equal to the spot price. We now give some examples for the calculation of option prices using Black's formula.

Example Sc(F,

•

When F

I 80, K

=

100, T

=

0.25,

=

«

0.2 and r

=

0.08, the call price, c, is

=

zero.

or

• •

lo2F2

[FN(d,)

e

with

(41)

At maturity, the value of a commodity option is given by the maximum of zero and the difference between the spot price and the contract price. Since at that date, the futures price equals the spot price, it follows that c(F, t*)

'

=

(40)

-

•

and using equations - t

c(F, t) =

This equation says that the forward contract's value is zero when the contract is initiated and the contract price, K, is always equal to the current futures price, F(t, t*) The main difference between a futures contract and a forward contract is that a futures contract may be assimilated to a series of forward contracts. This is because the futures contract is rewritten every day with a new contract price equal to the corresponding futures price. Hence when F rises, i.e. F > K, the forward contract has a positive value, and when F falls, F < K, the forward contract has a negative value. When the transaction takes place, the futures price equals the spot price and the value of the forward contract equals the spot price minus the contract price or the spot price:

•

t*

=

PRICING

OPTION

as v(F, t, F, t*)

•

Denoting T

EUROPEAN

' -

rc(F,

t)

' =

0

(45)

When F 100, K When 120, F 19.7754. c =

=

=

=

0.08, then c 100, T 0.2 and r 0.25, « 3,9087. 0.25, 100, T 0.2 and 0.08, then K r o =

=

=

=

=

=

=

=

108

FUTURES

OPTIONS,

Example

AND EXOTIC

DERIVATIVES

2

Table 5.7 simulates European call and put futures prices using Black's model.

TABLE 5.7 10% r

European

Futures

Values

by

Black's

Model:

S

=

100

'

=

Time to MoturitY Coll Volatility

Striking Price

Put

OPTION

PRICING

109

MODELS

the valuation of these options. However, when these options are of the European type, Black's model is often used in pricing them. The various strategies applied with options on individual assets can also be used for options on index futures. Index futures and options on index futures are often used in asset allocation and refers to the structuring of a multi-asset portportfolio insurance. Asset allocation with weightmg scheme of asset classes. Strategic and the the respect type to folio asset allocation is the construction of a portfolio such that long-run objectives are attained when the different classes of assets are transacted at their long-run equilibrium values. allocations

known also as market timing, involves short-term .

0.I

0.5

I.0

0./

0.5

l.0

90 100 I l0

9.90 I.25 0.00

9.70 2.68 0.29

9.69 3.6! 0.86

0.00 1.25 9.90

0.19 2.68 9.80

0.64 3.6 I 9.91

25%

90 I00 110

10.22 3. I2 0.46

12.22 6.70 3.27

\ 3.82 9.00 5.60

0.32 3. I2 10.36

2.70 6.70 12.79

4.77 9.00 14.65

50%

90 100 I \0

12.14 6.24 2.75

17.99 13.35 9.76

21.86 17.86 14.56

2.24 6.24 12.65

8.47 |3.35 l9.27

I2.8I 17.86 23.61

Options

EUROPEAN

Tactical asset allocation,

10%

5.3.2

ANALYTICAL

Applications on Equity index Futures

Options on index futures require upon exercise the exchange of a long position in the future contract for a call and a short position in the future contract for the put. Hence, a call is exercised into a long position in the future contract and a put is exercised into a short position in the same contract. The underlying futures contract does not requite a physical delivery but rather is settled m cash. The amount received corresponds to the difference between the current and the future level of the underlying index. In this context, the futures contract is regarded as an agreement to pay or receive a cash payment based upon the difference between the current and the future values of a specified index.

Options on index futures are often treated as options on an asset paying a continuous stream of dividends, regardless of whether the underlying spot index pays a continuous or a discrete dividend. Since the assumptions used by Black are similar to those in B-S, some of them are also questionable, such as the constant volatility and the certainty of interest rates. In fact, the non-stationarity of volatility causes some problems in the pricing of options on index futures. As we will see later, some extensions on Black's model are more appropriate for

toward rismg markets and away from fallmg markets. Portfolio msurance refers to a group of techmques that msure a portfolio against fallmg in value below a certain specified level, the floor level. This level does not eliminate the potential profits from a rise in the asset value.

Options

on Currency

Forwards and Options

on Currency

Futures

Options on currency forwards are traded in the OTC market. This market is regarded as the major market for currency options. The growth of the OTC market is due to its flexibility, since many banks and financial institutions offer options with tailor-made characteristics in order to match their chents' needs. Options on currency futures have been traded since 1982. These options are standardized contracts and can be inflexible. They are priced off the underlying futures contract. When exercised, the call buyer receives a long position in one futures contract marked to market at the current price. In the same way, when exercised, the put holder receives a short position in one futures contract. These options must have the same value as European options on the spot currency since they are not exercised before the maturity date, at which the futures price is equal to the spot price. The similarity between forward and futures prices suggests the use of Black's model for the valuation of these options. Currency futures and currency forwards are often used to hedge currency risks. Options strategies of on currency futures are applied m currency risk management. The basic volatility spreads, and calendar vertical, the calls and diagonal, selling and buymg puts, used m markets. also be options They also the applied futures can currency are m portfolio msurance. of currency futures and options are used to manage currency Several other applications .

.

.

.

.

.

.

risks. Examples mclude basket options, average rate options, cylinder options and many other exotic options. .

Futures Margined

.

American

Options

and

Options

on Bond Futures

Traded

on the LIFFE

There is an important difference between options traded on the LIFFE and other markets. For these options, the premium is not paid up-front and the option contract is margined like the futures contract.

'

I10

FUTURES

OPTIONS,

AND EXOTIC

Even if these options are of the American type, there is no incentive for early exercise. of the margined Hence, the European Black's model can be used for the valuation American options traded on the LIFFE, Black's model for the valuation of a European option on a forward or a futures contract when the underlying asset is an interest rate is given by c

-

KN(d2)]

' =

e

[-FN(-di)

EUROPEAN

OPTION

MODELS

PRICING

I11

important volume of transactions implies the use of an option pricing model. A simple and interesting analytic model was provided by Garman and Kohlhagen (1983).

The Model

(50)

and p

4

5.4.1

e "[FN(di)

=

ANALYTICAL

DERIVATIVES

+ KN(-d2)]

(51)

+(jo')T

(52)

Foreign currency options are priced along the lines of B-S (1973)and Merton (1973). Specifically, Garman and Kohlhagen (1983)and Grabbe (1983)presented models for riskless hedge portfolio can be currency options which are based on the assumption that a and the option. bonds domestic in formed by investing foreign bonds,

with

di and d2

=

di - od,

l =-

in

F -

The Currency

Using the same assumptions as in the B-S (1973)model, Garman and Kohlhagen presented the following formula for a European currency call:

where:

N(d2)

(1983) (53)

with

di

Note that the interest rate does not appear in this formula except for the discounting. This implies that the problem of the volatility of interest rates is solved within the context of this model. There is a simple adjustment to the model which needs to be multiplied by e" to account for the payment at the future date Options on bonds traded on the LIFFE are also margined in the same way as short¯ term interest rate options. Since there is no incentive in exercising these American options before the maturity date, Black's model also applies with the same modifica tion

GARMAN AND KOHLHAGEN, MODELS

N(di) - Ke

Se

=

c

F is the futures price, K is the option's strike price, r is the short-term interest rate, a is the volatility of futures prices, Tis the option's time to maturity.

5.4

Call Formula

AND GRABBE

and d2

=

da - a

,

ln

=

s' -

+

r*

(r -

+

(a2)T

(54)

where:

Sis the spot rate, K is the strike price, r is the domestic interest rate, r* is the foreign interest rate, o is the volatility of spot rates, Tis the option's time to maturity Adopting an approach similar to that in Black formula for the currencv call: c

=

B(T)FN(di)

(1976),Grabbe (1983) gives the foËowing - B(T)KN(d2)

(55)

with a formula for the valuation of foreign options. These traded options currency are on the foreign exchange market, which is fundamentally an interbank market where transactions are conducted over the telecommunications The foreign exchange market, called also the FX market, system. internationally 24 hours a day; the major participants are commercial banks operates around the world and treasury departments of large corporations. risks, the various participants search for hedging exchange As in other markets, arbitrage speculation and the implementation of strategies. options satisfy Foreign currency some of the needs of these participants and the

Garman and Kohlhagen

(1983) provided

d, and d2

=

di - o vi,

1 =

E (F' Iln

K,

+

(a T

where:

F is the forward price of the foreign currency, B(T) is the price of a domestic currency discount bond, a is the volatility of the foreign forward exchange rate.

(56)

f 12

OPTIONS,

The Currency

FUTURES

AND EXOTIC

DERIVATIVES

ANALYTICAL

Put Formula

Table 5.8 shows

The formula for a European currency put is p

EUROPEAN

"

Kc

=

N(-d2)

- Se

N(-di)

Ii3

MODELS

values

Garman call and put option

various

Garman

TABLE 5.8 ,·r

PRfCING

OPTION

Values:

Option

Currency

(57)

S

=

for S

100, r

=

=

100.

10%, r,

=

4%

Timeto Maturity

Note that the main difference between these formulac and those of B-S for the pricing of equity options is that the foreign risk-free rate is used in the adjustment of the spot rateThe spot rate is adjusted by the known i.e. the foreign interest earnings, whereas the domestic risk-free rate enters the calculation of the present value of the strike price since the domestic currency is paid over on exercise.

Put

Coli Strikmg Price

'dividend',

-

volot¡¡ity

-

0.5

l.0

0.;

0.5

l.0

90 100 i10

I0.50 1.57 0.00

12.48 4.4| 0.69

14.82 7.17 2.41

0.00 0.98 9.31

0.07 l.52 7.30

0.I8 I,57 5.87

25%

90 |00 i 10

10.76 3.44 0.53

\4.47

0.27 2.84 9.84

2.06 5.45

4.3\

18.00 I2.34 8.10

10.92

3.36 6.74 1I.55

90 100 I 10

12.6| 6.56 2.94

20.01 15.05 I I.IS

25.77 21.34 17.63

2,\ \ 5,97 12.24

7.60 12.\5 17.76

I l.13 15.75 21.08

Currency

Options

Examples Assume that the US dollar/sterling spot rate is 1.8, the time to maturity is three months, the three-month dollar interest rate is 7% and the sterling interest rate is 10%. When the volatility is 20%, the option price is 6.3817, calculated as

0.I

%

·

50%

8.35

follows: C di

=

=

"'°2"

'

180 e

N(d

)-

"'

180 e

N(d2)

5 0.5(0.2)2)0.25]/0.2v'Ï -0.05

[ln(180/180)+ (0.06- 0.10

=

The interest

Rate Theorem

and the Pricing of Forward

-0.15

d2

=

N(di)

C

=

di - 0.2 0.4801, N(d2)

84.284 - 77.896

=

=

6.3817

=

Since the option's delta or partial derivative with respect to the underlying price is given by Ae

=

'

e

The interest rate parity theorem states that the forward rate is equal to the spot tate compounded by the differential between the foreign and domestic interest rates. Using continuously compounded interest rates, the forward exchange rate is

0.4404

=

asset

/

c Ao

'

e

"'°2"0.4801

Se"

'

(58)

which means simply that the formula for the pricing of a European call on a spot currency can be rewritten as

N(di)

its value is =

=

0.4682

=

e

=

f N(d ) -

KN(d2)]

(59)

with

The delta for the currency put is A, which

=

e

N(- di)

=

e "(N(di)

gives a value of A,

and d2 =

e

"'

(0.4801 -

1)

=

0.5070

Note that the value of N(d ) is discounted to the present using the foreign interest rate. This is because this rate is assumed to correspond to a continuous dividend stream on the underlying asset. In the same way, we can calculate the risk parameters of other options. These calculations are held over to the next chapter.

di

- 1) =

- o

K

(jo2)T

(60)

di - ov'¯7. The formula for the European currency put is p

=

e

fN(-di)

KN(-d2)]

(61)

where all the parameters have the same meaning as before, except for the spot exchange rate S, which is replaced by the forward exchange rate f. Note that the mterest rate differential is not explicitly taken into account in the above formula. This is because all the available information about spot rates and the interest rate differential is integrated in the forward exchange rate via the interest parity theorem.

\ 14

5.4.2

OPTIOftS,FUTURES

AND EKOT1C

ANALYTICAL

DERIVATIVES

EUROPEAN

PRICING

OPTION

can be constructed and adjusted continuously, be satisfied by the option price, c, is

Applications

Currency options were traded on the spot currency for the first time in 1982 at the Philadelphia Stock Exchange. Since that date, currency options have been traded in many other financial centers. However, trading on the OTC market seems to be more important. The strategies discussed for stock options also apply to currency options and currency futures options, As we will see in the second part of this book, several financial OTC currency products can be valued using the Garman and Koh1hagen model, It is convenient to note that in the model proposed by Garman and Kohlhagen (1983), the foreign and domestic interest rates are constant, while in Grabbe's (1983) model these rates are stochastic. In the latter model, where the approach parallels that in Merton (1973a),bond price dynamics follow geometric Brownian motion. The assumption of Brownian motion for the bond returns may be criticized since it may imply negative interest rates, i.e. the bond's price may be greater than its face value and the variance will not be nil at maturity Chiang and Okunev (1993) presented another model for the pricing of foreign currency options where the domestic and foreign bond processes follow Brownian bridges. This process may also be criticized since there is also a possibility that the bond's price may be greater than its face value. Yet, it has the advantage of describing better the characteristics of the bond. In fact, with this process, the price of the bond is known with certainty at the maturity date and its variance is nil. The formula presented by Chiang and Okunev resembles that of Grabbe (1983), except that the variance and covariance terms (1993) estimated using approach. different are a

jo'S2

82c(S, t)

I 15

the partial differential equation

Oc(S, t)

+ bS

OS2

MODELS

Oc(S, t)

- rc(S, t) +

that must

0

=

(65)

This equation first appeared indirectly in Merton (1973a). When the cost of carry b is equal to the riskless interest rate, this equation reduces to that of B-S (1973). When the cost of carry is zero, this equation reduces to that of Black (1976). When the cost of carry is equal to the difference between the domestic and the foreign interest rate, this equation reduces to that in Garman and Kohlhagen (1983). It is convenient to note that the short-term interest rate, r, and the cost of carrsing the commodity, b, are assumed to be constant and proportional rates. Using the terminal boundary condition

c(S, T)

(1973b)showed

Merton

=

max

[0, Sr - K]

(66)

indirectly that the European call price is

c(S, T)

=

"N(d2) - Ke

Se" '"N(di)

(67)

.

with

d' .

and d2

=

S'

i oy'¯†

=

+(b+¼o2)T

(68)

K .

di - o v'T. Usmg the following boundary condition: p(S, T) max [0, K - Sr]

(69)

=

5.5

the European put price is

THE MERTON, BARONE-ADESI AND WBUM.EY MODEL AND ITS APPLICATIONS

p(S, T) Table 5.9 shows various

5.5.1

=

Ke-,rN(-d2)

Se

-

'1"N(-di)

BAW European option values for S

(70)

100.

=

The Model -

The model presented m Barone-Adesi and Whaley (1987), known as the BAW model, is a direct extension of models presented by B-S (1973), Merton (1973a) and Black (1976). The absence of riskless arbitrage opportunities implies that the following relationship exists between the futures contract, F, and the price of its underlying spot commodity, S: F

Sear

=

(62)

where T is the time to expiration and b is the cost of carrying the commodity. When the underlying commodity dynamics are given by dS/S=adt+odW

TABLE 5.9

a is the expected instantaneous relative price change of the,codamodity and a is its standard deviation, then the dynamics of the futures price are giventy the following differential equation:

Assuming

that a hedged portfolio

F

=

(a

-

containing

b)dt + od W the option and the underlying

(64) commodity

S

Values:

=

100, r

=

10%, b

=

4%

Put

call Volatility

25%

where

dF

Option

Time to Maturity

10%

(63)

BAW European

50%

Striking Price

0.I

0,5

l.0

0.I

0.5

l.0

90 100 Il0

10.30 1.46 0.00

\ 1.53 0.52

13.03 5.82 1.76

0.00 \.06 9.5\

0.10 1.86 8.11

0.28 2.13 7.12

90 100 110

10.58 3.33 0.51

13.70 7.77 3.94

16.53

0.28

I l.15 7.l9

2.93 10.0!

2.26 5.85 11.53

3.79 7.45 12.54

90 100 !!0

12.45 6.45 2.87

19.32 14.47 10.67

24.41 20.13 16.56

2.15 6.06 l2.38

7.88 !2.54 18.26

|1.67 16.44 2f.91

3.78

5.5.2

AND EXOTIC

OPTIOMS,FUTURES

116

Application

of the

DERIVATIVES

EUROPEAN

OPTION

MODELS

PRICING

tion of forward and futures prices on securities which and other income to the underlying asset.

Model

Forward and Futures Commodity

ANALYTICAL

117

provide

(and do not

provide) cash

Contracts

Sehr, is useful. The proposed relation between the futures price and the spot pricc, F Recall that F is the futures price, S is the spot price, b corresponds to the carrying cost applies to futures prices and to forward and T is the time to maturity. This relationship prices as well. =

When the cost of carrying an asset refers to the storage of commodity such silver costs a as or gold and when these costs are proportional to the commodity price, the futures price is given by

FuturesContractson Commodities

F

Se

=

,_or

(76)

where a stands for the storage costs. Forward Prices and Futures Prices Futures prices and forward prices are often regarded as being equivalent. However, this is true only if the risk-free interest rate is constant or a known function of time. The proof of this equivalence is based on a rollover strategy proposed in Cox, Ingersoll and Ross (1985a). In this strategy, the investor buys e' futures contracts at the end of the first day of trading (initialtime, day 0), e2; EUtUTCS contracts at the end of day 1, e futures contracts at the end of day 2, etc., and e" futures contracts at the end of day i - lSince, at the beginning of day i the investor has e" contracts in his position, the position on that day shows a profit (loss)of (F, - F, i)e". When this amount is invested until the day N corresponding to the contract's maturity date, where the number of days i is between 0 and N, this amount will be (F, The sum of the amounts of profit

F,_i)c"c(N-"'

=

(F, -

F,_i)eN

(loss)from day 0 until day N turns

(71) out to be

Fo)eNr

(FN

Fo)eNr -

is equal to the spot

Fo)eN'

(S, -

=

(73)

corresponds to this strategy and an investment Using a portfolio in a risk-free bond, gives the following pay-off at time T: which

(Sr -

Fo)eNi

N'

=

(74) the result

SveNr

with no-arbitrage they must have the same value in capital markets result, SreN', the opportunities, arbitrage futures price Fo profitable opportunities. In the absence of relation applies for futures proposed price the Hence, the equal forward to must be fe. and forward prices and we have

f

=

2'

400e'

=

Sc^r

Some examples are given below to illustrate the use of this relationship

'

=

410.126

Futures Contracts on a Security with No Income When there is no distribution to the underlying asset, the cost of carry is equal to the riskless interest rate. The futures price is given by F More generally, this relation represents spot price S. It applies to the valuation provides no income.

Se"

=

(77)

the forward or futures price F as a function of the of forward or futures contracts on a security that

Example

Since for a strategy in futures contracts no funds corresponds to the investment of Fo in the risk-free bond. Another strategy can be constructed to give the same pay-off as the preceding one. In fact, if fo stands for the forward price at the end of day 0, then a strategy which consists eN' contracts also 10 fOrward of investing this amount in a riskless bond and an amount gives a final pay-off at T equal to SreNr Since the two strategies require an investment of an amount Fo, (fo) and yield the same

=

F

of an amount Fo

SveN'

are invested,

F

Consider a six-month futures contract on silver or gold. Assume that the asset price is 400, the risk-free rate is 7% p.a. and the storage costs correspond to 2% of the commodity price. In this case, the futures commodity price is given by

(y )

-

However, since at the contract's maturity date, the futures price F, price, Sr, the terminal value of this investment strategy is (FN

Example

(75) in the determma-

Consider the valuation of a forward contract on a non-dividend paying stock. Suppose the maturity date is in three months, the current asset price is 100 and the three-month risk-free rate is 7% p.a. In this case, S 100, T 0.25 year and r 0.07, so the futures or forward price is given by =

=

=

F

=

"

100e'

25

=

101.765

Futures Contracts on a Security with a Known Income For dividend paying assets, the cost of carrying the stocks is given by the difference between the riskless rate and the dividend yield, d. This gives the following relation: F This relationship

=

Sc"

(78)

gives the forward price F as a function of the spot price S for a forward contract on a security that provides a known dividend yield.

OPTIONS,

I 18

Example

AND EXOTIC

FUTURES

DERIVATIVES

ANALYTlCAL

EUROPEAN

OPTION

PRICING

MODELS

I 19

bonds and some In the case of stocks paying known dividends, coupon-bearing commodities for which there are storage costs, the formula for future prices becomes

i

Consider the valuation of a three-month forward or futures contract on a security that provides a continuous dividend yield of 5% p.a. Suppose that the current 100, rate is 7% p.a. In this case, S asset price is 100 and the risk-free T 0.25 year, r 0.07 and d 0.05, so the futures or forward price is given

F

(S - I)ecr

(80)

to an income and is negative when it refers to a cost.

where I is positive if it corresponds

=

=

=

=

=

-·

by F

Example

100e'°°'

=

26

"

Example

100.501

=

Consider the valuation of a one-year forward contract on a two-year bond. The two-year bond's price is 800, the delivery price is 820 and two coupons of 50 will be paid in six and 12 months. The riskless interest rate is 8% p.a. for six months and 9% p.a. for 12 months. In order to apply the formula, the value of I must be discounted to the present at the appropriate interest rate. In this case, I is given by

2

Consider the valuation of a three-month forward or futures contract on the CAC 40 stock index. The index provides a continuous dividend yield of 4% p.a. Suppose that the current index price is 1000 and the risk-free rate is 7% p.a. In 1000, T 0.25 year, r this case, S 0.07 and d 0.04, so the futures or forward price is given by =

=

=

F

I

F

1000e(0.07-0.04)0.25 1007.528

=

50e

=

"*°*

"

+ 50e

=

48.039 + 45.696

=

93.735

and the forward price is given by

=

=

(800

'"

93.735)e

-

=

'O

706.265e

=

772.777

=

Futures Contracts on Foreign Currencies The cost of carrying a foreign currency is given by the difference between the domestic riskless rate and the foreign riskless rate, r*. This gives the following relation between the futures price and the spot price of the currency: F

=

(79)

Se"

This relationship also gives the forward price F (or foreign exchange rate F) as a function of the spot price S for a forward contract on a currency. It is often known in international finance as the interest-rate

I

parity theorem.

Example

2

of a one-year forward contract on a stock with a Consider now the valuation price equal to 100. When the dividend is 2 and the interest paid at the end of the year is 10% p.a., the present value of dividends is I

2e

=

""

=

1.8096

and the forward price is given by F

=

(100-

1.8096)e

""

=

108.516

Example

forward or futures contract on a foreign currency. If the spot price is 180, the domestic risk-free rate is 7% p.a. and the 180, T 0.07 and 0.25 year, r foreign risk-free rate is 6% p.a., then S is given 0.06. The future forward by r or price

Consider the valuation of a three-month

=

=

.

=

.

=

F

=

180e'

63°26

=

180.45

Example

3

.

Consider the valuation of a one-year futures contract on gold. If the cost of carry is 3 per ounce paid at the end of the year, the spot price is 500 and the risk-free rate is 10% p.a., the value of I is given by I

Futures Contracts on a Security with a Discrete lncome When the cost of carrying the commodity is not a constant proportional rate, the formulas above must be slightly modified.

=

3e

""

=

2.7145

and the futures price is given by F

=

(500 + 2.7145)e

""

=

555.585

OPTIONS,

120

Commodity

Options

and

Commodity

FUTURES

AND EXOTIC

DERiYATIVES

Futures Options

A commodity call gives the right to its holder to buy a specific commodity at a specified price within a specified period of time. A commodity put gives the right to its holder to buy a specific commodity at a specified price within a predetermined period of time. A commodity futures option is an option on the futures contract having a commodity as an underlying asset. The commodity may be a precious metal such as silver or gold. It may be a financial instrument such as a common stock, a treasury bond or a foreign currency. For example, if the commodity option is written on a foreign currency, the option refers to a currency option. If the commodity option is written on a stock index, the option is an index option. Since all the analytical models for European options presented in this chapter can be obtained by modifying the parameters in the Merton (1973a) and BAW (1987)model, all the applications presented here are also true for this model. For example, when the option's underlying asset is an index which is constructed to pay continuous dividends, this version is a particular case of the Merton and BAW model for commodity options. When the continuous dividend yield is d, the formula for European commodity options is rewritten for index options with b r - d. For a European index call, the formula is =

c(S, T)

=

Se-dTN(d

)-

ANALYTICAL

POINTS

• • •

•

ln

=

a and d2

=

di - on.

K /

+

2)

(r + (0

T

(82)

• • •

For a European index put, the formula becomes

p(S, T)

=

Ke "N(-d2)

"N(-di) - Se

•

(83)

• • •

SUMMARY This chapter presents in detail the basic concepts and techniques underlying rational derivative asset pricing, in the context of analytical European models along the lines of B-S, Black, Garman and Kohlhagen, and Merton-BAW. First, an overview of the analytical models proposed by the precursors is given. Second, the simple model of B-S is derived in detail for the valuation of options on spot assets and some of its applications are presented. Third, the Black model, which is an extension of the B-S model for the valuation of futures contracts and commodity options, is analyzed. Also, applications of the model are proposed. Fourth, the Garman and Kohlhagen model is presented for the valuation of currency options, and some applications of the model are provided. Fifth, the Merton and BAW model is applied to the valuation of European commodity options and commodity futures options. Again, some applications contracts, commodity

MODELS

121

FOR DISCUSSION

What is wrong What is wrong

• •

•

di

PRICING

natural extension of these models must introduce the possibility of early exercise and discrete distributions. However, before making some extensions of the basic analytical models, it is useful to study, in this simple context, the option price sensitivities and the use of the so-called Greek-letter risk measures in the monitoring and the management of an option position. The question of managing an option position is as important as some issues regarding option pricing. This is the subject of the next chapter.

•

with

OPTION

of the model are given. Note that this model reduces to the models in B-S, Black, and Garman and Kohlhagen for some values of its parameters. Since all these models are concerned with the valuation of European-style options in a continuous time framework (withoutdiscrete distributions to the underlying asset), a

•

Ke-rrN(d2)

EUROPEAN

•

• •

•

• • • •

• • •

with the Bachelier formula? with the Sprenkle formula'?

What is wrong with the Boness formula? What is wrong with the Samuelson formula? What are the main differences between the Black and Scholes model and the precursor models? for options on spot assets? How can we obtain the put-call parity relationship How can we obtam the put-call parity relationship for options on futures contracts? What are the main specificities of index options and their markets? How is the Black and Scholes model adjusted for index options? What are the implications of arbitrage for index option markets and their assets? How are indexes constructed? What is the main difference between zero-coupon bonds and coupon-paying bonds? What are the different types of bonds? What are the main specificities of short-term options on long-term bonds? What are the main specificities of bond options? Is the Black and Scholes model appropriate for the valuation of derivative assets whose values depend on interest rates? Justify your answer. of derivative assets whose values Is the Black model appropriate for the valuation depend on interest rates? Justify your answer. 1s there a difference between a futures price and the value of a futures contract? How can we obtain the put-call parity relationship for futures options from that of European spot options? What are the specificities of options traded on the LIFFE? What is the appropriate model for their valuation? What does the interest rate theorem tell us'? What are the main characteristics of currency options and their markets? What is the main difference between Black and Scholes' model and Black's model? What is the main difference between Black and Scholes' model and Garman and Kohlhagen's model? What is inappropriate in the derivation of Garman and Kohlhagen's model? Black and Scholes' model? What do you think of the assumptions underlying underlying Black's model? What do you think of the assumptions

122

• • • • • • • •

What What What What What What What How

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

'

do you think of the assumptions underlying Garman and Kohlhagen's model'? are the main differences between futures and forward contracts?

is a commodity option? is the formula for a futures contract on a security with no income? is the formula for a futures contract on a security with a known income? is the formula for a futures contract on a foreign currency? is the formula for a futures contract on a security with a discrete income? can we obtain the formulas in Black and Scholes' model, Black's model and Garman and Kohlhagen's model usmg the formula m Merton and BAW?

e

4

Monito rmg an dM an age ment Of Ûpt|On

CHAPTER

OUTLINE

ŸOSitiODS

.

This chapter is organized as follows: 1. In section 6.1, option price sensitivities are presented and the formulas are applied. 2. In section 6.2, the Greek-letter risk measures are simulated for different parameters. The issue of monitoring and managing an option position in real time is studied for the different risk measures with respect to an option pricing model. 3. In section 6.3, some of the characteristics of volatility spreads are presented. 4. In Appendix 6.A, we give the Greek-letter risk measures with respect to the analytical models presented in the previous chapter. 5. In Appendix 6.B, we show the relationships between some of these Greek-letter risk measures. 6. In Appendix 6.C, we present the relationships between these parameters of a more general equation for the pricing of derivative securities.

in the context

INTRODUCTION in a derivative asset price with respect to its to monitor the variations which the determinants or enter the option formula. These variations are often parameters named Greek-letter risk measures. The most widely used measures are known as the delta, charm, gamma, speed, color, theta, vega, Rho and elasticity. The delta measures the absolute change in the option price with respect to a small variation in the underlying asset price. Charm corresponds to the partial derivative of the delta with respect to time. The gamma gives the change in the delta, or in the hedge ratio asset price changes. Color corresponds to the gamma's derivative with as the underlying the change in the option respect to the time remaining to maturity. The theta measures price as time elapses, namely the time decay of the option. The vega or lambda is a measure of the change in the option price for a small change in the underlying asset's It is important

volatility. In this chapter, we show how to calculate some of these parameters

within

the context

124

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

MONITORING

of each analytical model presented in the previous chapter. Also, we develop examples to show how to use the Greek-letter risk measures in the monitoring and the management of an option position in response to changing market conditions.

6.1

OPTION

.

with respect to the parameters

sensitivities

POSITIONS

entering

125

the option formula. We begin our

discussion with the delta.

6.I.I

PRICE SENSITIVITIES

The sensitivity parameters are important in managing an option position. The delta measures the absolute change in the option price with respect to a small change m the price of the underlying asset. It is- given by the option's partial derivative with respect to the underlying price. It represents the hedge ratio, or the number of asset options to write or to buy in order to create a risk-free portfolio. Call buymg involves the sale of a quantity delta of the underlying asset in order to form the hedging portfolio. Call selling involves the purchase of a quantity delta of the asset to create the hedging portfolio. Put buying requires the purchase of a quantity delta of the underlying asset to hedge a portfolio. Put selling involves the sale of delta stocks to create a hedged portfoho. The delta varies frorn zero for deep out-of-the-money options to l for deep in-themoney calls. This is not surprising, since by definition the delta is given by the first partial derivative of the option price with respect to the underlying asset. For example, the value which the of a deep in-the-money call is nearly equal to the intrinsic value (S - K), for with partial S first derivative respect to is 1. Charm is a risk measure that clarifies the concept of carry in financial instruments. The concept of carry refers to the expenses due to the financing of a deferred dehvery of commodities, currencies or other assets m financial contracts. Charm is given by the derivative of the delta with respect to time. Even though charm is used by market participants as an ad hoc measure of how delta may change overnight, it is an important measure of risk since it divides the theta into its asset-based constituents (seeAppendix 6.C). The gamma measures the change in delta, or in the hedge ratio, as the underlying asset price changes. The gamma is greatest for at-the-money options. It is nearly zero for deep in-the-money and deep out-of-the-money options. Roughly speaking, the gamma is to the delta what convexity is to duration. The gamma is given by the derivative of the hedge ratio with respect to the underlying asset price. As such, it is an indication of the vulnerability of the hedge ratio. of an option The gamma is very important in the management and the monitoring position. It gives rise to two other measures of risk: speed and color. Speed is given by the gamma's derivative with respect to the underlying asset price. Color is given by the gamma's derivative with respect to the time remaining to maturity. The theta measures the change in the option price as time elapses, since the passage of time has a negative impact on option values. Indeed, options are wasting assets. Theta is given by the first partial derivative of the option premium with respect to time. the change in the option price for a change in the The vega or lambda measures volatility. underlying's given It is by the first derivative of the option premium with asset respect to the volatility parameter. option price is not sufficient and the for the monitoring The knowledge of the management of an option position. Therefore, it is important to know the option price

OF OPTION

AND MANAGEMENT

The Delta

The Call's Delta The delta is given by the option's first partial derivative with respect to the underlymg asset price. It represents the hedge ratio in the context of the B-S model. The option's delta is given by

.

A Applying this formula

N(di)

=

the calculation

needs

in d

(1)

of di:

Ky

=

(2)

5

a

Example 15, the short-term 18, the strike price K Let the underlying asset S 15%. 0.25 and the volatility a 10%, the maturity date T =

=

.

rate r

in di

18

interest

=

=

=

(0.1+

((0.15)2)0.25 =

=

0.15VT25

2.8017

N(2.8017) Hence, the delta is Ae 0.997. This delta value means that the hedge of the purchase of a call requires the sale of 0.997 units of the underlying asset. When the underlying asset price rises by l unit, from 18 to 19, the option price rises from 3.3659 to (3.3659+ 0.997), or 4.3629. When the asset price falls by one unit, the option price changes from 3.3659 to (3.3659 0.997), or 2.3689. =

=

-

The Put's Delta

The put's delta has the same meaning as the call's delta. It is also given by the option's first derivative with respect to the underlying asset price. When selling (buying) a put option, the hedge requires selling (buying)delta units of the underlying asset. The put's delta is given by A,

'true'

=

A, - 1

Using the data fiom the call example, A, 0.997 - 1 -0.003. The hedge ratio is 0.003. When the underlying asset price goes from 18 to 19, the put price changes from =

4

(3) =

*

AND EXOTIC

FUTURES

OPTIONS,

6.1.3

The Theta

with respect to the time the option's first partial derivative The option's theta is given by by is given model, the theta remaining to maturity. In the B-S San(di) (7) N(d2) + rKe oc Oc 2d BT =

The Calfs Gamma option's second partial derivative with respect The option's gamma corresponds to the derivative with respect to the asset price partial the underlying asset or to the delta's In the B-S model, the call's gamma is given by

Son

BS

to

(4)

n(di)

=

=

(5)

di)

(

exp

=

-

-

in the example above: or using the same data as Oc 0.2653 1.4571

option price decreases by maturity is reduced by one day, the the call's price When the option's time to maturity is shortened by 1% of a year, 1.1918 units. When the time to 3.3659 to from changes 0.011918, and its price decreases by 0.01 (1.1918),or (3.3659 0.011918), or 3.3540,

The Put's Theta put's theta is given by In the B-S model, the Son(di) op 8, 2d ST =

1 e

=

"23°"

0.09826

=

the example above, or using the data from

=

5

18(0.5)

(0.09826)

8

0.0727

=

0.997, a fall in the asset price by one When the underlying asset price is 18 and its delta is 0.0727), or 0.9243. Also, a rise in 0.997 to from (0.997 delta in the unit yields a change the delta from 0.997 to (0.997+ 0.0727), the asset price from 18 to 19 yields a change in deeply in-the-money, and its value is given by its the option is or 1. This means that apply to put options. The cal and the put arguments K). The same mtrinsic value (S .

Using the same 0.002

reasoning,

=

(

b¯"N(-d2)

=

-

-0.2653

Ee

-1.1918

=

-

=

--+

Using the same data as in the example: n(di)

=

-

-

with n(di)

187

POSITIONS

The Call's Theta

The Gamma

Ye

OF OPTION

AND MANAGEMENT

DERIVATIVES

18 to 17, the put price rises 0.0045 to (0.0045-0.003), or 0.0015. When it falls from from 0.0045 to (0.0045+ 0.003), or 0.0075.

6.1.2

MONITORING

-0.2594

+ 0.0058

the put price changes

=

from 0.0045 to

(0.0045 0ß025), -

or

-

have the same gamma.

6.1.4

The Vega

The Cau's Vega win the option price derivative The option's vega is given by given by model, the call's vega is parameter. In the B-S

The Put's Gamma

ve

The put's gamma is given by

f"

=

BA" 8S

1 =

-

So

(6)

n(di)

R

Er

=

(0.09826)

=

0.0727

unit, the put price changes by the When the asset price changes by one the delta chances bv an amount eaual to the samma,

dekt

amount and

-

oc SRn(di) =

volatility

(9)

da

or using the above data: De

Using the same data as in the example,

=

respect to the

=

18v'n5(0.098

26)

=

0.884 34

by 0.884 34. The rises by I point, the call price increases 3.3659 to Hence, when the volatility from price the option by 1% changes changes from increase in volatility price the put context, the same or 3,37474. In volatility falls by 1% the [3.3659+ 1%(0.884 34)], 434. When the 0.133 34)], 1%(0.884 or 0.0045 to [0.0045+ 3.361 56. In the same 1%(0.884 34)], or call's price changes from 3.3659 to [3.3659 -

OPTIONS,

FUTURES

way, the put price is modified from 0.0045 to option prices can not be negative.

[0.0045-

128

AND EXOTIC

DERIVATIVES

MONITORING

OF OPTION

ANDFIANAGEMENT

l%(0.884 34)], or zero since

Elasticity

=

S

POSITIONS

N(di)

=

c

129

(13)

c

or usmg the above data:

The Put's Vega

8

Elasticity In the B-S model, the put's vega is given by vp -

-

SÑn(di)

(10)

or using the above data: v

18NZŠ(0.09826) 0.88434 =

=

0.997

=

3.3659

5.3317

=

The elasticity shows the change in the option price when the underlying asset price varies by 1%. Hence a rise of the asset price by 1%, i.e. 0.18, induces an increase in the call price by 5.33%. The put price changes by 12%. Hence, when the asset price changes from 18 to 18.18, the call's price is modified from 3.3659 to [3.3659(1 5.33%)], or 3.545. In the same way, the put price changes from 0.0045 to [0.0045(1 12%)], or 0.0504.

It has the same meaning as the call's vega. The Put's Elasticity 6.1.5

The put's

The Rho

elasticity

is given by Elasticity

The Call's Rho

Oc

Rhoc

=

=

-

K(t*

or

- t)e

(11)

t is the current time and t* is the maturity date. Using the above data: Rhoc

15e

=

"

2"(0.996)(0.25)

=

3.64

The Put's Rho the put's Rho is given by Rho,

=

-K(t*

=

ßr

t)e

"'n(d2)

or using the above data: -l5e-0

Rho

=

Elasticity

-3.64 1(0.25)(0.996X0.25) =

(12)

-[N(dt)

-

1)

(14)

p

18 =

0.0045

[0.997-

The knowledge of the variations in these parameters of an option position. and the management

6.2

The Rho does not affect call and put prices in the same way. In fact, a rise in the interest rate yields a higher call price (positiveRho) and reduces the put price (negativeRho).

S

=

or using the above data:

"'n(dg

where

6.1.6

Ap Sc

The option's Rho is given by the option's first partial derivative with respect to interest rates. In the B-S model, the call's Rho is given by

In the B-S model,

=

MONITORING AND MANAGING POSITION IN REAL TIME

l]

=

-12

is fundamental for the monitoring

AN OPTION

Since option prices change in an unpredictable way in response to the changes in market conditions, traders, market makers and all options users must rely upon some model to monitor the evolutions of their profit and loss accounts. Such a model allows them to quantify the variations in option price sensitivities and their risk exposure. With such of option positions are more easily quantities the monitoring and the management achieved. We will illustrate the management of an option position in real time using the model proposed indirectly in Merton (1973a) and derived afterwards in Black (1975a) and BAW (1987)for the valuation of European futures options. First, option prices are simulated and the sensitivity parameters are calculated. Second, we study the risk management problem in real time with respect to option price sensitivities. .

Elasticity 6.2.1

Simulation

and

Analysis

of Option

Price Sensitivities

The Call's Elasticity For a call option, this measure

is given by

Recall that the commodity call price and the commodity futures call price ifrthe context of Merton's (1973a)and BAW's (1987)model is given by

130

OPTIONS,

c

Se"' '"N(di)

=

-

AND EXOTIC

FUTURES

DERIVATIVES

Ke "N(d2)

ORING AND MANAGEMENT

(15)

TABLE 6.3 In-the-money in the time to maturity 0.I, S a= 0.2, r= 0.08, b

where b stands for the cost of carrying the underlying commodity. By the put-call parity relationship or by a direct derivation, the put's value is given in the same context by p

=

Ke

'"

N(-d2)

-

Se"

'"

N(-di)

=

callprice 9.875 9.849 10.08 10.563 10.946

(16)

where

all the parameters have the same meaning as before. For a non-dividend paying asset, b r. For a dividend paying asset, b r - d where d stands for the dividend yield. For a currency option, b r - r* where r* stands for the foreign riskless rate. Tables 6.1-6.12 simulate option values and sensitivity parameters for calls and puts in the context of the above model. =

=

=

=

=

Colíprice 0.095

0.439 l.392 3.294 6.250 10.080 i 4.473 19.l59

=

=

Gommo

Theto%

Vego%

75 80 85 90 95 100 105 i10

0.037 0.127 0.294 0.507 0.705 0.844 0.92 I 0.959

0.007 0.019 0.033

-1.056 -2.891 -5.023 -5.842

2.8i8 7.836 14.07\ 17.509 I 5.988 I1,080 6.295 2.912

0.039 0.034 0,022 0.0 I2 0.005

-4.567 -\.997 -0.579

-0.477

call price sensitivities TABLE 6.2 In-the-money asset prices using the following in the underlying 0.08, b 0.25, K 0.2, r « 0. I, T I 00 =

=

=

=

0.004 0.034 0.175 0.631 I 3.659 6.567 10.287 .705

assetprice 75 80 85 90 95 100 105 I 10

0.988 0.950 0.844 0.744 0.685

0.006 0.0 I6 0,022 0.020 0.018

=

5.774 6.403

Theta % -1.584 -0.485 -1.997 -1.750 -1.322

call

price sensitivities the following 100 100, K

using =

for changes parameters:

for changes parameters:

Gamma

Theta %

Vego %

0.05

0.542 0.526 0.507 0.489 0.475 0.46I

0.088 0.062 0.039 0.027 0.021 0.018

-16.378 -!!.221 -6.491 -4.0 I 3 -2.879 -2.I90

8.876 !2.49 19.455 26.834 32.053 36.098

=

=

Volatility 0.05 0.10 0.20 0.30 0.40

9.3 I3 9.352 10.08 I 1.404 12.963

i 1.224 20.316 26.674

Delto

=

Callprice

0.569 3. I6

=

TABLE 6.5 call price sensitivities In-the-money using the following in the volatility parameter 0.08, b 0. I, S I 00, K 90 T 0.25, r =

Vega %

Maturity

0.l0 0.25 0.50 0.75 0.75

4.925

for changes parameters:

=

°/o

Delto

Gamma

Thetc %

Vega

0.975 0.959 0.844 0.762 0.716

0.000 0.00| 0.022 0.020 0,0160

-2.694 -2.I59 -1.997 -6.420 -10.504

0.006 2.5i7 I 1.224 14.783 16.252

=

Underlying

Callprice

0.05

=

3.659 Delto

Gammo

0.75

for changes parameters:

90

=

Deka

0.50

1.727 2.402

=

the following

100, K

=

13 I

sensitivities

price

Moturity 0. I0 0.25

callprice

POSITIONS

using

At-the-money in the time to maturity 0.2, r 0.08, b 0.I, S a

for changes parameters:

Underlying ossetprice

call

TABLE 6.4 =

TABLE 6. I At-the-money call price sensitivities in the underlying asset prices using the following 0.08, b 0.f, T 0.25, K 90 0.2, r «

OF OPTION

Delta

Gammo

Theta%

0.002 0.014 0.056 0.154 0.3 I4 0.507 0.687 0.82 I

0.001 0.003 0.0 I 0

-0.092 -0.499 -1.672 -3.713 -5.766

0.022 0.034 0.039 0.035 0.025

-6.491

-5.306 - 2.83 \

Vega%

0.243 l.325 4.477 10.112 I6.229 !9.455 18.152 \ 3.643

"t

call price sensitivities TABLE 6.6 At-the-money using the following in the volatility parameter 0.08, b 0.1, S 100, K 100 T 0.25, r =

Call price 0.750 |.7 I5 3.659 5.604 7.546

=

=

=

Volatility

Delto

0.05 0. IO 0.20 0.30 0.40

0.492 0.497 0.507 0.5l7 0.526

for changes parameters:

=

Gomma 0.15!

0.077 0.039 0.026 0.0 19

Theto %

Vega %

-l.Ol6 --2.8 I 5 -6.491 -l0.l72 3.836 - I

19.!19 I9.40 I 19.455 (9.437 I9.399

OPTIONS,

132

FUTURES

AND EXOTIC

TABLE 6.7 At-the-money put price sensitivities asset prices using the following in the underlying 0.I, T 0.25, K 90 « 0.2, r 0.08, b =

=

Put price

=

=

for changes parameters:

AND MANAGEMENT

=

Vega %

Put price

-0.938

0.007

-0.849 -0.682 -0.468 0.27 I -0.132

0.019 0.033 0.039 0.034 0.022

- I.3 I 3 -3.636 -6.256 -7.562 -6.775 -4.692

2.8 I8 7.836 14.071 I7.509 I5.988 11.224

0.000 0.039

Put

price

Underlying asset

24.876 20.029 I5.293 I0.873 7.07 I 4.148

TABLE 6.9 Out-the-money es in the time to maturity 0.I, S 0.2, r 0.08, b o =

=

=

changes meters: Put

Delto

Gamma

Theto %

Vego %

-0.973 -0.961 -0.9I9 -0.82 1 -0.662 -0.468

0.001

-0.435 -0.460 -2.I2I -4.648 -7. I90

0.243 1.325 4.477 i 0. I I2 I6.229 19.455

price

75 80 85 90 95 100

133

Gommo

Theto %

Vego %

0.000 0.001 0.022 0.020 0.0 I6

-0.00 I -0.537 -4.692 -9.I I5 - I 3. I99

0.006 2.517 I 1.224 i4.783 I6.252

sensitivities 6.12 price At-the-money put using the following in the volatility parameter 100 0.08, b 0.I, S 100, K T 0.25, r =

=

0.014 0.127 0.767 I.9I I 2.93 \ 3.829

Moturity 0.05 0.\0

0.25 0.50 0.75 l.00

0.003 0.0I0 0.022 0.034 0.039

-8.402

put price sensitivities the following 100, K 100 using

=

price

Volatility

l.239 2.204 4.148 6.093 8.035

Sensitivity Parameters

for changparameters:

0.05 0.10 0.20 0.30 0.40

Delta

Gommo

Theto %

Vega %

0.151 0.077

-2.927 -4.726 -8.402 -I2.083 - I 5.747

l9.I 19 19.40\ 19.455 I9.437 I9.399

-0.483 -0.478 -0.468 -0.459 -0.449

for Call

0.039 0.026 0.0 I9

Options

Table 6.1 gives call prices, delta, gamma, theta and vega when the underlying commodity price varies from 75 to l 10 in steps of $5. For example, when the volatility 20%, a 8%, b 10% and T 3 months (0.25year), the price of an at-the-money call for r K 90 is 3.294 The call has a delta of 0.507, a gamma of 0.039, a theta of and a vega of 17.504. Note that an at-the-money option has more theta and vega than in-the-money and out-of-the-money calls. Using the same data except for the strike price which is modified from 90 to 100, Table 6.2 shows that an at-the-money call has more gamma, theta and vega than in- and out-ofthe-money calls. Table 6.3 shows call prices and sensitivity parameters for in-the-money calls (S> K) when the time to maturity varies from 0.05 to 0.75 year. Note that the call price, the vega and the theta increase with the time to maturity. However, the delta falls when the time to maturity is longer. Table 6.4 gives call prices and sensitivity parameters for at-the-money calls when S K 100 and the time to maturity varies from 0.05 to one year. Note that the call price, its delta and vega are increasing functions of the time to maturity. However, the gamma and the theta are more important for near-maturities. Table 6.5 gives in-the-money call prices (S 100, K 90) for different levels of the volatility parameter. When the delta is equal to 1, the gamma is nearly equal to zero. Also, the vega is nearly ml and the theta is weak. Table 6.6 gives the same information as Table 6.5, except that calculations are done for at-the-money options (S K 100). =

=

=

Delto

Gamma

Theto %

Vega %

-0.007 -0.04 -0.132 -0.207 -0.243 -0.263

0.006 0.016 0.022 0.020 0.0 I8 0.016

-l.l94 -3.242 -4.692 -4.344 -3.8 I8 -3.375

0.569 3.I6 1 l.224 20.316 26.674 3\.448

TABLE 6.10 At-the-money in the time to maturity «=0.2,r=0.08,b=0.I,S= es

=

=

-5.842

for changput price sensitivities the following parameters: 100,K= 100 using

=

Put price I.826 2.600 4.148 5.88 I 7.176 8.231

Delta

Moturity 0.05 0.10 0.25 0.50 0.75 l.00

.

for para-

=

=

=

Put price

para-

=

=

0.000 -0.016 -0.132 -0.2I3 -0.259

=

for

=

=

=

=

for changes parameters:

Delta

0.05 0.10 0.20 0.30 0.40

0.767

=

=

Volatility

2.09I 3.650

TABLE

=

POSITIONS

sensitivities TABLE 6.1 I Out-of-the-money put price in the volatility parameter using the following meters: T 0.25, r 100, K 90 0.08, b 0.I, S

Theta %

TABLE 6.8 In-the-money put price sensitivities in the underlying asset prices using the following 0.25, K 100 0.2, r 0.I, T 0.08, b a

OF OPTION

changes

Gomma

Delta

75 80 85 90 95 100

MordTORING

=

Underlying asset price

I5. I65 10.632 6.708 3.734 I.8 I 3 0.767

DERIVATIVES

-0.453 -0.464 -0.468 -0.462 -0.453 -0.443

Gammo 0.088 0.062 0.039 0.027 0.021 0.018

Theto %

Vego %

-18.360 -13.184 -8.402 -5.839 -4.622 -3.853

8.876 12.49 19.455 26.834 32.053 36.098

=

=

-

.

t

=

,

=

=

OPTIONS,

134

Sensitivity Parameters

for Put

FUTURES

AND EXOT1C DERIVAT!

AMOBRMiBIGEMENT

=

35

-

=

3o

-

2o

-

15

-1.0

.

io

-

=

0.7 0.6

5

e

_

Monitoring

and

0.1

12o

S: Underlying asset price

=

Position

T: Timelomaturity

0.2

92 96 100 104108 112 116

are the sensitivity parameters. Table 6.12 shows at-the-money put prices (S 100) for different levels of the K volatility parameter. Note that the put price, the delta (in absolute value), the vega and the theta are increasing functions of the volatility parameter.

the Option

0.4 0.3

(

80 84

varies from 0.05 to Table 6.11 gives out-of-the money put prices when the volatility 0.40. When the volatility is respectively equal to 0.05 and 0.10 the put price is nil and so

and Adjusting

=

-

25

maturitieS'

Monitoring

,

=

=

=

K 100 r = 8% b 10% o 25% =

=

=

6.2.2

135

Parameters =

=

POSITIONS

Options

Table 6.7 gives put prices and the sensitivity parameters when « 0.2, r 0.08, b 0.1, T 0.25 and K 90. For an at-the-money put, the gamma and the vega are important. The put theta increases when the option tends to parity and decreases afterwards. The same behavior applies for the put gamma and vega. Table 6.8 gives the same information as Table 6.7, except for the strike price which is changed from 90 to 100. For in-the-money puts, the delta approaches 1, the gamma and vega are not important and the theta is very weak. 100, K Table 6.9 gives out-of-the-money 90) and the put price put prices (S sensitivities when the time to maturity varies from 0.05 to one year. The values of the delta and gamma are weak. However, the vega and theta are greater for longer maturities Table 6.10 shows the same information for at-the-money put prices (S K 100). Note that the deltas and gammas are decreasing functions of the time to maturity. For an at-the-money put, there is more theta on short maturities and more vega on longer =

OF OPTION

it

*The

call Values

by Underlying

Asset

Price and Time

to Maturity

in Real Time

Managing the Delta

Parameters

The call delta lies between zero and 1 and the put delta between zero and - L When the delta is 0.5, the call price rises (falls)by 0.5 point for each increase (decrease)in the asset price by l point. A delta of0.5 corresponds to an at-the-money call. For a deep in-the-money call, the variation in the asset price by one unit implies an call is almost equivalent variation in the option price. The delta of an out-of-the-money zero and of a deep in-the-money call is almost 1 Figures 6.1-6.4 are drawn according to the data in the preceding tables. and zero, a rise in the asset price Since the put delta lies in the interval between implies a fall in the put price and vice versa. The delta is for an at-the-money put, -1 for a deep in-the-money put and zero for a deep out-of-the-money put. Using the data in the preceding tables, Figure 6.2 shows the put price as a function of .

-

=

-\

18

=

16 14

12 io

-l

the asset price The call delta is often assimilated to the hedge ratio. Since the underlying asset delta is and that of an at-the-money call is 0.5, the hedge ratio is 1 0.5 or 2/1. Hence, we need 1 at-the-money calls to hedge the sale of the underlying asset. two The delta corresponds also to the (risk-neutral)probability that the option finishes in the money at the option's maturity date. Hence, when the delta is 0.6 for a call (-0.6 for a put), this means that there is a 60% chance in the risk-neutral economy that the option

K 100 r 8% b = 10% ci 25% =

20

o

6 6 4

o

-

-

-

-

¯ -1.0

0 9

_

0 7

-

a 80 84 66 92 96 100 104108

:)

i

Time

to nakisty

2

112 116 120

S: Underlying asset price

FIGURE 6.2

The Put Values

by Underlying

Asset

PrtgegiendTiitmaW9taturity

OPTIONS,

136

FUTU

EG AIIIBMIIR Ild IlialVATIVES

MONITORING

AND MANAGEMENT

OF OPTION

POSITIONS

137

finishes in-the-money at maturity. Note that the delta is calculated K 100 r 8% b = 10% a = 25% =

1.00

=

-

0.90

-

o.80

-

0.70 o

0.60

¯

0.50

-

0.40

-

0.30

1.0 0J30.9

-

0.20

-

o.1o

-

0.7

0.6

o.5 0.4 0.3 0.2

0.00

so

84 88 92 96 100104108112 116 120

T: Time . to maturity

0.1

S: Underlying asset price

PIGWila 6.3

The BAW Call's Delta

Parameters K r

o.oo-

=

100

=

8%

o

-0.20 -0.40-0.60

_

-o.80

_

-1.00-

-1.20¯

1.0 0.9 0.8 0·7

-1.40¯ -1.60-1.80-2.00

_

80 84 88 92 96 100 104 108

0.1 112 116 120 S: Underlying asset price

FIGURE

6.4

The BAW Put's

Delta

0.5 0.4 0.20.3

in practice using the volatility. the implied or Delta-neutral hedging requires the adjustment of the option position according to the variations in the delta. When buying or selling a call (put), the investor must sell or buy (buyor sell) delta units of the underlying asset to represent a hedged portfolio. In practice, the hedged portfolio is adjusted nearly continuously to account for the variations in the delta's value. An initially hedged position must be rebalanced by buying in the delta through time. and selling the underlying asset as a function of the variations The delta changes as the volatility, the interest rate and the time to maturity are modified. Table 6.13 shows how to adjust a hedged portfolio in order to preserve the main characteristics of delta-neutral strategies. It is important to note that delta-neutral hedging strategies do not protect completely the option position against the variations in the volatility parameter. Table 6.14 shows when volatility the position changes. adjust the how to It is also important to note that delta-neutral hedging does not protect the option position against the variations in the time remaining to maturity. Table 6.15 shows how to adjust the position when the time to maturity changes. Figure 6.5 shows the effect of the volatility parameter. When the volatility rises or the time to maturity is lengthened, options which are out become at parity or in-the-money and vice versa. Note that deltas are additive. For example, when buying two calls having deltas of 0.2 and 0.7, the investor must sell 0.9 units of the underlymg- asset in a delta-neutral strategy. Option market makers often implement delta-neutral hedging strategies in order to maintain a nil delta (in monetary units). When the delta of an option position is positive, this means that the market maker is long the underlying asset. It can be interpreted as a bullish position. If the asset price rises, he makes a profit since he will be able to sell it at a higher price. However, if the asset price decreases, he will lose money since he will sell the underlying asset at a lower price. When the delta is positive, the investor is over-hedged with respect to delta-neutral strategies. When the delta (in monetary units) is negative, the investor is short the underlying The holds investor asset. asset price rises, the investor a bearish position. If the underlying 10Ses money since he adjusts his position by buying more units of the underlying asset. However, when the asset price falls, he makes a profit since he pays less for the underlying asset.

observed volatility

Parameters

°

TABLE 6.13 Adjustment lying Asset Price

of the

Hedged

Portfolio

as a Function

Asset Price Rises: SÎ

Asset Price Falls:Si

Oþtions

Delta Hedging

Delta Hedging

Longa call Short a call Longa put Short a put

Delta Delta Delta Delta

T: Time to matufüy

increases: sell more buy more decreases: sell more decreases: buy more increases:

S S S S

Delta Delta Delta Delta

of the

decreases: buy more decreases: sell more increases: buy more increases: sell more

S 5 S S

Under-

OPTIONS,

138

TABLE 6.14

Adjustment

of a Hedged

FUTURES

Position

AND EXOTIC

when the Volatility

EMENT OF OPTION

MO9NT

DERIVATIVES

POSITIONS

139

Changes

Parameters

Adjustment of o Hedged Position Options Long a call: in-the-money at-the-money out-of-the-money

1.2

VolatilityDecreases

Volatilitylacreases

=

1.0

Delta increases: buy more S Delta non-adjusted Delta decreases: sell more S

Delta increases: sell more S Delta non-adjusted Detta decreases: buy more S

Delta decreases: resell of S Delta non-adjusted Delta increases: buy more S

Delta increases: buy more S Delta non-adjusted Delta decreases: sell more S

Delta decreases: resell of S Delta non-adjusted Delta increases: buy more S

Delta increases: buy more 5 Delta non-ad¡usted Delta decreases: sell more S

0.8

o

Short a put: in-the-money at-the-money out-of-the-money

0.4

0 Jiim 0.4$ 0.40 0.35

-

Delta increases: buy more S Delta non-adjusted Delta decreases: sell more S

0-

I

80 84 88 92

Delta increases: sell more S Delta non-adjusted Delta decreases: buy more S

96

100 104 108

112 116

0.30 0.25 a: VolaWW 0.20 y 0.15 0.10 1200.05

S: Underlying asset price

¥lGURE

6.5

The Call's Delta and the Volatility

When a portfolio is constructed by buying vj, the portfolio value is given by

(selling)the

securitiesandderivgive

Adjustment

of a Hedged

Position

as a Function

of Time

Options

Adjustment of the Hedged Position as a Function of Time

Longaco. in-the-money at-the-money out-of-the-money

Delta rises: sell more s Delta non-modified Delta decreases: buy more S

Short o coll: m-the-money at-the-money out-of-the-money

Delta nses: buy more S Delta non-modified Delta decreases: sell more 5

Long a put: m-the-money at-the-money out-of-the-money

Delta rises: buy more 5 Delta non-modified Delta decreases: sell more S

shorta put: m-the-money

at-the-money out-of-the-money

where n, stands for the numbers of units of the assets bought or sold. The delta's position, or its partial derivative with respect to the securities and derivative assets, is _BS

Sv3

Sv,

Ovi

Delta nses: sell more S Delta non-modified Delta decreases: buy more S

assets

(17)

P=niv,+n2v,+n3v3+...+nyv TABLE 6.15

10%

0.5

-

0.2~

Longa put: in-the-money at-the-money out-of-the-money

_

100

8%

0.6 -

Short a call: in-the-money at-the-money out-of-the-money

K= r= b= T

-

n2

BS

n3

--BS

+...+ny

Sv; -BS

(|8)

+ n,A niAi + n:A2 nsA3 (19) Delta-neutral hedging is convenient for an investor who does not have prior expectations about the market direction. However, if the investor expects a rising market, he can have a positive delta, i.e., long the underlying asset, so he can self at a higher price when the market effectively rises. If the investor expects a down market, he can have a negative delta, i.e., short the underlying asset, so he can buy it at a lower price when the market effectively goes down. =

Monitoring

---

and Managing

.

.

.

the Gamma

The gamma is given by the second derivative of the option price with respect to the . underlying asset price. A high value of gamma (positiveor negative) shows a higher nsk for an option position. The gamma shows what the option gains (loses) in delta when the underlying asset price rises (falls).

140

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

For example, when the option's gamma is 4 and the option's delta is zero, an increase by 1 point in the underlying asset price allows the option to gain 4 points in delta, i.e. the delta becomes equal to 4. When the delta is constant, the gamma is zero. The gamma varies when the market conditions change. Figures 6.6 and 6.7 show the relationship between the call's gamma, the underlying asset price, the time to maturity and the option's volatility. The gamma is highest for an at-the-money option and decreases either side when the option gets in- or out-of-the-money. The gamma of an at-the-money option rises significantly when the volatility decreases and the option approaches its maturity date When the gamma is positive, an increase in the underlying asset price yields a higher delta (Table 6.16). The adjustment of the position entails the sale of more units of the underlying asset. When the asset price falls, the delta decreases and the adjustment of the option position requires the purchase of more units of the underlying asset. Since the adjustment is made in the same direction as the changes in the market direction, the monitoring of an option position with a positive gamma is easily done When the gamma is negative, an increase in the underlying asset price reduces the delta. The adjustment of a delta-neutral position implies the purchase of more units of the underlying asset. When the asset price decreases, the delta rises. The adjustment of the option position requires the sale of more units of the underlying asset. Hence, the adjustment of the position implies a rebalancing against .the market direction which produces some losses. In general, the option gamma is a decreasing function of the time to maturity (Table 6.17). The longer the time to maturity, the weaker the gamma and vice versa. When the option approaches its maturity date, the gamma varies significantly.

POSITIONS

Parameters K = 100 ° br

o.12-

T

0.10-

=

o.S

0.06E O

0.04550 0.040.30

0.40 35

0.025 a: Volatility

_

o.oo

80 84 88 92 96 100

4

112 116120

0.05

S: Underlying asset price

FIGURE 6.7

The Gamma

and Volatility

TABLE 6.16

Adjustment

of a Hedged

Position

and

the Gamma

Adjustmento¶the Hedged Position and the Delto

Parameters K 100 r 8% b = 10%

OF OPTION

AND MANAGEMENT

MONITORING

Ef¶ect on the Position

Position

Gammo

Long options

Positive

Market-up: sell more S Market down: buy more S

Easy adjustment,

Short options

Negative

Market up: buy more 5 Market down: sell more S

Difficult adjustment,

=

0.050¯

=

0.045-

a=25%

0.040¯ 0.035-

yields profits

yields losses

0.0300.025-

0.020-

1.0 0.9 0.8

0.0150.010

0.60.7

-

0.005¯ 0.00080 84 88 92 0.2 96 100 104 108 112 116 1200.1 S: LJnderlying asset price

FIGURE 6.6

The Gamma

and

Time to Maturity

0.5 0.4 0.3

TABLE 6. I 7

T: Time to maturity

Gamma

The Variations

in Gamma

and the Time

to Maturity

Longer Maturity

Short Moturity

Near Moturity

Low

High

High for at-the-money low for out-of-the-money

Effecton the option position

Low

Easy adjustment position

of an option

options; very options

Delta very sensitive to the asset price; gamma used with care

142

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

The variations in the gamma for in-, at- and out-of-the-money options are explained in Table 6.18. The management of an option position with a positive gamma is simple. When the market rises, the investor becomes long and must sell some quantity of the underlying his delta-neutral position. When the underlying asset decreases, the asset to re-establish investor becomes short and must buy more units of the underlying asset to re-establish his delta-neutral position. This yields a profit The management of an option position with a negative gamma is more difficult when the underlying asset's volatility is high. When the market rises, the investor becomes short and must buy some quantity of the underlying asset to re-establish his delta-neutral position. This produces a loss. When the underlying asset decreases, the investor becomes long and must sell more units of the underlying asset to re-establish his delta-neutral position. This yields a loss.

Examples If E 30 an increase in the asset price by $1 yields an increase of the delta and produces a gain of $30. If f -30 an increase in the asset price by $1 yields a decrease in the delta and produces a loss of $30 145, A When the underlying asset price S 32, an increase in 97 and f 146) gives a new delta: A' 129. However, a A + T 97 + 32 S by $1 (S decrease in S by $1 gives a new delta: A' A - T 65. The delta-neutral strategy implies the sale of 65 units of the underlying asset. When S 135 and T -26, an increase in S by $1 gives a new 360, A 135 - 26 109. A decrease in S by $1 gives a new delta: A + F delta: A' 161. The delta-neutral strategy implies the sale of A' A- F 135 + 26 161 units of the underlying asset. It is possible to have a negative delta and a negative gamma. For example, when S and T 378, A -8, an increase in Sby $l gives a new delta: 167 - 8 -175. The delta-neutral strategy implies the purA' A+ T chase of 175 units of the underlying asset. A decrease in S by $1 yields a new A delta: A' F -167 + 8 -159. The delta-neutral strategy implies the purchase of 159 units of the underlying asset. =

=

=

=

=

=

In general, one should be careful when adopting a positive gamma strategy since the option position loses from its theta when the market is not volatile. In this context, it is volatile, a position with a better to have a negative gamma. However, when the market is underlying the asset positive gamma allows profits, since the adjustment requires buying the market rises. when selling it when the market falls and The gamma of an option position with several assets is Foss

For delta-neutral strategies, change rapidly and a negative

DA, =

ni

-

BA2

+ n2 -

BA;

.+

+

.

n,

.

and Managing

Monitoring

the Theta

The theta is given by the option's partial derivative with respect to the remaining time to maturity. Figures 6.8 and 6.9 show the relationships between the call and put prices and the time to maturity. As the maturity date approaches, the option loses value. An option is expressed as a function of the number of points lost a wasting asset. The theta is often each day. A theta of 0.4 means that the option loses $0.4 in value when the maturity date is reduced by one day. In general, the gamma and the theta are of opposite signs. A high positive gamma is associated with a high negative theta and vice versa. By analogy with the gamma, as a high gamma is an indicator of a high risk associated

=

Parameters

35

=

K br

-

=

=

-

-

conditions

-

-167

=

=

(20)

-

a positive gamma allows profits when market gamma produces losses in the same context.

=

=

=

BA =

=

=

=

143

POSITIONS

=

=

=

=

=

=

=

=

=

=

OF OPTION

AND MANAGEMENT

MONITORING

=

25

¯

20

-

-

1510

°/

25%

_

=

100

_1.0

0.9 0.8

-

0.7

TABLE 6.18

Effect of the Gamma Out-of-theMoney Options

Gamma

Near zero

Effect on the option position

Weak

on an Option

Position

5 In-the-Money Options

At-the-Money Options High for shorter longer maturity

maturity; stable

Gamma is fundamental the option position

for

to managing

Near zero for near maturity

0

0.6 0.5 T: Time to ma 0.4 0.3

-

-

80 84 88 92 96 100 104108

0.2

0.1 112 116 120

S: Underlying asset price

Weak

FIGURE 6.8

The Call Price and the Time

to Maturity

:Ry

144

OPTIONS,

FUTURES

AND EXOTIC

AND MANAGEMENT

MONITORING

DERIVATIVES

OF OPTION

POSITIONS

145

-1500

When f for an option position, theta may be 10 000, i.e. a gain of $10000 each day. However, the position implies a loss on the underlying asset since the adjustments are made against the market direction when the gamma is negative. =

Parameters 0 20 18

-

b

_

=

10%

-

a=25% 16

14 12 10

6 4 2 0

-

-

The theta of an option position with several assets is

-

-

Oposition

0.80.9

-

,

0 5 04

-

+ n3

.

(21)

ny

.

T Time to maturity

y Parameters

100 104 112

1Ñ6 120

0.1

S: Underlying asset price

6.9

n2

20 3

80

FIGURE

ni

O7 06

-

-

.+

=

25

The Put Price and the Time to Maturity

20

B

with the underlying asset price, a high theta is an indicator of a high exposure to the passage of time. An at-the-money option with a short maturity loses value much more than a corresponding option on a longer term. The theta of an at-the-money option is often higher than that of an equivalent in-the-money or out-of-the-money option having the same maturity date. Figure 6.10 shows the relationship between the theta, the asset price and the time to maturityThe option buyer loses the theta value and the option writer the theta value (Table 6.19).

15

K r b

-

a

10% 25%

-

19.0

o.e

'

0.60.7

-

o.5

'gains'

g

=

=

100

8%

-

10

5

=

=

0.4 0.3 0.2

_

so e4

88 92 96 100104108112 116120

T: Time to maturity

0.1

S: Underlying asset price

Examples

FIGURE 6.10

A theta of $1000 means that the option buyer pays $1000 each day for the holding of an option position. This amount profits the option writer. The theta remains until the last day of trading. When a position shows a positive gamma, its theta is negative. In general a high gamma induces a high theta and vice

TABLE 6.19

The Theta

The Option

versa.

For example, when C 1500 theta may be -$10 000, i.e. a loss of $10 000 each day for the option position. This loss is compensated by the profits on the positive gamma since the adjustments of the position imply sellmg (buymg) more units of the underlymg . asset when the market rises (falls). =

Value and the Theta

--

Loss in time value Effecton an option position

Longer Moturity

Shorter Moturity

Near Moturity

Low

High

Very high

Needs passive monitoring

Profit for the seller, loss for the buyer

Profit for the seller, loss for the buyer

146

OPTIONS,

Monitoring

and Managing

FUTURES

AND EXOTIC DERIVATIVES

,

,...n...a

'T OF OPTION

emmam

147

POSITIONS

the Vega

Parameters

The vega is given by the option's derivative with respect to the volatility parameter. It shows the induced variation in the option price when the volatility varies by 1%. The vega is always positive for call and put options since the option price is an increasing function of the volatility parameter (Figures 6.11 and 6.12). A vega of 0.6 means that an increase in the volatility by 1% increases the option price by 0.6. For a fixed time to maturity, the vega of an at-the-money option is higher than that of an in-theoption. money or out-of-the-money Figure 6.13 shows a decrease in the vega when maturity is shortened, i.e. a longer-term option is more sensitive to the volatility parameter than an otherwise identical shorterterm option. Since all the option pricing parameters are observable, except the volatility, buying (selling)options is equivalent to buying (selling)the volatility. When monitoring an option position, a trade-off must be realized between the gamma and the vega. Buying options and hence having a positive gamma is easy to manage. However, when the implied volatility falls, the investor must adopt one of the following two strategies (seeTable 6,20):

K = 100

'

°/

35

-

T 25

2

Ë o

o.5

=

-

20 15

-

¯

19.0

0.8

10

-

5

-

601

O

O 0.5 0,4 a: Volatility 3

/

.

80 84 88 92

T -'

im

112 116

120

S: Underlying asset price •

He can preserve a positive gamma if he thinks that the loss due to a decrease in will be compensated by adjusting the gamma in the market direction. He can sell the volatility (options)and re-establish a position with a negative gamma. volatility

•

The Put's Price

ggi ggligggt

and

the Volatility

Parameters Param1e0ters

45-

r=B% b 10% T 0.5 =

40

35 30

b=10% 35

-

a

30

-

15

1.0

-

0.9 0.8 0.7

1.0

15

0.9 0.8 0.7 0.6 0,5 0.4 a: Volatilitý 0.3

10

0

-

_

80 84 88 92 96 100 104108

1200.10.2

10

5

¯

0.6 0.5

-

-0.4

0

-

80 84 88 92 96 100 104108

112

S: Underlying

S: Underf ying asset price

FIGURE

25%

-

25 -

-

20

5

=

=

6.1 I

The Call's Price and the Volatility

FIGURE 6.13

The Vega

0.3 0.2 112 116

asset price

1200.1

T: Tirne to maturity

MS

OPTIONS,

TABLE 6.20

Effect of the Volatility Volatility

Options

FUTURES

AND EXOTIC

on a Portfolio

DERIVATIVES

MONITORING

AND MANAGEMENT

The vega of an option position

of Options

Ef¶ect

Op

Vega Long

Long

Profit

Short

Short

Loss

(loss)when the (profit)when the

volatility rises volatility

(falls) rises (falls) 6.3

In this case, the losses due to the adjustments compensate for the decrease in volatility.

of the delta must be sufficient to

Tables 6.21 and 6.22 show the effects of the vega on the option position with respect to time to maturity and option type. The impact of the vega on an at-the-money option is

highest.

Example When the vega is $500, this means that a rise in the volatility by 1% produces a profit of $500. However, when the volatility decreases by 1%, this implies a loss of $500 When the vega is $500, an increase in the volatility by 1% implies a loss of $500 and a decrease by 1% yields a profit of $500.

TABLE 6.2 I

Effect of the Vega with

Respect

e,

to Time to Maturity

Longer Maturity

Shorter Maturity

Near Maturity

Vega

High

Low

The implicit volatility is affected by factors other than time

Effect on the position

Very sensitive to volatihty

Little sensitivity to volatility

The implicit volatility is affected by factors other than time

=

n,

OF OPTION

with several assets

Ov;

+ n2

-

149

POSITIONS

is

+ n3

Ov3 -

+

.

.

OF VOLATILITY

THE CHARACTERISTICS

,

+ ny

Üvj

-

(22)

SPREAD

theta and As a simple standard option, a spread is also characterized by its delta, gamma, positions option as a vega. These sensitivity parameters allow the investor to manage his of strategies based consequence of the changes in market conditions. The implementation the use of delta-neutral strategies to be able to predict on volatility spreads often implies the variations in the market conditions. When the changes in the underlying asset value give more value to the spread, the negative when the variations gamma is positive. On the other side, the spread's gamma is value. spread underlying the price reduce the in asset Since the effects of the changes in the underlying asset price and the time to maturity negative theta operate in opposite directions, a spread with a positive gamma exhibits a various sensitivity effect of the the summarizes on parameters Table and vice versa. 6.23 spread strategies. The investor can implement delta-neutral strategies when he has no prior anticipations resort to bullish and bearish spreads as to where the market is going. However, he can abilities. market timing This leads him to be long or short his about confident when he is the underlying asset. when When the options used are over-valued (accordingto the investor), for instance volatility normal level), and the its volatility historical high is the implied (withrespect to the investor can sell some puts if the market rises and some calls if the market falls. when When the options are under-valued (againaccording to the investor), for instance normal level), volatility and its the implied volatility is low (withrespect to the historical the investor can be long calls and puts when the market falls.

TABLE 6.23

Characteristics

of Volatility

Spreads

¯

Nature of the Spread

Short a call spread Short a put spread TABLE 6.22

Vega

Option position

Effect of the Vega on the Option

Price

Out-of-the-Money Options

At-the-Money Options

to-the-Money Options

Depends on the time to maturity

For a given maturity, the vega is higher

to maturity

Low

High

Low

Depends on the time

Long a straddle Longa strangle Short a butterfly Long a call (ratio)spread Long a put (ratio)spread Short a straddle Short a strangle Long a butterfly

Position in Delta

Position in

0 0 0 0 0 0 0 0 0 0

+

Gamma +

Position in Theto -

-

Position in Vego

OPTIONS,

I50

FUTURES

AND EKOTIC

DERIVATIVES

SUMMARY

MONITORING

• •

This chapter presents the main Greek-letter risk measures, i.e. the delta, gamma, theta and vega in the context of the European analytical models developed in the previous chapter. These risk measures are simulated for different parameters which enter the option formulas. Then the magnitude of these risk measures is evaluated in connection with the management and the monitoring of an option position. The knowledge of the changes in these risk parameters is necessary for the manage' ment of an option position and for the determination of the profits and losses associated with the position. To put it differently, the pricing of a European call option can be viewed inputs (the underlying asset and the treasury bill) and a production as requiring technology (the hedge portfolio and the Greek-letter risk measures). In a B-S world, by tracking continuously the hedge ratio (beingdelta-neutral), the investor makes sure that the option. In the duplicating portfolio does mimic the call option, namely does the course of doing so, the investor controls his production costs and protects his mark-up on the option. However, these risk measures depend on the theoretical model used for the valuation and the management of the option position. There is what one could call a technological risk. This position must be adjusted nearly continuously in response to the changes in

AND MANAGEMENT

6.A.1

Call Sensitivity

Parameters

N(di)

F

• • • • • • • • • •

•

•

•

•

•

son(d,)

oc

BT

Rhoc

Put Sensitivity

n(di)

BS ¯ So

Da

FOR DISCUSSION

What is an option delta? What is the charm? What is the gamma? What does spread mean? What does color mean? What does theta mean? What does vega mean? What does Rho mean? What does elasticity mean? Why is the knowledge of Greek-letter risk measures important? How does the delta change in response to the changes in the option valuation parameters? How does the gamma change in response to the changes in the option valuation parameters? How does the theta change in response to the changes in the option valuation parameters? How does the vega change in response to the changes in the option valuation parameters? asset in response to changes in the underlying How is a hedged portfolio adjusted price? How is a hedged portfolio adjusted in response to changes in the volatility parameter?

tM

B-S Model

ve -

•

RISK MEASURES

APPENDIX 6.A: GREEK-LETTER ANALYTICAL MODELS

conditions.

POINTS

IN

POSITIONS

How is a hedged portfolio adjusted in response to changes in the time to maturity? What are the main characteristics of volatility spreads?

'produce'

market

OF OPTION

=

rKe-rN(dy)

(23)

2R -

=

SÖn(di) K(t*

"'

- t)e

N(d2)

Parameters

A

f

Ae - I

=

n(di)

=

=

BS

So

Son(d ) _

UT

_

Rho

=

=

Or

-

2

rKe¯"N(,di)

SRn(d ) -K(t*

- t)e

"

N(d2)

-

ear

(24)

152

6.A.2

OPTIONS,

Black's

Call Sensitivity

FUTURES

AND EXOTIC

DERIVATIVES

Model

e

N(d,)

BA, BS

c¯

Oc

A "

=

8 BS

n(di)

rN(-di)+rKe¯"N(-d2)+

_ Se

'on(di)

+ rSe¯"N(d,)

2

-

rKe¯"N(d2)

v

on(A

(28)

=

Br

TKe

N(-di)

and Barone-Adesiend

Merton's

Whaley's

Model

'

Call Sensitivity

BA

Parameters

n(di)

Ac

o n(di)

Se

N(-di)

e'

N(d

) n(di)

rKe "N(-di)

Op

8

Model

Ke--r Da -

Rhoe

s

Oc - -

N(da) + rKe-,rN(d2) +

(b - r)Se'

BT

v

and Kohlhagen's

=

(26) - rSe

n(di)

S

=

'

n(da)

Ke

-8¤

Rho'

n(di)

l

Op

Ke

(25)

6.A.4

A - Ae - e

Call Sensitivity

e-

Sa

Op

Agt gensitivityParameters

Garman

) + 1] r

*

e

Oc

6.A.3

r[N(-d

-e

Sonn(ds)

Oc - -BT -

=

=

y

Se ve - Ba -

vp

153

OSITIONS

put Sensitivity Parameters

Parameters Ae

OF OPTIOM

AND MANAGEMENT

MONITORING

Ke-cran(d2)

2R

(29)

n(d2)

N(di)

- - TSe

Parameters Put SensitMty he

, =

E

e

Parameters

r N(di)

A

BAe

e

OS

Sa

n(di)

Oc Be=-=-r*Se

rN(di)+rKe¯"N(d2)+

BT

=

-et

[N(-d e
BA, BS ,Ke

(27)

8

Op -

n(d

So

Ton(d2)

=

l]

)

(b -

r)Se

Ke

r

) N(-di)

""

ve

Rhoe

=

---

=

Oc Dr

Ke

-TSe

n(d2)

,arN(di)

=

v Rho

Op -Dr

n(d2)

TKe crN(-d2)

rKe "N(-d2)

Ke +

ron(d2)

(30)

154

FUTURES

OPTIONS,

AND EXOTIC

DERIVATlYES

S.A,

02c(S, t)l

Oc(S, t) DS

«Q, O

(31)

DS2

N

-rc(S, t) + rSA +

=

POSITIONS

155

jo2S2F

(32)

=

C

(41)

This equation shows that an outright purchase of the security is equivalent to the replicating strategy which consists of buymg or selbng the amount dictated by the delta. The generalized B-S equation can also be expressed as a function of the Greek-letter risk measures as follows:

or

-8

OF OPTION

N

Using the deñnitions of the delta, gamma and theta, the B-S equation can be writtena, --

AND MANAGEMENT

BETWEEN

APPENDIX 6.B: THE RELATIONSHIP HEDGING PARAMETERS

Oc(S, t) Di ]

giggilTORING

2

N

x

o,a,pçS,Syfy +

(r -

r,)S,A, + 8

=

rC

The use of the charm allows one to write a potentially more separable equationforderivative

or

rc(S, t)

=

8 + rSA+)a2S'T

For a delta-neutral position, the following relationship

rc(S, t)

=

(33)

applies:

8 + (a2S'F

(34)

Using the definitions of the hedging parameters, the Black equation can be written as (Oc(F, i

t)

L 8t

j

82c(F,

=rc(F,t)-jo'F

t)

(35)

OF-

or

8

rc(F,

=

t)

o2F2F

(36)

Using the defmitions of the delta, gamma and theta, the Merton and BAW written as

a2S2

82c(S, t) Sc(S, t) t) + bS - rc(S, + 8 OS DS

=

(1987)s

[uation can be

0

(37)

or

rc(S, t) For a delta

position, the following

=

8 + bSA+¼o'S2F

relationship

rc(S, t)

APPENDIX BETWEEN

=

(38)

applies:

(a2S2F + 8

(39)

6.C: THE GENERALIZED RELATIONSHIP THE HEDGING PARAMETERS

In this appendix we use the following notation: S, is the price of asset i, C is the value of the contract, o, is the volatility of the underlying asset S,, pg is the correlation coefficient between assets S, and Sy, r is the instantaneous rate of return on accounting commodity, ry is the instantaneous rate of return on asset i. If we denote by H, the charm associated

8

=

with asset i, the theta given in German(1992)ts S, H,

The proof of this result relies on the repheation

with

H,

ojo pyS,S,fy + =)

=

equation which

-' states that

(Al

S,[H - r,A,]

=

0

(42) assets:

(43)

7 Extension to American Options: Dividends and Early Exercise

CHAPTER

OUTLINE

This chapter is organized as follows: 1. Section 7.1 is an introduction to the general issue of pricing of American options with and without distributions to the underlying asset. 2. Section 7.2 studies the valuation of American spot and futures options in the context of a constant proportional dividend rate. 3. Section 7.3 deals with the valuation of American options when there are discrete distributions to the underlying asset.

INTRODUCTION

«

There are several extensions to the basic Black and Scholes model. We have already seen the first extension by Black (1976)who took into account the specificities of futures contracts. The second extension was made by Garman and Kohlhagen (1983)who derived analytical valuation formulas for European options on currencies. In the same year Grabbe (1983)implemented a similar approach to that used in Black (1976)to provide the value of options on foreign currencies. Merton (1973a) indirectly, and Barone-Adesi and Whaley (1987),hereafter BAW (1987),provided the values of European commodity options and commodity futures options. All these models apply only to European options. Since all the proposed analytical models deal with the pricing of European options in the absence of discrete distributions to the underlying assets, the extension of the analytical European models proposed by B-S (1973), Black (1976), Garman and Kohlhagen (1983),Merton (1973a) and BAW (1987) to the valuation of American options is covered in this chapter. The two important extensions and contributions to the literature of American options are those of Merton (1973a) and Geske (1979a, on the valuation 1979b).

lš8

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

The pricing of American options was first analyzed in Merton (1973a)who showed the difficulties in obtaining closed-form solutions when there are discrete distributions to the underlying asset. However, he provided closed-form solutions for American options when the time to maturity is infinite· Geske (1979a, 1979b) provided analytical formulas for the valuation of options on options or compound options. He used a valuation by duplication technique for the pricing of American options. A compound option can be defined as an option on the firm's equity. For a levered equity firm (a firm with debt in its capital structure), equity can be seen as a call on the value of the assets. This was first noted by B-S who considered the option on equity as an option on the value of the firm's asset. Geske's approach is interesting since it allows the valuation of American options and includes the question of dividend. American option pricing models have been proposed by several authors. However, given the difficulties in obtaining closed-form solutions, it became quite natural to resort techniques which to analytical approximation models, binomial methods and numerical will be discussed in another chapter. When there are continuous distributions to the underlying asset, the literature on the valuation of American options provides some analytical approximation formulas. BAW (1987)presented simple analytical approximations for the pricing of American options on commodities and commodity futures contracts. These formulas are accurate and computationally more efficient than compound-option pricing models, binomial models and finite difference methods. The approach relies on a quadratic approximation method similar to that used in McMillan (1986)for the valuation of the American put option on a non-dividend paying stock. However, the formulas apply only for a constant proportional rate. Adopting the same approach, Chateau (1990)presented analytical approximations for the valuation of American cap options implicit in credit lines. The model applies to fixed and variable rate commitments with a covenant stípulating a cap on the commitment rate When there are discrete distributions to the underlying asset, the valuation of American options is more complex. In this context, models were proposed by several authors including Roll (1977), Geske (1979a,1979b), Whaley (1981)and Whaley (1986)for call options. Put formulas were proposed by Johnson (1983),Geske and Johnson (1984), Geske and Shastri (1985),Blomeyer (1986)and BAW (1987),among others. Most of these models, if not all, are based on the concept of compound options.

EXTENSION

TO AMERICAN

DIVIDENDS

OPTIONS:

AND EARLY EXERCISE

159

asset. The non-existence of a put-call parity theorem for American options implies a distinct treatment for call and put options. The problem is different according to the is pattern of distributions to the underlying asset. In fact, if the dividend stream continuous, it may not induce the optimal early exercise of American options. However, when the dividends are discrete, this may induce early exercise of American options.

7.1.1

Early Exercise

of American

without

Distributions

Early Exercise

Calls

When there are no distributions to the underlying asset, there is no incentive to exercise maturity date. Hence the value of an American call is an American call option before its equal to that of a European call. This result is due to theorems I and 2 in Merton (1973a)where it is shown that m the absence of dividend payments, an American call will never be exercised prior to expiration. In this context, the value of an American call is equivalent to that of a European call when interest rates are constant. The intuition of this result is simple. In the absence of dividends, the option is worth than because of its time value. Killing the option would mean more value. The investor is better off m pocketing its intrinsic value while losing its speculative 'alive'

'dead'

selling the option rather than killing it.

Early Exercise with Continuous

Distributions

When dividends are paid continuously at a constant rate of d donars per unit time, and is that when the interest rate is constant, a sufficient condition for no prematmeesercise )) r

This condition shows that the call strike price must be greater continuous dividend rate to the short-term riskless rate.

than the ratio of the

Early Exercise with Discrete Distributions

7.1

THE VALUATION OF AMERICAN GENERAL PROBLEM

OPTIONS: THE

The option pricing literature has evolved since the work of B-S (1973)and Merton (1973a). Even if models now exist for the valuation of European and American call and put options on a variety of underlying assets and commodities, includmg common stocks, bonds, agricultural futures contracts and financial futures, there are still no known analytical solutions for options written on assets that have known discrete cash payments during the option's lifeTwo problems arise in the valuation of American options. The first is associated with the possibility of early exercise. The second is linked to the distributions to the underlying

.;

.

1, n, When the amounts of dividends per share d, paid at different dates ry, for j are known and the interest rate is constant, then a sufficient condition for no premature exercise is =

K>

P(y

)

d(r,)

.

(2)

where P(r) is the price of a discount bond paying one dollar in r years. This condition shows that the net present value of future dividends must be less than the present value of earnings from investing the strike price for r periods. The intuition of this result is simple. In fact, if the losses from dividends are less than the gains from

160

OPTIONS,

FUTURES

AND EXOTIC

EXTENSION

DERIVATIVES

investing the required funds to exercise the option and hold the stock, the option is exercised When the dividend per share is D(S, t), the B-S differential equation is slightly modified. In fact, the instantaneous rate of return is no longer µSdt but rather (µ - D(S, t)/S)dt. This gives the following partial differential equation, which first appeared in Merton (1973a):

o'S2

\ßS2

+ [rS - D]

- rC - 0 -

St)

ßS/

(3)

C(S, r, K) is the American call value. This partial differential equation must satisfy the following boundary conditions: .

C(0, r, K) C(S, 0, K) C(S, r, K)

=

0

max [0, S - K] max [0, S - K]

=

>

.

'dead'

(4) (5) (6)

'alive'.

-

C(I(r),

r, K)

I(r) - K

=

=

h

(7)

r, K) satisfies the partial differential equation for GSG

0

AND EARLY EXERCISE

161

equal to 1. Intuitively, this condition corresponds to the point of tangency between the call function and its intrinsic value, since only in-the-money options are exercised. The proof of condition (10) is relatively simple. In fact, for the function f(x, I), differentiable and concave with respect to its second argument, the total derivative with I is given by respect to Ialong the boundary x =

df/dI

=

dh/dI

fi(I,

=

I) +

f2U,

I*, f2(X, I Û. When the function has a maximum for a level I I*) fi(I*, I*) dh/dI 1. This I and h K, so fi(I dh/d! Since df/d! is the proof that the derivative must be equal to 1. The solution to this problem gives the value of the American call when there are dividends. Samuelson (1972),McKean (1969)and Merton (1973a)analyzed this boundary' problem and did not provide solutions for a finite-life option. However, a solution for a infinite-life option (a perpetual warrant) was provided in Merton (1973a)

C(S, r, K)

=

maxy

f(S, r; K, I)

(9)

'high

BS that the option's

r, K)

=

-

,

=

(equation46, 7.1.2

p. 172).

Early Exercise

of American

Puts

It is important to note that the valuation of American calls is simpler than the valuation of American puts with and without discrete cash distributions to the underlying asset. While European and American calls have the same value when there are no distributions to the underlying asset, this result does not hold for European and American puts. In fact, while there is no incentive to exercise an American call option before its maturity date in the absence of dividends, there is always a probability of premature exercise of an American put. This is due first to the arbitrage condition, according to which the value of an American put must be greater than its intrinsic value at each instant, and second to the fact that the value of a European put with an infinite time to maturity or is

zero.

(8)

The optimal I(r) is independent of the current underlying asset price, and the following contact' condition, which is at the boundary, must be accounted for: SC(I(r),

=

=

=

a perpetual put

I(r)

Since I(r) is an unknown function of time, the structure of the problem is complex. Indeed, the boundary conditions are semi-infinite and time dependent. The value of I(r) must be determined as part of the solution. As shown in Samuelson (1972)and Merton (1973a), this is rather a difficult problem since the unknown function I(r) must be determined from the behavior of the call's holder, who maximizes the option value at each instant, by adding a condition of regularity. To make this point explicit, consider a function f(S, r; K, I(r)) which is a solution to this problem for a given I(r), or:

This means

DIVIDENDS

'free

Condition (4) shows that the option is worthless when the underlying asset is zero for any time to maturity r. Condition (5) indicates that the option price is equal to the greater of zero and its intrinsic value at the option maturity date. Condition (6) is an arbitrage condition. It indicates the American option value at any time during the option's life. It shows also that at each instant r, there is a positive probability of early exercise. This implies that there exists a certain level I(r) of the underlying asset price such that for each S> I(r), the option is worth more than Since the value of an immediate exercise is (S K), the structure of the problem implies the additional condition

C(I(r),

OPTIONS:

=

where

where

TO AMERICAN

Early Exercise without

Distributions

Since there is no put-call parity relationship for American options, American puts must be treated separately. The problem with the rational pricing of European or American put options is somewhat different from that of European or American calls. This results from the work of Merton (1973a)who showed that an American put option can be exercised early even in the absence of distributions to the underlying asset. The American put must satisfy the B-S (1973)or Merton's (1973a)partial differential equation: 2S2

- rP

rS

1

partial derivative with respect

(10) to the underlying

asset price

is

0

(12)

under the following boundary conditions: P(oo,

=

=

I,

P(S, 0, K)

=

P(S, r, K)

>

K) max max

=

0

[0, K [0, K

-

-

S] S]

(13) (14) (15)

f 62

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

(13) shows that the put is worthless when the underlying asset price tends to infmity. This result is rather intuitive. In fact, the put intrinsic value is given by the difference between a constant (the strike price) and the innnite underlying asset price. Condition (14)is standard and gives the put value at the option's maturity date. Condition (15) shows that it is sometimes optimal to exercise the put before its maturity date. This happens when the underlying asset price tends to zero. In this situation, it is interesting t° exercise the put as soon as possible to benefit from the investment of the strike price until the option's maturity date. From the analyses of McKean (1969),Samuelson (1972),and Merton (1973a),there is no closed-form solution for a finite-life put when the above boundary conditions are applied to the partial differential equatíon. However, for an mfinite time to maturity, a closed-form solution is given in Merton (1973a)(equation 52, p. 174). Condition

The structure of this problem implies the existence of a certain level of the underlying asset I(r), referred to as the critical asset price level, for which the exercise depends on the maximization behavior of the put's holder. The determination of this critical level imphes adding the followmg condition on the put's derivative with respect to the underlying asset: .

-

.

.

-

8P(I*,

oo,

K) =

dt

This condition is Samuelson's

-1

'high

contact' boundary conditionThe analysis of this problem (the free boundary problem) by Samuelson and McKean (1969)and Merton (1973a)allows the derivation of the put's value when the time to maturity is infinite·

Early Exercise with Continuous

Distributions

When there are continuous distributions to the underlying asset, the pricing of American puts is rather difficult. However, an interesting approach was proposed in BAW (1987)for the valuation of American options. This approach, referred to as the quadratic approximation method, will be presented in detail.

TO AMERICAN

EXTENSION

OPTIONS:

DIVIDENDS

AND EARLY EXERCISE

163

(1985),Blomeyer (1986) and BAW (1987)among others. However, most of them do not work as well as binomial models and finite difference methods.

7.2

VALUATION OF AMERICAN COMMODITY OPTIONS, FUTURES OPTIONS WITH DISTRIBUTIONS CONTINUOUS

futures are of the American type. Most options written on commodities and commodity Hence, an early exercise premium is embedded m Amencan call and put prices. Analytical solutions for the American option pricing problems with several dividends have not yet been found. efficient. The quadratic approximation method used by BAW (1987)is computationally It provides an accurate, inexpensive method for the valuation of American call and put options traded on commodities and commodity futures. .

.

.

.

.

7.2.1

American

Commodity

Options

The following analysis applies to commodity options for which the cost of carrying the underlying commodity is a constant proportional rate, to commodity futures options and to stock options. The model used here is an extension of the Merton (1973a)and BAW (1987)model to American options. The pricing of American options involves the valuation of the early exercise premium attached to the possibility of early exercise. It is possible to show that, under certain conditions, American options should be exercised early. In fact, when the cost of carrying the underlying commodity, b, is less than the riskless interest rate r, and the option becomes deep-in-the-money, the values of N(d,) and N(d2) approach 1. The European Ke ". Since an American option may be call value tends towards the quantity S e"' exercised for its intrinsic value, S K, this value may be higher than S e"' - Ke When b r, there is no possibility of early exercise for an American call option. This argument does not hold for American put options since for puts, as shown by Merton (1973a),there is always some probability of early exercise. 0, then all formulas Since the cost of carrying any futures position is nil, i.e. b proposed here can be used to price commodity futures options by substituting the futures 0. Also, when the option's underlying price F for the commodity price S and setting b paying stock, b r and the formulas can be applied to stock asset is a ñon-dividend options. European and American option values must satisfy the following partial differential equation: '

-

'

=

Early Exercise with Discrete

Distributions

When there are no dividends, the American put option may be exercised early. In fact, since interest income can be earned on the exercisable proceeds of the option when exercised, this is suffìcient to justify early exercise. When the American option holder delays exercise, this means that he forgoes the interest income on the exercisable proceeds. When there are dividends, the American put option holder must compare the effect of dividends on the put value and the interest income. If the holder does not exercise his put, he forgoes the interest income. If he exercises before the ex-dividend date, he will not profit from the increase in the exercisable proceeds when the underlymg stock goes exdividend. There is a trade-off between the interest income and the dividend. Hence, between dividend dates, the option holder is always in a dilemma. Some interesting approximations of American values are given in the literature. These approximations include Johnson (1983), Geske and Johnson (1984), Geske and Shastri

=

=

=

2S2

82C(i 0 OS2

+ bS ,

BC BS

,

O

+

BC(i - St

0

applies This equation to American options, European options premium, This premium is given by the difference in value European option values, i.e.

-

rCCL 0 -- 0

(KB

and the early exercise between American and

Ec

=

AND EXOTIC

FUTURES

OPTIONS,

164

DERIVATIVES

C(S, t) - c(S, t)

(17)

P(S, t) - p(S, t)

(18)

EXTENSION

TO AMERICAN

OPTIONS:

-(N q2

DIVIDENDS

(N - 1)2 +4M/k 2

- 1) +

=

and =

o

Using this notation,

824¶ Q Ogg 0 + bS OS OS2

2

=

(16) can be

equation

la2S2 Let M

rewritten

2r a2, N

2b/a2

=

and r

for

the early exercise

&(S, 0 St

+

(19)by (2/o

T - t. Multiplying

=

2)

0

=

S2

+ NS

BS2

Be(S, t)

M

Be(S, t)

BS

r

or

- Me(S, t)

S* and rearranging

=

0

(20)

Now define the early exercise premium as

s(S, k)

k(r) f(S, k)

(21)

=

S2

1- e

",

S*

- K

c(S*, T)+-{l

=

"

1- e

(30) can be determined

price which

-el"¯'"N[di(S*)]}

(31)

q2

The value of S* corresponds to the value of S above which the call's value is equal to its value can be determined by exercisable proceeds, S a Newton-Raphson - K. This procedure or usmg the efficient algorithm presented in BAW (1987). When b is less than r, the American call value is given by formula (27). Otherwise, when b is greater than or equal to r, C(S, T) c(S, T) smce the call will never be exercised early. In the same context, the value of an American put option presented by BAW (1987) is .

.

P(S, T)

(22)

K(S T) + Ai(S/S*)

=

(32)

with - Mf

If we choose k(r)

kf + kk

=

(20),NS

When substituting

=

=

and

k

2b/a2, k

=

165

(29)

.

=

Hence =

N

(19)

gives 82e(S, t)

2r/o2,

=

In this formula, S* stands for the critical commodity iteratively using the following system:

premium as

- re(S, t)

M

AND EARLY EXERCISE

equation

82f

=

Ai

(23)

Mf

NS

\8S2)

(1 -

-

-(N

M(1

k

=

-

q -

=

(24)

0

order to eliminate

(24) in

0 (k approaches

k

the

M

=

2r/r;2,

1)

N

e"

'"

N[-di(S

)}

(2N - 1)2 + 4M

-

2b/<>2, k

=

=

(33)

k

1- e

(34)

*

(35)

price S* is determined iteratively using the following system:

The critical commodity

K - S*

1). Hence,

=

p(S*, T) -

41

{l

-

e" ""N[-di(S*)]}

(25)

0

=

\8S)

8

k)

Now, the quadratic approximation is applied to equation Sk]. term (1 k)M[8f 0 (oc), [8f/ßk] approaches In fact, as r approaches equation (24)becomes a second-order differential equation:

S2

0

(23)becomes

8f \8S)

Ò/

1

1+

Examples

The above equation presents two linearly independent solutions of the form

(S)

(26)

aiSo + a2S

=

Solving this equation, the American call option value presented in BAW

c(S, T) + A2(gf S K

C(S, T) _

-

F

ŒS

<

if S

>

f S*

(1987)is

=

(27)

with -[1

2

=

S* -

e"'

'

N[di(S*)]]

.

and TABLE 7.1 European Prices: K 100, r 0.08, T

(28)

=

=

American Call Option 0.25, b 0.04, « 0.2 =

Underlying asset

European call

American call

price

price, c

price, c

90

0.84 4.432 I 1.45 I 20.891

0.841 4.441 I L662 20.898

100 I I0 I20

=

criticalunderlying asset

priceS*

214.952 214.952 2 I4.952 2\4.952

(36)

f 66

FUTURES

OPTIONS,

AND EXOTIC

DERIVATIVES

EXTENSION

TO AMERICAN

OPTIONS:

American Option Call TABLE 7.2 European and 0.2 Prices: K 100, r 0.08, T 0.25, b 0.08, « =

=

=

=

M

European coli

American call

price, c

price,C

Criticol underlying osset price 5*

90 100 i 10 120

0.698 3.909 10.737 19.748

0.705 3.934 10.823 20.009

20.323 20.323 20.323 20.323

and k

=

<72

=

Underlyingosset price

DIVIDENDS

=

Underlying asset

=

European

-(N

price, cp

90 100 i 10

9.765 3.455 0.777

l20

0.!12

American put price, P

Criticalunderlying asset price S'

10.80 3.544 0.798 0.!18

87,48 87.48 87.48 87.48

M

=

=

European and American 0.08, 0.08, T 0.25, b =

=

asset underlying

Europeonput

price

price,cp

90 100 110 120

10.50 3.909

«

Put Option 0.2

U2

N

,

=

U2

10.565 3,921 0.938 0.145

83.501 83.50! 83.501 83.50\

(42)

(N - 1)2 + 4M k 2

(43)

and k

A2(F F*)q2. The critical futures price, F*, is calculated

(l

=

- e

")

(44)

- K

c(F*,

=

F* 92

T)

using

the following equation:

{l

e"

-

N[d;(F*)]}

'

(45)

When the futures price is below F*, the American option price is given by the European price plus the early exercise premium. When the futures price is above F*, the American K). option price is given by the intrinsic value, (F In the absence of carrying costs, the formula for the American futures put is

=

anderlying criticat

0.93 0.\44

=

Prices:

American put price, P

(41)

The price of an American call futures option C(F, T) is equal to the price of a European futures option c(F, T) plus a term corresponding to the probability of early exercise

F* TABLE 7.4 K 100, r

if F < F* if F > F*

'"N[d,(F*)]]

-e

- 1)

Ñ2

=

put

price

(40)

with

European and American Put Option Prices: 0.2 0.08, T 0.25, b 0.04, « =

e-,r)

-

c(F, T) + A2(F/F*)* F - K

92

=

(1

=

167

When there are carrying costs, the formula becomes

A2=-[1 TABLE 7.3 K 100, r

AND EARLY EXERCISE

-

osset price S*

p(F, T) + Ai(F K - E

F*)

if F> F* if F é µ*

(46)

with

7.2.2

American

Futures

From the analysis in BAW are no carrying costs is

C(F, T)

F*

--{l

--

A,

-e

=

"N[-d

(F*)]}

(47)

Options

(1987),the

American

c(F, T)+A2(F F - K

futures call option formula F

if F F*

)

when

there

M

2r/o2

=

and k

=

(1 -

e

")

(49)

carrying costs, the formula for the American put is 37) P(F,

T)

p(F, K-F

=

T) + Ai(F

F*)*

if

F > F ifFGF

(50)

with F* A2=-[1-e 42

*N[d(F*)]] 1+

42

l+4M

=

2

k

(38) (39)

--{1

A, qi

=

=

([-(N

y*

-e" - 1) -

'orN[-d,(F*)]} (N - 1)2 + 4M/k]

(51) (52)

Agg

OPTIONS,

M

2r

=

«2,

N

=

The critical futures price corresponding the following equation: K - F

=

p(F

2b/o2

and k

AND EXOTIC

(1 -

=

DERIVATIVES

e-,r)

(53)

to an optimal early exercise is calculated

F* T) - -

,

FUTURES

{l

-

e" "'N[-di(F*)]}

usin 8

(54)

When the futures price is above the critical price, the American futures option price is j given by the sum of the European price and the early exercise premium Ai(F F*)q,. 4 When the futures price is above the critical futures price, the American futures option value is equal to its intrinsic value, (K - F).

Examples

=

=

=

N(-di)

N(-d2)

p

P

90

0.8522 0.5 i99 0. I 826 0.0364

0.8375 0.480 I 0. I57 I 0.0289

10.2300 3.9087 0.9607 0.l7l4

10.6500 4.0 I99 0.9807 0.1795

I00 I l0 120

OPTIONS:

DIVIDENDS

AND EARLY EXERCISE

commitments

Chateau

fo no+

nio

=

is a random

(55)

where è financing is a forward mark-up. For example, a forward rate of9% is composed of an uncertain cost of funds of 7% and a forward mark-up of 2%. Capping the commitment rate and then allowing for early exercise represents the question of valuing American capped variable rate commitments. This cap is introduced by a constant, k > 1, say for example 1.16, in the form cost and E

F*

kfa

=

f,

_

86.530 86.074 86.732 86.664

169

represent an optimal risk-sharing scheme in competitive extends this valuation literature in two ways, first by intro(1990) ducing a cap on the maximum rate charged by a bank on a variable rate commitment, and second by allowing the commitment holder to exercise the cap prematurely. period during which an intermediary writes an American Let [0, T] be the commitment commitment option for a 100 dollars, one year credit. The forward variable rate at the period's beginning is given by

(1987),find that

equilibrium.

k[ëo + mo]

=

This yields a ceiling rate, which applies if the forward We define the differential rate as

=

5

TO AMERICAN

o

and American TABLE 7.5 European Futures 0.25, b 0, 0.08, T 100, r Put Prices: K 0.2 « =

EXTENSION

=

kno - 2, +

(56)

variable

(k -

rate moves above this rate.

1)Ro

(57)

Equation (57) renects the difference between the forward variable rate at date t and the initial capped rate. This rate differential implies the following debt instrument: -

S,

Lexp

=

2, +

[(kno-

(k -

1)mo)(ti

-

t)]

(58)

where

TABLE 7.6 European and American Futures 0.08, T 0.25, b 0 Call Prices: K 100, r 0.2 « =

=

=

value of the borrower's indebtedness if he borrows L, L is the maximum amount to be borrowed, ti - t is the duration of the loan.

S, is the economic

=

=

5

I I10 120

N(di)

N(da)

0

0

0.8428 0.9710

0.8173 0.9635

C

c 360 10.7600 19.7750

0 10.8380 20.0250

F*

Using the same approach as in BAW (1987), Chateau formula for the pricing of the American commitment put:

0 122.02\ 122.021

P(S, t)

Ai

The difference between European and American call option prices corresponds early exercise premium attached to American options.

to the

-(N

Capped

Variable

Loan Commitments

An isomorphic correspondence between loan commitments and put options on stocks has been shown by Ho and Saunders (1983). Some authors, including Thakor and Udell

(59)

SGS

N[-di(S*)]}

- 1) -

(60)

(N - 1)2 + 4k

(6 1)

2 N

7.2.3

if

{l -

=

the following

S>S*

MS0 L- S

=

(1990)presents

=

2r a2, k

di(S*) where r and a correspond

respectively

=

ln (S* =

2r

(o'(l -

L) + (r +

e

"))

(62)

(a2)t

to interest rate and the volatility

(63) of in (S)

L is the

OPTIONS,

170

FUTURES

AND EXOTIC

EXTENSION

DERIVATlYES

strike price and S* is the indebtedness value critical boundary below which the American put should be exercised. The value of S* is given by solving the following system:

S* L - S

=

p(S

,

t, L) - -

{l

-

N[-di(S*)]}

"

7.3.2

(64)

The difference between the American put price and the European put price (P p) corresponds to the value of the early exercise premium. The simulations of this model and the effect of the changes in its parameters on option values are reported in Tables 7.8-7.11 in Appendix 7.A.

TO AMERICAN

Compound

DIVIDENDS

OPTIONS:

Option

AND EARLY EXERCISE

171

Approach

A compound option is an option whose underlying asset is an option. Since an option may be a call or a put, we may fmd four types of compound options: a call on a call, a call on a put, a put on a put, and a put on a call.

~

7.3 7.3.1

Geske's Approach

The theory of options on option, or compound options, was initiated and applied by Black and Scholes (1973),Cox and Ross (1976), Geske (1977),Roll (1977)and Myers (1987)

VALUATION OF AMERICAN OPTIONS WST&i DISCRETE CASH DISTRIBUTIONS Early Exercise

of American

Call on a Call

for a

Options

Most of the traded stock and index options around the world may be exercised before their expiration dates and are therefore of the American type. Early exercise of these options is often induced by the payment of dividends on the underlying asset. For stock options, it may be optimal to exercise a call on the last cum-dividend day to receive the dividend. In this case, the adapted versions of the B-S model proposed by Roll (1977) Geske (1979a, 1979b) and Whaley (1981)can be used. For an index option, when the dividends are distributed fairly evenly over time, B-S type models may be used. However, when the dividends tend to be clustered, early exercise may be triggered at many points in time and B-S type models cannot be applied Another potential dividend-induced reason for early exercise is asset carrying costs. When these costs are different from the riskless interest rate, early exercise is also possible. Hence, for many reasons, the B-S model does not price adequately American commodity and futures options since, by definition, the model is to European paying stocks. options on non-dividend Whaley (1981, 1982, 1986) and Whaley and Harvey (1992)showed that ad hoc valuation procedures sometimes produce large pricing errors because of the discrete and seasonal pattern of dividends on some indexes like the S&P 100 index portfolio. Such ad hoc valuation procedures for index options include European-style formulas approximation methods where it is assumed that the index pays and American-style dividends at a constant proportional rate as those presented in the previous section. The most commonly used methods are the B-S model adjusted for dividends because of its ease of computation and the American-style option pricing approximations with a constant proportional dividend yield. These include the quadratic approximation of BAW binomial method under the assumption and the Cox, Ross and Rubinstein

among others. To understand the nature of compound options, consider a firm whose capital structure is composed of stocks, S, and bonds, B. Assume that bonds are discount bonds, paying M dollars in T years. Assume also that the firm plans to liquidate at that date, paying off the bonds and any remaining value to stockholders as a liquidating dividend. In this setting, bondholders have given the stockholders an option to buy back the assets at date T. In fact, if at that date the firm's value, V is less than M, bondholders get the assets V and stockholders receive nothing. If V is greater than M, bondholders get M and stockholders receive ( V - M). The pay-off to stockholders is then max (0, V - M). Hence, a call on the firm's stock is an option on an option, or a compound option Following Geske (1979a),the compound option is written as

'

C(V, t)

(1987)

(1979)

proportional dividend yield rate. The appropriate approach for the valuation of American options when there are discrete dividends is based on the compound option approach. Several models are proposed in this option approach to the pricing of context. Given the contribution of the compound American options, this approach is now presented in detail.

f(S, t)

=

f(g(V,

t), t)

(65)

Hence, changes are expressed as a function of the changes in the firm's value and time. The dynamics of the returns on Vand C are described by the following equations: in the call's value

dV/V=a,dt+o,dW, d C/C

'reserved'

of a constant

=

=

(66)

ac dt + ac dW

(67)

'

respectively

where as, as, ao and ¤c correspond to instantaneous expected rates of return on the firm (on the call) and instantaneous volatility of return on the firm (onthe call). Using Itö's lemma gives the following dynamics for the call option:

dC

=

Ot,

dt+

\BV}

dV -E

V2a dt

2\8V2,

(68)

Followmg Merton (1973a), a three-security riskiess hedge portfolio, H, containing the firm, the call and a riskless asset can be created for zero net investment. If dH is the mstantaneous dollar return to the hedge portfobo, then dH

=

nt

(d V/

V) + na(d C/C) + vns di

(69)

where the n, terms correspond to portfolio weights. Substituting for the stochastic return on the firm and the call, choosing n so that the Wiener term corresponding to V is zero implies that dH 0. Simplifying yields the familiar partial differential equation =

172

OPTIONS,

BC Ot

Sincethe eaWe value

rC - rV

=

AND EXOTIC

FUTURES

ßC

1

82C

DV

2

BV2

V2a 2 *

DERIYATIVEs

nr "

C,.

=

(0, S,. - K)

max

OPTIONS:

TO AMERICAN

using

The value Veris calculated A

t*, is either zero or the intrinsic value

at expiration,

EXTENSION

S,-K=VN(h2+o,f)-Me"N(h2)-K=0

(71)

dynamics are given by

173

the following equality:

with

this boundary poses a problem from the perspective of the stock as an option on the value of the firm. In fact, the variable determining the option value in equation (70)is not S but V However, since the stock is an option on V it follows a related diffusion and again its

AND EARLY EXERC1SE

DIVIDENDS

I

=

r,

=

(78)

T - /* t* - t

r2

0 or T oo, the stockholder's option to repurchase the firm from Note that when M and the formula reduces to that oñB-S applied to a call disappears bondholders the wntten on the equity of an all-equity-financed firm. =

=

.

BS -8t

rS - rV

=

BS

1

82S

BV

2

ßV2

The solution to this equation subject to the condition equation:

S

=

VN(h2 + a

)

S,

V2

'

'

Generalization max [0, Vr - M] is the B-S

=

"

Me

-

2 "

N(h2)

(73)

(1993),the

Following Rubinstein

Option

Approach are used:

following notations

the value of the compound option, the time t value of the underlying option, Ke the strike price of the compound option, K the strike price of the underlying option, option, t the time-to-maturity of the compound t* the time-to-maturity of the underlying option with t* > i, R one plus the riskless interest rate, d one plus the payout rate, value 1 (-1) when the underlying option is a call (put) r¡ a binary variable taking the variable taking the value 1 (-1) for a call on a call and a call on a put (a binary a ¢ put on a call and a put on a put).

C, c

with

of the Compound

=

=

=

h2

=

ln ( V M)

=

o )(T - t)

(r

(74)

=

=

At the call's expiration date, exercise will depend on the relationship between S and K. t*, the firm's value, Ver, that makes the holder of an option on the stock When t indifferent between exercising or not is the solution of the integral equation S* - K 0 with r T - t*, where S * is given by equation (73).For values of the firm less than V., the call on the stock remains unexercised. Given equations (70) and (72) and their boundary conditions, the compound option value given by Geske (1979a)is =

=

=

=

=

=

=

C, C

=

VN hi + a

h2 + o,¾,

,

(75) Me

"2

N

h

,

- Ke¯"'

h2,

=

as yK|t*])

[0, ¢PV,(max (0, QSc -

max

C,

R 'E[max

=

{0,¢c(S,,

(79)

- ¢Ke]

where PV,(.) is the present value after time t of (.). The current value of the compound option is given by its expected appropriate probability measure:

N(hi)

value under the

t) - ¢Ke }]

(80)

expectation operator and c(S,, t) is the B-S call formula. where E is the mathematical This can be written in an integral form as

with

hi and

option can be written

The pay-off of a compound

ln(V

V

=

)+(r-jo

)ri

(76)

C,

=

R

'

max{0,

t) -¢Ke}f(u)du

¢c(Se",

(81)

with

. In(V/M)

h2

=

(r-jo

)r2

u

=

In(S,/S)

(82) "'72

where

N(A, B, p) stands for the bivariate cumulative

normal

distribution.

f(v)

=

o

vdii

e

(83)

174

OPTIONS,

v

(u

=

FUTURES

AND EXOTIC

µt)/o

-

µ=ln(R/d)

a

The above integral can be valued using a decomposition corresponding off variables, S, K and Kc, denoted respectively by I(l), I(2) and /(3): I(l)=¢SR'

I(2)

¢KR

N(z,

=

¢qKR

TheValuationFormula

I(3)

=

'

¢KeR

¢ya

f(u)du

=

gy KR

-

175

Options with Dividends

Consider the following portfohos: purchase of a European call with strike price K and maturity date T European call with strike price See and maturity date t-e (b)A purchase ofa where 2 > 0, S, is the ex-dividend stock price above which the option will be 0), e (e exercised early and (c) A sale of a European call on the call given in (a) with strike price Se, + D - K maturity date (t E).

(a) A (86)

-¢>1

-

are equivalent to that of an American call, the absence of costless arbitrage implies that the American call's value is equal to that of these portfolios. Hence the application of the B-S (1973)formula for the first two options, (a) and (b), and the Geske (1979a,1979b) formula for the third option, (c), gives the American call option pricing formula. The formula was derived by Whaley (1981) for the valuation of American call options

Since the pay-offs of these portfolios

yoÑ, ¢p)

'N(¢>¡\

AND EARLY EXERCISE

DIVIDENDS

OPTIONS:

of American

Valuation

(85)

(87) N(¢yx -

TO AMERICAN

7.3.3

)f(u)du

a

EXTENSION

(84)

to du three pay-

¢ySd'"N(¢yx,yy,¢p)

e"N(z,)f(u)du=

=

DERIVATIVES

E)

(88)

where

on stocks with known dividends: 0

in

(89)

=

call

American

ce

In

=

SN(ai, bi,

b - qKR

N(z

)

Ke

0

di

z

=

Se d KR

by ¢qSd -

¢KeR

N(¢qx, 'N(¢Qx

gy, ¢p) - ¢qKR -

¢ya0)

+

In(S/Se,) =

(r

)-

+ (a2)T

(r+)a2)t

(96)

N(d2)

(Se, + D) e

N(a2, b2,

In(S (Sc, + D)) =

(Sc, + D - K)e

and

a2

and

b2

(r + la2N

and

(97)

=

N(b2)

60

a

(99)

bi - o

(100)

a,

=

"

d2

=

-

di

(101)

a

where N(a, b, p) stands for the bivariate cumulative normal density function. Using the N(a) - N(a, b, p) and gathering the terms in S and K property that N(a, -b, gives the following formula: =

c(S, T, K)

=

S[N(bi) + N(ai, bi, -

=

(95)

Cc

-p)

(93)

This equation gives the value of the critical underlying asset price, Se,(S,). It can be calculated using an iterative procedure. The current value of a compound option is given

Cap

Ke

-

(92)

with

ln

"

=

and Se, is the solution to the following equation: =

SN(di )

ln(S/K) a

-

=

)-

(91)

QSc,d

ca + cb ¯

=

N(aa)

SN(aa) - Ke

=

c, cb

"N(z)

call(c)

-

with

Sd

(90)

"

call(a) + call(b)

=

'N(¢yx -

¢yan,

r¡y -

Ke

"[e

"

N(b2) + N(a2, -b2, -

The critical stock price is found by applying following equation:

r¡oÑ, ¢p) (94)

C(So, T, K)

)] + De

=

a numerical

Se, + D - K

"

N(b2)

(102)

E

search procedure

for the

(103)

176

OPTIONS,

FUTURES

AND EXOTIC

EXTENSION

DER1VATIVES

Simulations

•

• • • •

(1981) is used

to generate call option prices. Using the -

Strike price, K 100 Ex-dividend date, t 6 months Time to maturity, T 1 year Interest rate for 6 months, r 0.04 Volatility of the stock (6 months), « Dividend, D 5,

DIVIDENOS

AND EAREYRKERCISE

177

This chapter presents in detail the basic concepts and techniques underlying rational derivative asset pricing. This is done in the context of analytical European models along the lines of B-S, Black, Garman and Kohlhagen, and Merton-BAW.

=

This chapter extends the European analytical models presented in the previous chapters

to the valuation of American options. Our attention is focused on the question of dividend and the distinguishing feature of early exercise. First, the general problem of valuing American options is analyzed in three different asset, when there is a contexts: when there are no distributions to the underlying constant proportional distribution rate, and in the presence of discrete cash distribu-

=

=

=

=

0.2

=

Option prices are reported

in Table 7.7.

TABLE 7.7 Price

*

Option

Price

and

Critical

Stock

S

5 ex-dividend

Se,

C(S, T, K)

82 85 87 90 92 95 97 100

77.196 80.196 82.196 85.l96 87.196 90.196 92.196 95.196 97.196 100.196 102.196 105.196 107.196 i I0. I96 I12.196 I \5.196 I 17.196 ]20.196

123.5818 123.5818 I23.58\8 123.5818 123.5818 123.5818 123.5818 123.5818 123.5818 123.5818 !23.5818 123.58l8 123.5818 I23.58 I8 123.5818 123.5818 123.5818 l23.58l8

3.8050 4.8175 5.5758 6.8389 7.7645 9.2759 10.3636 I2.I I 13 13.3506 15.3155 16.6922 18.8509 20.3486 22.6759 24.2774 26.7476 28.4363 31.0255

102 105 107 I10 i 12 I I5 I17 \20 122 125

tions. The analysis covers American options on spot assets and American futures options. The main results reported in the literature regarding the pricing of American calls are reviewed. Some models are presented in detail and simulations are run. of a put-call parity theorem for American options implies a specific The non-existence options. The pricing of American puts is rather a difficult task, even in for put treatment the absence of distributions to the underlying asset. However, some interesting results are presented in the absence of dividends and when there is a continuous dividend rate. In the case of discrete distributions to the underlying asset, the formulas proposed in the literature are not so efficient as those proposed for American calls. In all cases, there is still no published analytical formula for the pricing of American calls or puts when there asset. Therefore, these problems can be are several discrete distributions to the underlying handled using binomial models or finite difference methods, which will be presented

later.

POINTS • • •

•

•

Applications

The different applications presented for European options in the previous chapters can be used with American optionsThe compound option approach also applies to the valuation of wildcard options, options. These applications will be studied in Part 3 of this some exotic and complex book

FOR DISCUSSION

What is meant by early exercise? What is an early exercise premium? Can an American call option be exercised

in the absence of distributions to the underlying asset? Can an American put option be exercised in the absence of distributions to the underlymg asset? When is an American call option exercised in the presence of continuous distributions to the underlying asset? When is an American call option exercised in the presence of discrete distributions to the underlying asset? distributions When is an American put option exercised in the presence of continuous asset? to the underlying When is an American put option exercised in the presence of discrete distributions to the underlying asset? Explain the incentives for early exercise of American call options on a spot asset. Explain the incentives for early exercise of American put options on a spot asset. Explain the incentives for early exercise of American call options on a futures contract. ·

•

7.3.4

OPTIONS:

SU MMARY

The formula obtained by Whaley following parameters:

•

TOlkWERICAN

•

•

• • •

178

•

OPTIONS,

FUTURES

AND EXOTIC

EXTENSION

DERIVATIVES

7.A.2

Explain the incentives for early exercise of American put options on a futures contract. What is meant by a contact condition'? What is a capped variable loan commitment? What is a compound option? 'high

• • •

OPTIONS:

TO AMERICAN

in the Underlying

Effect of a Change Time to Maturity

TABLE 7.10 30% «

DIVIDENDS

and

European

AND EARLY EXERCISE

Asset

American

Price Volatility

Cap Prices:

L

=

179

and

100, r

10%,

=

=

t

APPENDIX 7.A: SIMULATION AMERICAN CAP OPTIONS Effect of Maturity

7.A.!

TABLE 7.8 «

=

a Change

in the Underlying

European

Asset

and American

FOR Price and

Cap Prices:

L

ga

=

100, r

10%,

=

20% t

=

0.25

t

B-S

Cap

S

Put

Put

100 99 98 97 96 95 94 90

2.826 3.227 3.668 4. I49 4.672 5.236 5.840 8.640

TABLE 7.9 20%

«

RESULTS

3,064 3,552 4.008 4.544 5. I 32

5.797 6.500 10.00

European

and

=

0.50

t

=

0.75

B-S Put

Cap Put

B-S Put

Cap Put

3.400 3.749

3.92) 4.330 4.793 5.288 5.820 6.402 7.040 10.083

3.65)

4.457 4.859 5.292 5.76 I 6.266 6.810 7.397 10.259

4.124 4.528 4.959 5.420 5.910 8.\73

American

Cap Prices:

3.963 4.295 4.650

5.027 5.426 5.850 7.790

L

=

100, r

=

\

t=3

5

B-5 Put

Cap Put

B-S Put

100 99 98 97 96 95 94 90

3.573 4.036 4.336 4.654 4.990 5.346 5.722 7.432

4.658 5.228 5.649 6.10\ 6.587 7.\09 7.67! 10.374

3.156 3.309 3.469 3.636 3.8\0 3.993 4.183 5.033

B-S

Cap

Put

Put

2.694 2.8 i I 2.934 3.062 3.195 3.334 3.479 4.122

6.2\8 6.568 6.942 7.34| 7.766 8.220 8.707 11.018

6.052 6.38 I 6.762 7.\60 7.600 8.063 8.557 10.916

t

" S Put

Cap Put

B-5 Put

Cap Put

B-S Put

100 99 98 97 96 95 94 90

4.751 5.\69 5.6 13 6.084 6.582 7.108 7.066 10.l55

4.976 5.4\9 5.890 6.392 6.925 7.490 8.087 (0.8f 4

6.029 6.404 6.798 7.210 7.64\ 8.091 8.56 \ 10.643

6.560 6.975 7.4 I 3 7.873 8.356 8.863 9.390 I l.790

6.759 7.104 7.463 7.837 8.226 8.63\ 9.05 \

and American

European

t

×

100 99 98 97 96 95 94 90

t=4

Put

0.50

=

Cap Prices:

L

=

0.75

=

Cap Put 7.602

8.002 8.4 t 8 8.855 9.311 9.789 10.287

100, r

=

10%,

=

10%,

Cap

t

S

TABLE 7.1 I 30% a

=

t=

0.25

=

'

=

1

t

=

3

t

B-S Put

Cap Put

B-S Put

Cap Pt

7.2\7 7.538 7.870

8.391 8.678 9.I88 9,60! 10.039 10.497 i0.973 13.088

7.686 7.890

i \.358 \ \.697 |2.048 12.412 12.789 13.179 I3.584 l5.359

8.2lS 8.573 8.944 9.328 !!.003

8.099 8.3\5 8.536 8.763 8.996 9.994

B-S Put 7.240 7.409 7.582 7.760 7.942 8.129

8.320 9.136

=

4 Cap Put 12.0\2 12.34| |2.68i 13.033 \ 3.395 13.774 I4. I64 15.874

Generalization

to Stochastic Interest

CHAPTER

Rates

OUTLINE

This chapter is organized as follows: 1. Section 8.1 presents in detail Merton's option pricing model for the valuation of stock options under stochastic interest rates. 2. Section 8.2 develops the main results in Rabinovitch's model. It presents the model and some of its applications to the valuation of stock index options. 3. Section 8.3 reviews the main results in some of the more specific models for the pricing of bonds and bond options, in particular those of Vasicek (1977),Cox, Ingersoll and Ross (1985a)and Heath, Jarrow and Morton (1992).

INTRODUCTION Several authors have derived alternative formulas to the basic B-S model for the pricing of stock options, index options, bond options and foreign currency options, when interest rates are not constant. However, until 1989, all the proposed models, except Merton (1973a),used the assumption of a constant free rate prevailing during the option's life, i.e., the effects of the interest rate's variance and covariance with the underlying asset's return on the option's price were precluded from models other than that of Merton. This chapter presents some models for the analysis and valuation of derivative assets whose values depend on stochastic interest rates. Since Merton's model represents a convenient starting point for most of the complete option pricing models, it deserves a special treatment in this chapter. Using Merton's approach, Rabinovitch (1989)derived valuation formulas for options under the assumption that the interest rate follows a mean reverting process. However, Rabinovitch's formulas apply only to stock and index options. They do not apply for the pricing of bond options since there is a deficiency in his model. We present the necessary of bonds and contingent corrections and propose other specific models for the valuation claims whose values depend directly on the dynamics of interest rates.

l82

OPTIONS,

8.1

DERIVATION

FUTURES

OF MERTON'S

A940 EXOTICOERIVATIVES

MODEL

8.1.1

(dS)2

(dS)(dP)

The Model

Let C(S, P, r, K) be the option price function depending on the stock price S, the bond price P, the exercise price K and the time-to-maturity r The stock price dynamics are represented by the following stochastic differential equation:

(1)

T)

(4) (5)

JöSP(d

W)(dq)

öpaSPdt

=

coefficient between the returns

correlation

(2), equation (3) is rewritten

82C BS2

-

BC BS

OC -OP

82C

2S2dt+2

BSSP

(2)

orina

BC

+µP

82C

+)ö2,2

The bond price dynamics are given by the following equation: µ(r) dt + ö(r) dq(t,

aS

dC=

.

=

=

-OP2

on the stock and

as

BC

[µPdt+öPdql+

82 C

JöpSPdt+

dr

-87 -

ö2P2dt

(7)

OP2

BC

SP

+)o.2S2

Br BC

dt+aS

-OS

82C -BS2 BC

dW+õP

-BP

öpSP

82C -

OSSP

dq(t, r)

(8)

dC=ßCdt+yCdW+yCdq =

1

+ aS

dq(s, r)dq(t,

on any ofthe assets:

among the returns T)

dq(s, r)dW(t)

=

=

dq(t, r)dq(t,

for

0,

T)

( ß2C g) DC -

+ oöpSP + µP

=

s

__

pa dt

y

=

82 C

C

+ (ö2P

/ 82C\

OC

DC S

¢ t for s ¢ t

0,

(9)

with

1,

=

Assume that there is no serial correlation

(6)

simple form

where P(x) is the price of a discount bond with maturity r satisfying P(0) µ(r) is the instantaneous expected return, ö2(r) is the instantaneous variance with ö(0) 0, dq(t, r) is a standard Gauss-Wiener process.

are

which is equivalent to

the instantaneous expected return on the common stock, is the instantaneous variance of return, restricted to be a knowafettetienktime.

dP/P

+ öPdq)

OC [aSdt+oSdW]+ -DS 1

is

.

of (dS)2, (dP)2 and (dS)(dP)

(aSdt + «Sd W)2 __ 2S2dt

=

(aS dt + oSd W)(µPdt

2

·

a

=

dC=

where

02

183

(dP)2=(µPdt+öPdq)2=ö2P2dt

where p is the instantaneous on the bond. Substituting from (1) and

dS/S=adt+adW

RATES

calculus, the values

Using the properties of stochastic given by

Consider the basic case of a European option with no pay-outs to the underlying asset Assume that there are no transaction costs, that trading takes place continuously and that no restrictions are imposed on borrowing and short selling. Assume also that the borrowing rate is equal to the lending rate.

INTEREST

TO STOCHASTIC

GENERALIZATION

(10)

DC BS /

ö

(12)

where

p, may be less than 1 for r ¢ T. When the interest rate is constant over time, this means that ö 0, µ r and P(r) e Assuming that investors agree on the values of (ö, a), Itö's lemma gives the change in the option price over time: =

=

=

".

BC -OS

dC=

BC -

ßP

dP+

82C (dS)2+2 2 \DS2/ l

+

dS+

BC dr ßr

!

The expression for ß represents the instantaneous expected return on the option. Consider now a hedged portfolio with the underlying asset, the option and riskiess bonds where the portfolio weights wi, w2, w3 are chosen to eliminate market risk. The aggregate investment in the portfolio y is zero when investors are allowed to use proceeds from short sales and to borrow without restrictions to finance long positions, so that w,+w2+w,=0

-

82C -BSOP/

dSdP+

82C \8P2)

In this context,

(dP)2

(3)

the instantaneous

dollar return,

dy, may be

dy=wi(dS/S)+w2{ßdt+ydW+ydq}+{-wi-w2}(dP

written

as P)

(13)

184

PTIBMS;FUTURESAND

EKOTK

ERIVATIVES

GENERALIZATION

change in variables, x thecallpriceis

or

dy=wi{adt+adW}+w2{ßdt+ydW+r¡dq}+{-w,-w2}{µdt+ödq} Rearranging

(14)

y(x, T)

(15) h ' ¯- -

=

and whèté erfc is the derer complement

This linear system presents a solution if and only if

(ß -

µ)/(a

µ)

-

=

y a

erfc y)/ö

(ö -

=

(17)

y a

r¡ ö)

(1 -

afinitionof y and equation S(BC/BS)/C

(18)

(9)that (19)

(27)

2 e

y

dw

(28)

=

=

SN(di)

-

P(r)KN(d2)

(29)

with ln(S/K)

S(OC/OS) + P(BC/8P)

=

,

.

or

C

1- -

=

=

c

1 - [P(OC BP)/C)

=

,

function:

(h)

=

This implies from the

17 2

For more details about the solution method, see Merton (1973a). 1, equation (25)is identical to the B-S 0, a2 1 and K In the special case when r equation. The call price in the context of Merton's model can be written in a B-S form as

(17)holds, then =

(26)

.o

0

=

'

[o2+ö2-2paö]du

T=

(16)

(y - ð)

(25)

(h2)]

r

y=0

ö+ w

-w

erfc

¼T

ln(x)

ha - -

(ß-µ)=0

(a-µ)+w

w o+w

If

([xerfc (hi) -

=

(1973a)for

where

Now consider a strategy where the weights w, w are chosen so that the stochastic terms in (15)affecting dW and dq are always zero. The expected return on this strategy must be zero since it requires zero investment. Hence: w

185

RATES

S/KP(r) the solution presented in Merton

=

terms yields

{wi(a-µ)+w2(ß-µ)}dt+{wia+w2y}dW+{pw2-(w;+w2)ð}dq

dy=

INTEREST

TO STOCHASTIC

d'

(20)

=

- ln(P) + a,

(o2r

and

d2

=

di - a,f

(30)

The put price is given by

Using (17)gives

(ß Recalling the defmitions of 0

=

ß and

(o2S2

µS BS

µ)

y(a - µ) a

=

p

(21)

+ µP

ac

OP

p2

+

OSOP

ac

-SN(-di)

ln(S/K)-ln(P)+

ß2C

+ oöpSP

=

(3l)

+ P(r)KN(-d2)

with

y in (9),we have

82C -ßS2

ac

-

2

di

82C BP2

P(r)

- µC

87

Ur

and

=

(22)

,

1

d2

=

(32)

di - a

" =

(33)

e

'

[o2+ö2-2paöjdu

a;=-

(34)

0

Using

(9) and 0

rearranging

=

2

o2S

terms,

\8S2

(22)can be rewritten 2aöpSP

OSSP

as

+ö2P2

2C ßP2

BC Or

8.1.2

(23)

. This is a second-order linear partial differential equation of the parabolic type. The price of any option in Merton's economy must satisfy equation (23).In particular, the price of a European call must satisfy that equation and the following boundary conditions: .

Options

-

C(0, P, r, K)

C(S, 1, 0, K)

=

=

.

[0, S

- K]

(24)

on Bank Deposit

Rates and on Treasury

Bill Yields

.

Options on yields of short-term financial instruments are traded m the over-the-counter market. These fmancial products are in the form of bank deposits, certificates of deposit, Treasury bills, commercial paper, and so on. Treasury securities are backed by the faith and credit of the government. They are

0 max

Applications

se

regarded as having no credit risk. The interest rates on Treasury securities

are often

used

OPTIONS,

186

FUTURES

AND EXOTIC

DERIVATIVES

as a benchmark in national and international capital markets. There are two categories of government assets: discount and coupon assets. Treasury bills are securities issued with maturities of a year or less as discount securities, which do not make periodic payments. All the assets having a longer maturny are issued as coupon securitics. When the maturity is between two and ten years, the issued security is a note. When the maturity is greater than ten years, the Treasury security is a bond. An option on the yield of the most recently issued 13-week Treasury bill is traded on the Chicago Board Options Exchange. This option is based on an interest rate coraposite, i.e. 100 times the yield on the 13-week Treasury bill. Recall that a fundamental property of a bond is that its price moves in the opposite direction with respect to the required yield, i.e. when the required yield increases, the present value of the par value and the coupon decreases and vice versa. In general, the price yield relationship is convex. The yield on any investment corresponds to the interest rate which makes the present value of the cash flows equal to the cost of the investment. The yield is simply an implicit interest rate. In practice, two measures of yield are used: the current yield and the yield to maturity. The current yield gives the relationship between annual coupon interest and the market price. It is simply given by the ratio of the annual dollar coupon interest to the market price. The yield to maturity is the implicit interest rate that makes the present value of the future cash flows until the bond's maturity date equal to the bond's price. Merton's (1973a) model can be applied to the valuation of these options and is more appropriate than the B-S (1973)model since interest rates are stochastic. Short-Term

Options

on Long-Term

RABINOVITCH'S

187

RATES

where a is constant and o is a deterministic function of time. The dynamics of the interest rate r(t), defined as the yield to maturity on a bond paying one dollar in the next instant, are given by

dr

q(m - r) dt + vd W2

=

(36)

pdt for t with dWi(t)dW2(s) s, and zero otherwise, where p is the correlation coefficient between the two processes. The parameters m, q and v' are constants with the following meaning: =

=

riskless interest rate r, m is the long-run mean of the of speed of the adjustment is interest rates towards that mean, q q(m - r) is the instantaneous expected change in the short-term rate, v2 ÍS the instantaneous variance of interest rate.

The above process is an Ornstein-Uhlenbeck process, a Gaussian diffusion process, implying the long-run possibility of negative interest rates. As in Vasicek (1977), the bond price P can be written as a function of r(t) and time to maturity, r: P

(seeVasicek,

1977; Rabinovitch,

P[r(t),

=

r]

(37)

the following

(37)gives

Applying Itô's lemma to (36) and

solution for the bond price P

1989): P(r)

Ae"

=

(38)

with

Bonds ,

Transactions on short-term options on long-term bonds take place often in the OTC market. Some options are also traded on organized markets. These options may be European or American. A European short-term option on a long-term bond is traded on the Chicago Board Options Exchange. This option is similar to the option on yield of short-term financial instruments traded on the CBOE and is cash-settled. The interest rate corresponds to the average of the yields to maturity of the two most recently auctioned seven-year and two most recently auctioned ten-year treasury notes and also the two most recently auctioned 30-year treasury bonds. The average of the yield to maturity of these six instruments gives the interest rate on which the option is priced. This option can be priced using Merton's model which is more appropriate than the B-S model in the context of stochastic interest rates.

8.2

TO STOCHASTICtNTEREST

GENERALIZATION

,

B - B(r)

.

(1 -

=

exp [k(B

A

=

A(r)

k

=

m + v1

A

=

(y -

)

e

=

(v

q -

q

-

q)2

T)

-

(vB/2)2|q)

/2

r)/ö

correspond respectively where y and to the bond's expected return and variance and 2 l. (a constant) stands for the market price of risk. It is convenient to note that P(0) 0. A(0) 1 and B(0) From Itõ's lemma ö is a known function of r,

o2

=

=

=

ö(r)

=

vB(r)

(39)

According to equation (25),a call H, with a strike price K, a time to maturity r and an underlying asset S is given in the Merton (1973a)model by H

MODEL

T)

(40)

([xerfc(hi)-erfc(k2))

(41)

KP(r)y(x,

=

with 8.2.1

The Model for Equity Options

y(x, T) In this model,

=

the stock price dynamics are described by the famikar equation dS=aSdt+aSdW

(35)

x

=

KP(x)

(42)

I 8

OPTIONS,

ln

hi

=

-

h2

=

-

FUTURES

AND EXOTIC

D

TIVES

T

(x)

(h)

RATES

189

-1, the unanticipated returns on the stock and the bond move in opposite When p These trends are directions and the variance effect on the option's price is maximum. shown in Table 8.1.

TABLE 8.1 0.06, A

h

2

1

=

INTEREST

(44)

function:

erfc

TO STOCHASTIC

=

(43)

ln(x)-¼T

where erfc is the error complement

GENERALIZATION

y

""

dw

e

=

=

Rabinovitch 0.2

Call

Values:

S

100,

=

(45)

r

=

10%,

-0.25 Time to Maturity =

Jo

[a2(4 ö2(s) -

2pö(s)r>(s)] ds

(46)

Volatility

Under the assumption of a constant stock volatility, substituting the solution of T from (46),ö2 from (39)and P(r) from (38)into equation (40),Rabinovitch gave the following formula for a call option with a stochastic default-free rate: H

SN(di) - KP(r)N(d2)

=

0%

25%

(47)

where

di

=

ln

-

d2 T

=

a2r +

(r -

=

KP(r))

di -

+)T

- B)

solution is simple and useful, since it shows the effect of stochastic interest rates on option values. The difference between this formula and equation 29 of Merton (1973a)is that equation (47) reflects the effect of the mean reverting terni structure on options values. When compared with the B-S formula, this formula takes into account not only the stock's volatility, but also the volatility of the interest rate and its correlation with the stock's returns. It can be shown that partial derivatives with respect to the stock price, the strike price and the interest rate are given by Hs HK

N(di)>0

=

-Ae

H,.=

BKAe

"N(d2)<0 '

(51)

0.4,

0.25 Time to Maturity

0.\0

0.50

l.00

0.\0

0.50

!.00

90 100 i io

10.90 1.83 0.00

\4.55 6.05 I.29

19.07 10.91 4.88

10.90

\4.53 5.90 I.I2

l9.0I 10.63 4.38

90 100 i lo

I I.\4 3.67 0.60

16.25 9.76 5.28

21.67 15.53 10.7\

I l.14 3.65

0.59

16.13 9.58 5.09

21.34 15.08 10.19

90 \00 i I0

12.93 6.79 3.07

21.59

I6.43

29.10 24.40

I2.32

20.4|

12.92 6.77 3.06

2l.43 I6.24 I2.I2

28.68 23.93 |9.90

8.2.2

l.8\ 0.00

The Model for Bond Options

an explicit and closed-form the default-free rate is stochastic. He assumed that the current interest rate r(t) describes the whole-term structure, that the option's underlying asset is a default-free discount bond maturing at Ts, and that the option expires at Ts, such that Te > T,. He also assumed the existence of another bond paying one dollar with certainty at time T, and denoted by Pc[t, TL, r(t)] and T, - t. the prices of these bonds with r, P,[t, T,, r(t)] respectively T, - t and r, results the longer-term and Itö's lemma in Merton (1973a), he for bond Using some concluded that formula (40)also applies for the pricing of call options on the long-term

(1973a),Rabinovitch (1989)presented

Following Merton

formula for the pricing of bond options

when

=

=

bond, with x=

=

P,[t,

T,, r(t)]

KP,[t,

T,, r(t)]

(53)

and T,

T

=

[U2(To

O

ö2(T,, s) -

2pa(TL,

s)o(T,, s)]ds

(54)

«209

+ ö2 - 2pöu is lowest when compared with other values of p. --

=

K

N(d2)>0

1, the It is convenient to note that when the correlation coefficient is l, i.e. p unanticipated the same returns on the stock and the bond behave statistically as instrument. The total variance, given by v2

q

(50)

This closed-form

=

0.I,

(49)

:

- 2po(r

2B) +

50%

(48)

N

=

Rho

and T

m

Rabinovitch presented a solution in the form of the B-S formula for stock options where the long-term bond replaces the stock price. His solution is similar in form to that of Cox,

190

OPTIONS,

FUTURES

AND EXOTIC

Ingersoll and Ross (1985a)for options on discount bonds. However, his paper was the subject of a reply by Chen (1991)who showed an incoherence with the formula in Rabinovitch.

8.2.3

Chen's

=

P(t, T)

PRICING OF BONDS AND INTEREST OPTIONS

=

of

r(s) ds

(59)

),

the future interest rates, R(u, T), are wthaean and models of interestratesmustbeestablishedtofmdtherelationshipsbetweestibesergtes.

8.3.2

Instantaneous

Rates under

interest

Uncertainty

Under uncertainty, the instantaneous interest rate r(t) is a stochastic times t and t + dt. If we consider a riskless asset, then its price is given by B(0, t)

the dynamics

exp

In a context of uncertainty,

RATE

The pricing of bonds and interest rate options needs a specification interest rates under certainty and uncertainty.

191

RATES

INTEREST

Consider a zero-coupon bond, which is a bond paying 1 dollar with certainty at its l. The price of this bond at time t, P(t, T), in this context (in maturity date, P(T, T) with no risk, where all future interest rates are known) is given by economy an

Correction

The model presented by Rabinovitch for the pricing of stock and bond options when the short-term interest rate is stochastic and follows a mean reverting process, reduces to those presented by Chaplin (1987)and Jamshidian (1989). In fact, careful attention to the assumptions in Rabinovitch shows that the two factors d Wi and d W2 reduce to just one factor, since the bond price is determined by the same current interest rate r(t). Given the assumption of a mean reverting process for r, the bond price can only follow a log-normal process. Hence, there is no need for an additional assumption for the bond price, When the correlation coefficient equals 1, the model simplifies to those in Chaplin (1987)and Jamshidian (1989).

8.3

TO STOCHASTIC

GENERALIZATION

DERIVATIVES

=

exp

process between

r(s)ds

Lo

#90) -

Assumption H Consider a process P(t, u)os,ss satisfymg the boundary condition l. Then, as for the processes of stock prices, it can be shown, with no arbitrage P(u, u) opportunities, that there is a unique probability P* equivalent to the probability P under =

8,3.1

Instantaneous

interest

which the process

.

Rates under

Certamty

When an investor borrows $l at time t, he must pay F(t, T) at time T where T is the maturity date of the debt. This amount corresponds to an average interest rate R(t, T) which applies over the period [t, T]. It is given by F(t, T)

e""

=

"'

"

(55)

P(t, u)

=

F(t, u)F(u,

for all t
s) =

*

R(t, T)

=

=

exp

r(s)

ds

r(s)ds

P(t, u)

(61)

P(t, u)

=

u)\F,} E*{Ï>(u,

=

E*

exp

-

r(s)

P*, we have

ds

0

(62)

(56)

1 allows one to show that, there is an Using this equality and the fact that F(t, t) instantaneous interest rate r(t) such that in the absence of arbitrage opportunities, the amount F(t, T) is given by F(t, T)

r(s)ds

exp

is a martingale for each u in the time interval [0, T]. This is an interesting assumption since, under the new probability

In the context of certainty, when the interest rates are known over the period, the function F is given at each instant by F(t, s)

=

(57)

)

or P(t, u)

=

E*

exp

r(s)

ds

F,

If one compares this formula with formula (59),we notice that the prices of bonds depend only on the process P(t, u)os,,,, under the probability P*.

(63) zero-coupon

Assumption H allows the identification of the probability density of P*, denoted, LT, with respect to P. In this context, it is possible to show that there exists a stochastic process q(t)os,s, such that for all t in the interval [0, T]:

192

OPTIONS,

L,

=

FUTURES

q(s) dW

e×p

GENEllALTEM10N

AND EXOTltWVATIVES

(64)

o

Using equation

(64)and

the following property E*[X F,]

=

E[XLr|F,]

is

Li

possible to show that the price at time t of a zero-coupon given by

It is

P(t, u)

=

E exp

/

("

(65) bond

"

r(s) ds +

with a maturity

"

q2(S)

q(s) dW

Û',

Rate Processes

and

the Pricing

ds |Fr

=

(P(r,

Co×, ingersofi and Ross Model In the context of this model, the dynamics of the instantaneous interest rate are described by the following equation:

of Bonds and Options

T) - K)*

(67)

(K - P(r, T))*

(68)

For a put option, this pay-off is given by c

=

Yasicek Model The presentation here uses equations (63)and (66) Vasicek (1977)used the following process for the dynamics of the interest rate r(t): dr(t)=a[b-r(t)]dt+adW,

(69)

where a, b and o are constants. If we suppose that the process q(t) is constant equal to -1, then a dr(t)

=

a[b*

-

r(t)]dt

+ o

with

b*

=

b

2¤ a

dÑ7,

W, + At

(66)

Several interest rate models are based on the spot rates where the spot interest rate is the underlymg state variable. This is the case for the models of Vasicek (1977), Hull and White (1987,1988, 1990, 1993), Cox, Ingersoll and Ross (1985a),etc. These models require as inputs different parameters to describe the possible future paths of the spot rate. This implies that the models have to be calibrated in such a way that zero-coupon bond prices are close to market prices. Another way is to use the Heath, Jarrow and Morton (1992)model. Once these problems are solved, the option value can be calculated using the appropriate boundary conditions. Hence, a European bond option with a maturity date r bond with a maturity date T is an option characterized by its terminal on a zero-coupon pay-off. For a call option, this pay-off is given by c

=

The process in equation (69)can be shown to be an Ornstein-Uhlenbeck process. The valuation of bond options in this context can easily be done since the OrnsteinUhlenbeck process is a Gaussian process.

dr(t) Interest

193

date u

The probability is often referred to as the risk-neutral probability

8.3.3

REST RATES

and

q2(s) ds

-

TO15TOCMiliB%WO¾tW

(70)

=

a[b

-

r(t)]

where a and a are positive and b is a real number. In this context, the process q(t) takes the form q(t) number.

Heath, jarrow

and Morton

dW,

dt + o =

-aß

(71) where a is a real

Model

The main drawbacks of the Vasicek (1977)and the Cox, Ingersoll and Ross (1985a) models are that these models do not account for the observed term structure of interest rates. Ho and Lee (1986)proposed an interesting discrete-time model which is appropriate for the description of the whole-term structure of interest rates. The same ideas were used in a continuous-time version of this model by Heath, Jarrow and Morton (1989,1992), hereafter HJM. The HJM interest rate model is based on a new approach to interest rate option pricing. As in the BS model, the HJM model requires the underlying asset (the initial term structure) and a measure of its volatility as the only inputs. The HJM model uses all the available information in the term structure. The model accounts for several features of the term structure movements. It uses a methodology which is often applied in multi-factor models of the term structure risk. In this continuous-time model, the returns of zero-coupon bonds of different maturities are not perfectly correlated and volatility functions are directly calculated from data of changes in the term structure of interest rates. bonds, they are used Smce forward rates are more stable than the prices of zero-coupon in a two-factor model. The two factors are the changing level and the changing slope of to these two factors the term structure. The two stochastic processes corresponding prevent the perfect correlation between bond prices or forward rates. The model has many similarities with the B-S model since it needs only the knowledge The use of multiple of the underlying term structure and its associated volatilities. volatility functions gives the model a certain flexibility since it can accommodate the volatility of the term structure resulting from changes in the level, the slope and the curvature of the term structure. More formally, denote by f(t, u) the forward interest rate, i.e. the rate at which an investor can contract at date t to borrow and lend for a short period at a future date u. The

194

OPTIONS,

AND EXOTIC

FUTURES

initial forward rate curve is taken from market data and

used

DERIVATIVES

GENERA41IlllKTION

TO STOCHASTIC

as an input in the HJM

C(t)

model volatility

of each forward rate is specified in the model. The forward rate for date The u can change over the instant of time from t to t + dt in the following way:

f(t The volatility structure. When o(

+ dt)

function a(

f(t,

=

+ drift(-)dt

u) + o(-)dW

of each forward rate describes

)

with

=

INTEREST

a

In

h

of the term

and the entire a constant, all the forward rates have the same volatility forward curve is submitted to parallel shocks. In this context, the HJM model is the continuous-time limit of the Ho and Lee model. Consequently, the model shows the same drawbacks as that of Ho and Lee, namely, the possibility of negative interest rates. A more interesting model is obtained by HJM when o(-) is a function of the maturity (u - t) of the forward rate f(t, u). " then the model with a>0 and Jo>0, For example, when a(ut)=aae proposed in Vasicek (1977)and studied in Hull and White (1987,1988, 1990, 1993) is obtained-

195

P(t, T)N(h) - KP(t, t*)N(h - q)

(72)

the dynamics

RATES

P( t, T) KP(t, t*))

=

q q2

) is

=

(T

-

t')(l'

4a2 - t) +

[e

and where P(t, T) is the bond with an exercise t* 6 T.

'

-

e

]'[e

' -

e^']

price K and a maturity

t* where

Osts

"

For more details and other applications of the model to the pricing of an entire book of estimation, see Heath, Jarrow and Morton (1992) and Spindel options and volatility

(1992). Example

i

SUMMARY

f(t,

T)

f(0, T) +

=

a2t

t

T

In this chapter, the model developed in Merton (1973a)for the pricing of stock options in the context of stochastic interest rates is presented in great detail. This model represents a starting point toward the theory of option pricing when interest rates are uncertain. The main ideas in Merton's model were used later by many authors for the pricing of a large number of interest rate sensitive claims. For example, Rabinovitch presented an extension of Merton's model for the valuation of stock and index options when the interest rate is uncertain. This model is presented as an alternative to Merton's model for the valuation of long-term stock options. However, it is not appropriate for the valuation of interest rate options and bond options since it

+ a W(t)

In the HJM model, the price of a European bond option is given by C(t)

=

P(t, T)N(h) - KP(t, t*)N(h - q)

with

ln

h

=

is

As

+ '

if

-

o(T - t*) bond with an exercise price K and a maturity q

and where P(t, T) Ostst*GT

P(t, T) KP(t, t*)

=

t* where --

Example

2

df(t,

presents some deficiencies. The literature on the pricing of bonds and bond options is concerned mainly with the stochastic process describing the dynamics of interest rates. Since there are several approaches for the modeling of term structure dynamics, we give the main results from the work of Vasicek and of Cox, Ingersoll and Ross. We present also an interesting model of interest rates, the Heath, Jarrow and Morton model, which accounts for several features of term structure movements. Much work is currently under way on term structure modeling. .

POINTS T)

=

a(t,

T) dt + oi d Wi(t) + o exp

l

A

-(T

-

t) dW2(t)

• • •

In the HJM model, the price of a European bond option is given by

•

FOR DISCUSSION

What are the different categories of government What is a treasury security? What is the main property of a bond? What are the different measures of yicids?

assets?

OPTIONS,

196

•

• •

• • • •

Can t nesRabinovitch model

FUTURES

AND EXOTIC

DERIVATIVES

be used for stock options and stock index options? J

Can the Rabinovitch model be used for bond options? Justify your answer. Which of the models presented in this chapter is appropriate for the pricing of shortterm options on long-term bonds? What are the valuation parameters in Merton's model? What are the valuation parameters m Rabmovitch's model? What are the valuation parameters in Heath, Jarrow and Morton's model? What are the specificities of Heath, Jarrow and Morton's model?

riCin

CHAPTER

g

CO rpO

rate

BOn

ds

OUTLINE

This chapter is organized as follows: modeling along the lines of 1. Section 9.1 introduces the traditional contingent-claims Black and Scholes (1973)and Merton (1974).It is shown that default risk is equivalent to a European put option on the corporate assets. A risky discount debt is priced and the corporate spreads are given. 2. Section 9.2 shows the limits of this traditional approach which has three weaknesses: and the deviations the interest rate uncertainty, the bankruptcy-triggering mechanism from the strict priority rule. A survey of recent contributions is also given. 3. In section 9.3, the Longstaff and Schwartz (1995)model is developed. The building assumptions are given and the resulting price of a corporate zero-coupon bond. 4. Section 9.4 is devoted to the Briys and de Varenne (1997)model which tries to correct a defect of the Longstaff and Schwartz approach. The price of a risky discount bond is given by a closed-form solution. Corporate spreads and interest rate sensitivity are also derived and analyzed.

INTRODUCTION For corporate bondholders, default by the bond issuer is a possibility that cannot be ignored. Expectations of possible future losses are reflected in current bond yields. For angels' do command instance, bonds issued by so-called a higher yield than an otherwise identical Treasury bond. This extra yield, the corporate default spread, rewards the corporate bondholder for carrying the risk of not being repaid. Even though the contingent claims analysis has delivered many insights into the modeling of default, corporate bonds turn out to be more difficult to price than equivalent Treasury bonds. Black and Scholes (1973)and Merton (1974)have been the pioneers in the pricing of claim franiework. They modeled corporate liabilities corporate bonds using a contingent total value of the firm. Default is assumed to occur and debt) options the (equity as on when the debt expires and when the firm exhausts its assets. Because of the limited 'fallen

198

FUTURES

OPTIONS,

AND EXOTIC

gigiglMIO

DERIVATIVES

thus equivalent to a European put option on the corporate assets. Corporate zero-coupon bonds consist of a portfolio with an otherwise equivalent Treasury bond and a short position into a put-to-default. Since the seminal papers of Black and Scholes (1973) and Merton (1974),the contingent claims analysis has been extensively used to price corporate bonds. Black and Scholes (1973)and Merton (1974)have modeled corporate liabilities as options on the total value of the firm. Merton (1974)and its refinements (Lee, 1981; Pitts and Selby, 1983) analyze default spreads of pure discount corporate bonds and the risk term structure of interest rates. In these models, default is assumed to occur when the bond matures and when the firm exhausts its assets. The term structure of interest rates is assumed to be flat and deterministic. Of course, the contingent claim approach has been extensively used to price and to securities. Black and Cox (1976)examined the effects of some study more complex indenture provisions (subordination arrangements, safety covenants). Ingersoll (1977a) and Brennan and Schwartz (1980)valued callable and convertible bonds. Geske (1977) showed how coupon bonds can be viewed as portfohos of compound options. Smith (1979)is a good survey of the traditional contingent claim approach for the pricing of corporate debts. However, this traditional modelmg has shown three limits: interest rates are assumed to be constant, or deterministic, default can occur only at the maturity of the bond, and deviations from the absolute priority rule are ignored. As will be shown below, it is now well documented that these three points play a crucial role in determining the size of the risky spreads. Recent contributions in the default risk literature have proposed modeling frameworks where these issues are explicitly taken into account. The purpose of this chapter is to show how the contingent claim analysis is a useful framework for the study of corporate default risk. Throughout this chapter, we only consider the case of a corporation which issues only two classes of securities: a single homogeneous debt, consisting of a zero-coupon bond, and equity. .

.

TE BONDS

199

TABLE 9.1

The Corporate

Balance

Assets

Liabilities

y

Assets

Sheet

Equity

E,

Debt

D,

A,

bond) (zero-coupon

,

{A.I) Complete

Financial Markets

Financial markets are assumed to be complete and frictionless. Tradmg takes place continuously. There is no taxation, nor transaction cost. Under this hypothesis, - as shown in a previous chapter, one can show that there exists a unique probability measure Q, the risk-neutral probability, under which the continuously discounted price of any security is a

Q-martingale.

(A.2) Corporate

Asset Process

The total value A, of the assets of the firm at time t is governed under the probability Q by the following stochastic differential equation: dA, =

r dt + ogd W,

risk-neutral

(l)

where the continuously compound interest rate r is a constant and the constant a represents the instantaneous standard deviation of the return on corporate assets. W, is a standard one-dimensional Brownian motion.

9.1 9.1.1

THE TRADITIONAL MODE LING

COgg¶lM

NEAI LAINS (A.3) Modigliani-Miller

which issues at time t We consider a corporation 0 two different classes of securities: a single homogeneous debt consisting of a zero-coupon bond and the residual claim, namely equity. The debt has a face value F and a maturity E In this and the following sections, we derive the time t value D, of the corporate zero-coupon bond and the time t value E, of equity. The total value of the corporate assets is equal to A,. The balance sheet of such a firm is given by Table 9.l. valuation Black and Scholes (1973)and Merton (1974)developed a continuous-time risk. model for corporate debt which accounts for default They made the following four =

assumptions.

hnances its total assets at time t 0 by issuing equity. In the absence of taxes and bankruptcy costs, the total independent of this capital structure decision. In other words, (1958)proposition does hold. Cash outflows are assumed to be securities.

The corporation

Assumptions

Theorem =

a zero-coupon bond and value of the firm A, is the Modigliani-Miller financed by issuing new

(A.4) Limited Liability

It is assumed that sharehoMorshave a Manitedliability.ffthings they simply walk away.

go wrong in their firm,

200

9.1.2

OPTIOf¾¿FUTißlllES4hNO The Pricing

of Corporate

EXO

ERIVATIVES

PRICli

201

G CORPORAgg

Equity

Debt

An obvious advantage of the contingent-claims analysis is that it captures the share holders' option to walk away if things go wrong in their firm. At maturity T of the corporate bond, they assess the net worth of the corporate assets and check whether the company is solvent or not. In this traditional approach, if at maturity assets are found to be less than the face value F of the debt, the corporate assets are costlessly transferred to bondholders. Indeed, because of their limited liability, shareholders are not obliged to issue new shares or bonds to raise the cash to repay bondholders if the assets are less than the face value F. To clarify the valuation procedure, we first look at the cash flows which bondholders are entitled to under the various scenarios following potential cash flows.

at maturity

T They have a claim on the

Debt

a FIGURE 9.1

Shareholders'

As far as equity is concerned,

its value at time tsT E,

Solvency

Final Pay-offs

Bondholders'

and

=

is

given by

Cs(A,, F)

(7) T, with

call option

In the first case, the firm is able to fulfill its commitment to bondholders. Assets A, generate enough value to match the face value F, namely Ar a F. Bondholders receive the promised payment F: Dr

F

=

if

A, a F

maturing at date where Cs(A,, F) is a European From Black and Scholes (1973),the value of Ca(A,, F) is given by CE

=

A,N(di)

an exercise price E

o N(d2

- Fe

where

(2)

Insolvency

F)

I,

Ay

di

In(A,/F)

+

=

a /2)(T - t)

(r +

d2

=

A

¯

and N( ) denotes the cumulative normal distribution. Equity is equivalent to a long position in a European call option which characterizes to walk away if things go wrong the shareholders' limited liability. They have the in their firm. As far as liabilities are concerned, their final pay-offs indicate that they can be priced 'option'

In this case, the firm is totaly insolvent: the net worth of its assets is below the guaranteed liability F, namely A, < F. The company is declared bankrupt, and the bondholders receive what is left: Dr=Ar

if

Because of limited liability, the shareholders' Er are as follows: E,

Ar
0

=

(4)

if A, < F

Dr

=

max {Ar - F, 0} F - max {F - Ar, 0}

=

(5) (6)

This combined position can be summarized by Figure 9.1, which shows shareholders' and bondholders' final pay-offs. By applying the option pricing framework, the market value of both equity and liabilities can be assessed.

(10)

t,F)

where Pr(A,, F) denotes the price of the shareholders' put-to-default - that is, to walk away from their commitments. Ps(A,, F) is given by the following formula: pE

These pay-offs suggest that both the corporate debt and equity have the feature of a contingent claim written on the corporate assets. Indeed, their value at maturity T can be rewritten as Er

D,=Fe"TD-PE

(3)

stake is a residual stake. The final pay-offs

A, - F if A, a F

=

as follows:

i,

F)

=

-A,N(-di)

N(-d2

+ Fe

The corporate debt is thus a portfolio made of a long position on a risk-free zero-coupon 6 6 Fe and a short position on a put-to-default PE is simply the t, F). value of the equivalent Treasury bond (sameface value F, same maturity T). The put-todefault PE(Á,,F) C3ptufeS the effect of the limited liability. This effect is of course negative for bondholders. A discount is applied because in some cases the firm will be insolvent and unable to repay the debt owners. bond D, can be The instantaneous standard deviation op of the risky zero-coupon easily computed by applying Itõ's lemma: '"

bond Fe

.

N(di) oc where l,

=

Fe

'"

"|A,.

=

N(di) + l,N(d2)

.a

(12)

202

9.1.3

OPTIONS,

The Pricing

of Corporate

FUTURES AND EXOTIC

DEINVATIVES

203

OS

PRICtNG CO

eld spread

Spreads

(bps) that their investment will not perform as initially planned. If Bondholders face the firm is declared bankrupt, they are paid on what is left. Consequently, they ask for a risk premium to compensate them for the risk they are carrying. In this section, we derive the term structure of default spreads. For the sake of simplicity and without loss of generality, we let t 0. We denote by Yothe yield of a corporate zero-coupon bond De whose maturity is Tand face value F: the risk

Highly leveraged firms

=

1

Yo

=

Using the previous closed-form Yo

ln

Do

(13)

1

ln

r -

Low leveraged firms

N( -di) + N(d2)

.

.

=

1

Yo- r

=

ln

-

i

i

+ N(d2)

N(-di)

(15)

The spread So captures the risk premium of the risky corporate debt. For a given corporate debt, So is a function of the maturity 1" the leverage, as measured by the quasi-debt ratio lo, and the volatility ofthe assets as. The following results can be easily derived: BSc

1

ao

=

BS ßlo a ,

T

T

n(-di)

2

Tl --

-

N(-d

(16)

0

=

7

0

>

.

)

(17) So

/

§

,

(14)

The ratio la is the quasi-debt ratio. It is not equal to the true debt to asset ratio because the numerator is discounted at the riskless rate r. As a result, it is an upward-biased estimate of the real debt to asset ratio. The corporate spread So is defined as the difference between the yield Yoand the yield of an otherwise equivalent riskiess zero-coupon bond. The corporate spread is thus given by the following expresssion: So

Medium leveraged firms

solution for De gives l

=

-

_

0

(18)

where n() is distributed normally with mean zero and standard deviation one. The corporate spread So is an increasing function of the quasi-debt ratio lo and of the volatility of the assets a,,. Indeed, other things being equal, if lo or as are bigger, the debt is riskier because of a higher default risk. However, it is not possible to sign the effect of the maturity on the corporate bond. As shown by Figure 9.2, the term structure of the corporate bond is downward-sloping for highly leveraged firms, humped for medium leveraged firms and upward-sloping for low leveraged firms

FIGURE 9.2

9.2

The Term

Structure

,

,

,

of Corporate

,

,

,

,

,

,

Tim

yrrnaturity

Spreads

THE LIMITS OF THE TRADITIONAL

APPROACH

Although many theoretical papers have studied the pricing of corporate fixed-income securities, empirical investigation remains scarce. Jones, Mason and Rosenfeld (1984) interest rates is unable to showed that the Merton (1974)model with non-stochastic generate corporate spreads compatible with those observed in practice. Ogden (1987) obtained two main results. On one hand, the two traditional default-risk measures (the in the and the leverage) explain about 78% of the variation corporate asset volatility of ratings corporate bonds. On the other hand, default premiums are inversely agency models do not obtain. To avoid related to firm size, which the previous contingent-claims some complicated bond features, Sarig and Warga (1989) used pure discount bonds to test the relationship between risk premiums and time-to-maturity. The results appear to fit the predictions of theoretical models more closely: the term structure of risky spreads is downward-sloping for highly leveraged firms, humped for medium leveraged firms and upward-sloping for low leveraged firms. The traditional modeling of default risk along the lines of Black and Scholes (1973) and Merton (1974)suffers from three weaknesses: the term structure of interest rates is assumed to be flat and deterministic, the bankruptcy-triggering mechanism is simple and the strict priority rule applies.

9.2.1

The Three

Weaknesses

default is thus not really satisfying: actual credit The simplified approach of corporate and leverage spreads are too large to be accounted for, even when excessive volatility should be incorporated to fit real-world levels are used. Three refinements corporate spreads more closely.

IBMB;9UTURESANDEXOTIGhERIVATIVES

204

Interest

Rate UncertaintY

The stochastic

movements of the term structure of interest rates obviously play a crucial of constant rates is embarrassing when one deals with the pricing of interest rate sensitive instruments. Moreover, to price corporate bonds properly, the intertwined effects of interest rate uncertainty and corporate default cannot be ignored.

role. The assumption

Bankruptcy-Triggering

Mechanism

PRICING CORPORATE

9.2.2

Recent

Deviations

fromthe

Strict Priority Rule

The traditional

contingent-claims approach assumes that creditors receive their full seize any portion of the remaining assets. However, it is that strict priority is rarely enforced in financially distressed now well documented corporations. Franks and Torous (1989,1994), Eberhart, Moore and Roenfeldt (1990)and Weiss (1990)indicated that the absolute priority rule is enforced in only 25% of corporate bankruptcy cases. There is also strong evidence that bond and equity markets anticipate violations of the strict priority rule. Fons (1994) indicated that the Moody's recovery rates for senior secured, senior unsecured, senior subordinated, subordinated and junior subordinated debts are respectively 64.59%, 48.38%, 39.79%, 30.00% and 16.33%. Franks and Torous (1989,1994) showed that these recovery rates, based on a sample of 37 firms that formally reorganized under Chapter 11 between 1983 and 1990, for secured debt, bank debt, senior debt, junior debt and preferred stock were respectively 80.1%, 86.4%, 47.0%, 28.9% and 42.5%. This write-down of creditor claims is usually the outcome of a bargaining process which results in shifts of gains and losses among corporate claimants relative to their contractual rights (Milgrom and Roberts, 1992). Franks and Torous (1989, 1994), for instance, reported that, over 41 Chapter ll bankruptcies, junior claimants managed to extract $878 million that should have normally been received by senior claimants. Common stockholders gained a third of those $878 million although they had no valid claim on them. They also reported the same kind of pattern in the case of 47 workouts· under Chapter ll deviations They found that on a sample of 37 firms that reorganized from absolute priority for bank debt, secured debt, senior debt, junior debt, preferred stock and equity were respectively -0.96%, -1.67%, -1.44%, 0.94%, 0.80% and 2.28%. For 45 firms that restructured their debt informally, the deviations for bank debt, senior debt, junior debt, preferred stock and equity were respectively -3.54%, -3.44%, senior claimants -1.39% and 9.51%. In observations these that 0.95%, suggest any case, prefer to be sure to receive a slice of a larger pie than all of a much smaller one, Violations of the strict priority rule should be modeled to better reflect the bargaining game between stakeholders upon default. payments

before shareholders

-

205

Contributions

More recently, contributions have proposed modeling frameworks where the aforementioned issues are taken into account (Jarrow and Turnbull, 1992; Kim et al., 1993; Lando, 1993; Longstaff and Schwartz, 1994; Leland, 1994a, 1994b; Nielsen et al., 1993; Briys and de Varenne, 1997). All of these contributions introduced default risk and interest rate risk (exceptLeland, 1994a, 1994b). They all considered quite general default triggering mechanisms.

for instance, default as the first time when the value of stochastic level. This level is stochastically driven by both corporate assets is lower than a and the corporate assets uncertainty. Longstaff and the term structure uncertainty Schwartz (1994)and Kim et al. (1993)looked at default along the lines of Black and Cox (1976).Financial distress in their models occurs when the value of assets reaches a constant or deterministic barrier. Despite this difference in the definition of the default barrier, all contributions explicitly model the deviation from the strict priority rule. Upon given number of riskless bonds. More bankruptcy, bondholders receive an exogenously specifically, their pay-off upon default is limited to the product of an exogenous specified number and the value at the time that default intervenes of an equivalent (namelysame time-to-maturity to go) riskless bond. As noted by Longstaff and Schwartz (1994),such modeling has the advantage of being consistent with the usual practice whereby claimants are given new securities rather than cash. Leland (1994a, 1994b) derived closed-form bankruptcy-triggering value. solutions for the pricing of risky debts with an endogenous interest rates and timeBut he assumed a constant default boundary, non-stochastic homogeneous debt cash flows. However, some of these recent models suffer from some defects. Indeed pricing equations do not ensure that the payment to bondholders is no greater than the firm's value upon default. Indeed, nothing in Nielsen et at (1993) prevents bondholders from receiving more than the assets permit upon either early bankruptcy or bankruptcy at specified, maturity. Because in their model the pay-off upon bankruptcy is exogenously i.e. independent of the level of the stochastic barrier and of the value of the assets, the presumption is that an external guarantor provides the bondholders with an implicit put option. Consequently, the pricing of corporate spreads is affected by the presence of this implicit put option. As will be shown below, the same kind of problem arises in the Longstaff and Schwartz (1995)model at maturity. Briys and de Varenne (1997)develop an analysis of corporate spreads which corrects these defects.

Nielsen

The very notion of corporate default is somewhat fuzzy. Indeed, the traditional contingent-claims analysis in the spirit of Merton (1974)usually ignores the possibility of early default. The corporate threshold is not accurately modeled. To cope with this weakness, Black and Cox (1976), for instance, assumed a cut-off level whereby default can occur at any time. This cut-off is introduced by considering a safety covenant which protects bondholders-

BONDS

9.3

et al.

(1993)denned,

THE LONGSTAFF-SCHWARTZ

MODEL

is driven by a Longstaff and Schwartz (1995) assumed that the interest rate uncertainty In this framework, if r, denotes the short-term riskless Vasicek (1977) representation. interest rate, its dynamics are given by dr,

=

a(b

-

r,)dt

+ a dZ,

(19)

Brownian where a, b and a are positive constants and Z, is a standard one-dimensional motion. The coefficient b is the long-run mean of the riskless interest rate r,, and a is the speed of adjustment towards that mean. a is the instantaneous standard deviation.

206

OPTIONS,

FUTURES

AND EXOTIC

Vasicek (1977) showed that the time t value P(t, T) of the default-free bond maturing at date Tis a closed-form solution given by P(t, T)

=

A(T - 1)-exp(

-B(T

-\nXo

zero-coupon

t)r,)

-

207

IDS

PRICING CORPORATE

DERIVATIVES

a

(20)

=

M(jT

b

=

- M(iT/n'

n, T)

(28)

M(iT

T)

n,

(29)

where 1- e

A(T-

t)=

(o

exp

""

and

o\

b (T-

-

t)+

B(T-

b

t)+

B(2(T-

t))

a2a'

aa2

(22)

Longstaff and Schwartz (1995)assumed that a firm finances its total assets at time t 0 by issuing a zero-coupon bond, with a face value Fand a maturity T, and equity. The total value A, of the assets of the firm at time t is governed under the risk-neutral probability Q by the following stochastic differential equation: dA, =

r,

d W,]

dt + os[p dZ, +

(23)

(1

e

=

S(t)

=

\

a

a2

a

t-

a2

a'

/

'")

(30)

e

-

'")

(1

e

+

2a3

(1 -

e

2"')

(31)

Longstaff and Schwartz (1995)showed that, although Q(Xo, ro, T) is defined as the limit of Q(Xo, ro, T, n), the convergence is rapid. Numerical simulations showed that setting 200 resulted in values of Q(Xo, ro, T) and Q(Xo, ro, T, n) that were n 'virtually

=

o represents the instantaneous standard deviation of the return on corporate assets, p the correlation coefficient between the riskless interest rate and the value of corporate assets, and W, a standard Brownian motion, independent of Z, The value of the firm A, is assumed to be independent of the capital structure of the firm, which is the standard Modigliani-Miller (1958)theorem Following Black and Cox (1976),Longstaff and Schwartz (1995)assumed there is a threshold value v(t) for the firm at which financial distress occurs. As long as A, is greater than v(t), the firm continues to be able to meet its contractual obligations. If At reaches v(t), however, the firm immediately enters financial distress, defaults on all of its obligations, and some form of corporate restructuring takes place Longstaff and Schwartz (1995)assumed that the default threshold v(t) is a constant, namely, v(t) K. If a reorganization occurs during the life of the corporate bond, the bondholder receives (1 w) times the face value of the security at maturity. In other words, if F is the face value of the corporate bond, the pay-off at maturity T on this contingent claim is F if default does not occur during the life of the bond, and (1 m)F where

=

if it does Longstaff and Schwartz (1995)showed that the value of the risky discount bond given by the following expression at time t 0:

is

=

Do

=

FP(0, T)(l

-

wQ(Xo, ro, T))

(24)

Xo denotes the ratio Ao K and Q(Xo, ro, T) is the limit of Q(Xo, ro, T, n) as n where Q(Xo, ro, T, n) is defined recursively by Q(Xo, ro, T, n) q, q,

=

N(at) -

=

q,N(b,,)

=

cc

(25)

q,

(26)

N(at) for

-

i

=

2, 3,

.

.,

e

(27)

indistinguishable'. The value of the risky discount bond depends on Ao and K only through their ratio Xo. Thus, Xo is a sufficient statistic for the riskiness of the firm. The authors pointed out that the variable Xo can be viewed as an instrumental variable for the credit rating of the firm. However, this model suffers from the defect that pricing equations do not ensure that the payment to bondholders is no greater than the firm's value upon default when the corporate bond reaches maturity. Indeed, the corporation can find itself in a solvent position at maturity (accordingto the threshold) but, nevertheless, with assets insufficient to match the face value of the bond. For example, one can assume a fixed default threshold of K $50 and a promised repayment of F $100. If it turns out that the value of corporate assets until maturity has always been above the threshold (no early default) and that the final value of these assets is $80 at maturity, the firm is solvent' but unable to repay the $100. Moreover, it can be checked that the Longstaff and Schwartz (1995)closed-form solution converges to the price of a default-free zerocoupon bond when the default barrier goes to zero. It should, however, converge to the price of a risky corporate bond that can only default at maturity, along the lines of Black and Scholes (1973)and Merton (1974). The presumption is that an external guarantor provides the bondholders with an implicit put option. Of course, the corporate spreads are affected by the presence of this implicit put option which is not priced. =

=

'threshold-

9.4

THE BRIYS-DE

VARENNE

MODEL

The main objective of the Briys and de Varenne (1997)model is to correct this defect. They denne the default barrier as a fixed quantity discounted at the riskless rate up to the maturity date of the risky corporate bond. As soon as this barrier is crossed, bondholders receive an exogenously specined fraction of the remaining assets. Thus deviations from the strict priority rule are easily captured. As a result, the model is characterized by a stochastic

208

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

barrier and avoids the limitation of having a constant default boundary. Moreover, it ensures that bondholders do not receive a payment greater than the firm value upon default.

9.4.1

The Model and its Assumptions

The model

a

(1995)and

Q-martingale.

It is assumed that the short-term riskless interest rate r, at time t follows a Gaussian diffusion process and that its volatility structure is a deterministic function. r, is governed under Q by the following stochastic differential equation:

dr,

=

- r,) dt + o(t) dZ,

a(t)(b(t)

(32)

for some deterministic functions a(t), b(t) and a(t). a(t) is the instantaneous standard deviation of r,. Z, is a standard Wiener process. This general continuous-time model enables us to consider several different cases, for of Merton (1973a),Vasicek (1977), El Karoui and Rochet (1989), Jamshidian (1991)and Heath, Jarrow and Morton (1992).The only drawback is that negative mterest rates are not precluded in such a Gaussian environment. Nevertheless, it should be noted that for reasonable values of the parameters, this event has a quite low namely probability of occurrence. For constant parameters, a(t) b and a, b(t) a(t) o, the Vasicek (1977)model is obtained. Under this Gaussian framework, the dynamics of the default-free zero-coupon bond P(t, T) maturing at time T is given at time t under Q by the following stochastic differential equation: instance

those

=

=

=

dP(t, T) =

where

os(t,

T) is a deterministic

Q, the

r,

dt - ap(t,

T) dZ,

(33)

function defined by

T)

os(t,

Under

209

BONDS

bondholders the right to bankrupt or force a reorganization of the firm if its performance this benchmark. v(t) denotes does not match some prespecified benchmark. Let v(t) be As time which bankruptcy t. at soon as the value of corporate threshold occurs level at the and bankruptcy or the bondholders safety protects the below v(t), covenant falls A, assets and takes the workout is forced upon. We assume that v(t) is exogenously specified following form:

is rooted

of Black and Cox (1976), Longstaff and in the contributions Nielsen et al. (1993). Financial markets are assumed to be complete and frictionless. Trading takes place continuously. Under this hypothesis, there exists a unique probability measure Q, the risk-neutral probability, under which the continuously discounted price of any security is

Schwartz

PRICING CORPORATE

=

a(t)

-

exp

-

a(s)ds

du

(34)

total value A, of the assets is governed by the following stochastic process: dA =

r, dt + a

p dZ, +

d W,

(35)

'

where as represents the instantaneous standard deviation of the asset return, and p the correlation coefficient between the riskless interest rate and the value of corporate assets. W, is a standard Wiener process, independent of Z,. A, is supposed to be independent of the capital structure decision of the firm. In other words, the standard Modigliani-Miller (1958)proposition holds. Bondholders are assumed to be protected by a safety covenant that allows them to which gives mechanism trigger early bankruotcv. A safetv covenant is a contractual

v(t)

=

a·F·P(t,

T)

(36)

< l. where F denotes the face value of the corporate bond and 0 m a time is assumed to be terminology, the default To use Duffie and Singleton's (1995) accessible (predictable),In other words, the very first time the market value of the firm's surprise'. Another route, suggested by Madan and assets hits the barrier is not a Unal (1993),is to consider the default time as potentially inaccessible. For instance, when some default can be modeled as a Poisson arrival. Default occurs the first time when the objective is, convenient is framework l. Such from 0 to a Poisson process jumps value-based for example, to disentangle cash flow-based insolvencies from market the captured by could be occurrence of the jump insolvencies (suddencash flow shortages Indeed, in a distinction not this is assumptions, of necessary. process). Under our set when the situation is only insolvencies world flow-based cash occur can frictionless viewed as hopeless by stakeholders from a corporate net worth standpoint. To put it in the cash flows could be negative, but if equity value words of Leland l994b), 'sudden

'current

(1994a,

remains, the firm need not be forced into bankruptcy'. The default threshold v(t) can be viewed as an extension of the barrier of Black and Cox (1976)to a stochastic interest rate environment. This specification has three obvious advantages. First, because of the stochasticity of interest rates, the barrier v(t) is also stochastic. The second benefit is obvious when the bondholders' pay-offs upon bankregardless of the form ruptcy are defined. Franks and Torous (1989,1994) indicated that, each creditor receives a bundle of securities in exchange for his of reorganization, original claim in a distressed firm. As we will see, it is easy to relate the level of these pay-offs, namely the fraction of an equivalent riskless zero-coupon bond, to the level of with some of the barrier. Finally, the shape of the barrier enables us to draw comparisons value equal for corresponds to zero. to a a the previous literature. The Mertonian case equal where is stringent covenant is given a situation by a very The completely risk-free receive what they were promised in the first to that bondholders 1. In to are sure case, place. For values of a lying strictly between 0 and 1, intermediate cases of early default equal, the closer a get to 0, the less protective is the are considered. Other things being safety covenant for early default. When bankruptcy occurs, namely when A, crosses v(t) for the first time, corporate bondholders receive a fraction of the corporate assets which is exogenously specified and that is applied to the value that should be received by the write-down represents rule were enforced. Let fi (0 m fi 6 l) denote this priority strict bondholders if the fraction when default occurs before maturity and f2 (0 m fa < 1) when default occurs at maturity. In the limit case where fi f2 l.0, the strict priority rule is enforced and =

=

nothing. shareholders this write-down As already mentioned, receive

bargaining process among the bargaining

process

corporate

between

of creditor claims is usually the outcome of a is not to model explicitly The objective and its outcome (forced creditors firm the and the claimants.

210

FUTURES

OPTIONS,

AND EXOTIC

DERIVATIVES

bankruptcy and liquidation, Chapter l 1, informal debt workouts, etc.). Deviations froin the absolute priority rule are viewed as implicit bond covenants which are anticipated by market players: see, for instance, Eberhart and Senbet (1993).The fact that fi and f could take different values indicates that the nature of the bargaining process upon bankruptcy may be different before or at maturity.

PRICING CORPORATE

21I

BONDS

cash flows under the

coupon D, is thus given by the discounted value of future expected risk-neutral probability Q: r

D,

=

EU

exp

redu

-

-(fraF-1,

,

,ar,,,,,+

+ F·1r

,,a,<,)

f2 Ar·1,

(43)

within the expectations operator corresponds to the discount factor. The second term captures the pay-off conditional upon forced bankruptcy before maturity. The third term represents the best-case scenario for bondholders, namely solvency. The last term gives the final cash flow to bondholders conditional upon both no premature forced bankruptcy and final insolvency (withassets at maturity higher than the threshold value but less than the face value of the bond). By using the methodology of the change of numeraires and time change, it can be shown that D, collapses to the following closed-form solution which gives the price of a

The first term The Valuation of Risky Zero-Coupon

9.4.2

Bonds

The corporation at time t 0 issues two classes of securities: a single homogeneous debt consisting of a zero-coupon bond (withface value F and maturity T) and the residual claim (equity). In what follows, we derive the time t value D, of the corporate zero coupon bond with maturity Tand face value F To clarify the valuation procedure, we first look at the cash flows which corporate claimants are entitled to under the various scenarios. Bondholders have a claim on the following potential cash flows. =

corporate zero-coupon D,

No Default before Maturity the value A,, of total assets has always remained above the and bankruptcy can only occur at maturity T In that default barrier v(u) with tsusT case, bondholders receive a fraction f2 of the remaining assets. Let Ts, denote the first passage time of the process A,, through the barrier v(u):

=

FP(t,

=

inf

{u a t, A,,

v(u)

=

=

aF P(u, T)}

1-

PsQ,) l,

l f2)-(N(d:) ¯ (I -

Under the first scenario,

Ts,

T)

bond: Ps

- N(di) + q,(N(d4

where

l,

(37)

FP(t,

1 0

random variable

Vw é B

di

=

d,

=

=

-

otherwise

Under this second scenario, bondholders receive an amount =

fcAr

=

feaF

P(Ts,,

ds

T) if Ts,<

=

(1 -

+ (A, - F)-1,

72,,,

+

(1 -

f2)-Ar

1,

,

(41)

and

maturity T:

PE(q

/l,)

=

[(((>a,9

ap(u,

uF-1,,«,

+ F-1,,,,,

2,+

f2-Ar Ira,ar

om

d2 + Ï(t,

T)

(47)

d4 + Ï(t, T)

(48)

=

T))2 +

N(-di =

(46)

do + Ï(t, T)

0 -

p2)o

(49)

] du

denote the price as of time t of two European

Bondholders Dr

'

r

pE(l,)

fi)-aF-Ir,,
=

I,) + Ï(t, T)2 2 Ï(t, T)

-ln(q

Ï(t, T)2

Equityholders

ln l, + Ï(t, T)2/2 Ï(t, T) -ln q, + Ï(t, T)2/ if f, I)

(40)

T

To sum up, the total cash-now picture of the firm at maturity reads as follows:

Er

(45)

A,

and

defined by

Default before Maturity

Dr

(44)

(38)

(39) =

T)

=

aFPA

the indicator function Is for B is the real-valued =

¯ N(ds)))

__

a,,,+/21,47,,,,,,<,

la

+q,N(-d4

where

The pay-offs D, at maturity Tare thus equal to F-1,

(N(-dy)

0 - fi) /

-

\l,/

) + N(-d2)

(50) put options

of

(51)

(42)

We are now in a position to price the risky zero-coupon bond issued by the firm. To do pricing technique. The price as of time t of the risky zeroso we use the risk-neutral

P

=

'/

N(-ds)

-

9:

+ N(-ds)

(52)

OPTIONS,

211

FUTURES

AND EXOTIC

DERIVATIV

The value D, of the risky bond involves two ratios, namely l, and q,. The first is the classical Merton's quasi-debt ratio l,. It is not equal to the true debt to asset ratio because the numerator (i.e. the face value of corporate debt) is discounted at the riskless rate. As a result, it is an upward-biased estimate of the real debt to asset ratio. This quasi-debt to asset ratio can also be given another interpretation. Indeed, it is nothing but the forward price of assets, namely the value of assets that prevails under the Q-economy. The quantity q, can be defined as the bankruptcy or early default ratio. It is simply the ratio of the current default threshold to the current value of the firm. As soon as q, is equal to 1, bankruptcy is forced. The expression for D, lends itself to a rather intuitive interpretation. Indeed, the risky bond can be decomposed into five basic components. The first corporate zero-coupon bond. The second term is term corresponds to an otherwise identical riskless zero-coupon the usual put-to-default at maturity as derived in both Black and Scholes (1973)and Merton (1974).The third term, a long position on a European put, appears because of the possibility of an early default triggered by the safety covenant. As such it contributes to mitigating the effect of the previous traditional put-to-default. This interpretation is even more convincing when one considers the case whereby the absolute priority rule is strictly enforced (fi f: 1). The last two terms disappear. In the polar case cx 1, q, l, and the two put options cancel out. The bondholder's situation has become riskless. As soon as the value of corporate assets reaches the present value of liabilities discounted at the risk-free rate, early bankruptcy is forced. The bondholder is then sure to receive the face value Fat maturity T The last two terms in the expression for D, represent the effect of the deviations from the strict priority rule. This effect ís negative. A discount is applied: when fi < l or f2 < 1, bondholders are somehow because of the nonenforcement of the strict priority rule. Each term measures the impact of partially removing shareholders from their residual claimant positions. On top of being economically sensible, the formula for D, is also computationally efficient as it only involves normal univariate distributions. =

=

=

CORPORATE

PRICING

=

'spoliated'

213

BONDS

The corporate spread So is defined as the difference between the yield Yoand the yield of an otherwise equivalent riskless zero-coupon bond. The corporate spread is thus given by the following expression: So

=

(55)

In P(0, T)

Yo +

or equivalently

Se

=

In i

-

-

(I -

2)

Ps(lo)

- P

(N(-d3)

+ qaN(-d4

(N(d3)

-

-

N(di) + qa(N(d4

Under a flat term structure of interest rates, no safety covenant and no deviations from the absolute priority rule, So boils down to Merton's formula for corporate spreads. If the and the deviations from the strict priority rule are skipped but the safety covenant stochasticity of interest rates is preserved, the expression for So is similar to the one derived by Decamps (1994).As expected, the term structure of corporate spreads is affected by the presence of the safety covenant and the violations of the absolute priority rule. Two immediate implications can be drawn from the wider set of parameters influencing the term structure of corporate spreads. First, larger corporate spreads than those derived by Merton are to be expected. Spreads predicted by our model will thus be closer to those observed in practice. Second, it is reasonable to guess that corporate spreads will exhibit more complex properties than those derived in the previous literature. implementation of the To confirm these implications, we now turn to a numerical model. The numerical computations are done with the following basic parameter values. 0.05. The 0.02 and ro For the interest rate process, we fix a 0.2, b 0.06, o coefficient correlation 0.2 and the standard is to deviation set to corporate asset os unexpected negative coefficient of increase because correlation is -0.25 an (the p interest rates implies an unexpected decrease in asset prices). The coefficient a which 0.9 in most of the determines the level of the default barrier is set equal to a 0.9/o. The coefficients f simulations. Equivalently, the early default ratio is set to qa and f, are set to f f, 0.8. It is worth pointing out that, as expected, this model yields larger corporate spreads than Merton's model as witnessed in Table 9.2. Indeed, Merton's model corresponds to an early default ratio qa equal to zero and no deviation from the absolute priority rule (fi fa 1.0). Figures 9.3 and 9.4 illustrate the relationship between the level of the corporate spread and the time-to-maturity of the bond for various leverage levels. Two different leverages are examined: constant quasi-debt ratio lo (Figure 9.3) and constant face value to asset ratio (Figure 9.4). These figures resemble those drawn by Merton (1974): the term for highly leveraged firms, humped for medium leveraged structure is downward-sloping firms and upward-sloping for low leveraged firms. These results match the empirical results by Sarig and Warga (1989). of the bond for Figure 9.5 relates the corporate spread level to the time-to-maturity various values of the early default ratio. The main objective of this figure is to assess the =

=

=

=

=

=

9.4.3

of Corporate

The Valuation

Spreads

.

=

We are now in position to denve the term structure of default spreads. For the sake of simplicity and without loss of generality, we let t 0 and F l. We denote by Yothe yield of a corporate zero-coupon bond whose maturity is T: .

.

=

=

.

Yo

=

-

T

(53)

ln D

Using the closed-form solution for D, gives Yo

=

·

-

T -

-

ln P(0, T) L

(1 - f (I -

f2)

1 )-(N(-d3)

1 - Ps(lo)

lo

P

9

lo

+ qaN(-d4))

(N(d3) - N(di) + qa(N(d4

6

(54)

(56)

¯ N(do)))

.

.

=

.

=

=

=

=

GPTIONS,

14

UTURESANDEKOTIC

ATIVES

96MClilIGCOggPORATE

Yield spread

BONDS

215

Yield spread

(bps)

(bps)

1000

1000

800 800

0

-

=

-

1.1

600

-

400

600

-

qa 0 90 0.9/o Go= 10 =

200 400-

le=0.8

=

-

00

2

4

6

8

10

12

14

16

18

20Maturity

(yrs)

Yield spread

200

(bps) lo = 0.4 2

O

4

6

000 8

*

I

I

I

I

10

12

14

16

18

Maturity

20

500

(yrs)

-

400

IlŠlURE 9.3

Yield Spreads

of Time-to-Maturity

as a Function

Panel 8

:

to = 0.8

300 Ro= 0 qa = 0.9/o qa= le

200 100 Yield spread

00

(bps)

2

4

6

8

10

12

14

16

18

20 Maturity

(yrs)

1000

Yield spread

(bps) Boo

100

-

lo

=

P (0,T) / 1.0

80 -

qa 0 qa= 0.9/o =

600

-

60

-

Panel C : lo = 0.4 400

40 -

-

la 200

=

00

-

le

0

FIGURE

20 -

P (0,T) / 1.25

0

9.4

=

2

4

6

8

10

12

14

16

18

20

Maturity

(yrs)

P (0,T) / 1.5

2

4

Yield Spreads

6

8

'

'

'

'

'

10

12

14

16

18

as a Function

20

Maturity

(yrs)

=

=

=

=

Ja

=

0.2,

f,

=

f2

=

of Time-to-Maturity

useful complement in that violation of the strict priority rule there is no respect spread 1.0), the When any such deviations are absent ( , level is a decreasmg 2 function of the degree of protection provided by the safety covenant. As soon as violations of the absolute priority rule are possible, the situation becomes more complex

impact of the early bankruptcy threshold a. Table 9.2 is a since it also displays the case where

=

FIGURE 9.5 Yield Spreads as a Function of Time-to-Maturity: 0.06, « 0.02, ro 0.05, p 0.8, a 0.2, b -0.25

=

=

Indeed, long-term bonds exhibit a different pattern from short-term bonds. For long-term bonds, the lower the early default ratio (i.e. the less protective the safety covenant), the larger the corporate spread. For short-term bonds, this result does not carry over and an indeterminate relationship holds (see also Table 9.2, Panel A). Bondholders are confronted with what could be dubbed a dilemma'. They simultaneously hold a long position due to the early default covenant and a short position due to the violations 'bankruptcy

of the strict priority rule. These conflicting positions combine with the solvency situation

216

OPTIONS,

TABLE 9.2

f,

(2

=

AND EXOTIC

Yield Spreads as a Function of initial Quasi-Debt f, a 0.2, b 0.06, « 0.02, ro 0.05, p -0.25 =

=

FUTURES

=

=

=

=

DERIVATIVES

Ratio

lo: Ja

=

gdiltililì3COilIPORATE

Yield spread

0.2

basi' (spreads in

(bps) 800

points) go Io Panel

¶=l.0 A: T

=

0.4 0.6 0.8 l.0 1.2 I.4

=

lo

go

=

0.9/o

go

=

0.8/o

go

=

0

(=0.8

(=l.0

(=0.8

(=1.0

¶=0.8

(=l.0

(=08

2 86 493 II16 1598 l8l8

0 2| 148 389 622 768

I 67 452 i191 l972 2513

0 23 184 550 1007 \400

I 64 424 I\44 20l8 28\6

0 23 188 586 1150 1773

| 63

287

BONDS

Pane!A:qa=0.0

600

f,

400

'1=12

2 0 0 0 0 0 0

t,

200 00

4l8 Il\3 1941 2733

=

=

f,

=

0 6

o

t,

4

2

8

10

2

14

16

18

20

8

10

12

14

16

18

20

(yrs)

Yieldspread

(bps) 1000

Panel i 7

5

=

0.4 0.6 0.8 l.0 l.2 f.4

Panel C: T 0.4 0.6 0.8 \.0 I.2 1.4

0 0 0 0 0 0 =

800-

22 I27 290 446 560 627

7 44 109 179

233 270

20 129 327 546 735 870

8 6I 168 298 419 5l6

19 I27 334 588 836 1044

9 67 196 379

588 805

l9 124

329 589

·

90 ¯

0

''''2

400

866 !!35

200

fef2=LD

0

10 0 0 0 0 0 0

''=fp=06

600

43 109 173 223 259 282

I6 43 72 95 \\2 124

47 128 2l6 293 353 396

24 7l 125 174 2\5 245

48 137 242 342 428 495

29 93 180 278 378 477

4

2

6

Maturity

(yrs)

Yield spread

48 14l 258 383 506 624

(bps) 1400 1200

-

1000 os 600 400

I

'

of the firm, as measured by lo, to deliver an ambiguous picture of the relationship between the spread and the early default ratio. In any case, firms that have issued long-term bonds and are lo-insolvent (lo> l) are characterized by spreads inversely related to the level of qa. In such cases, bondholders discount negatively the absence of any safety covenant. In that respect, it is interesting to underline that, other things being equal, the less lo-solvent the firm, the larger the corporate spread. Violations of the strict priority rule also have a strong influence on the level of corporate spreads as displayed by Table 9.2 and Figure 9.6. The less bondholders receive upon bankruptcy, the larger the corporate spreads. Figures 9.5 and 9.6 also suggest that the corporate spread is more elastic to changes in i and/or 2 than to changes in qa. Figures 9.7 to 9.9 capture the effect of corporate asset volatility on the level of spreads. Merton (1974) showed that the corporate spread is an increasing corporate function of corporate volatility. Decamps (1994) proved that this result was a direct product of the constant interest rate assumption. According to Decamps (1994), when

200

.1

O

4

2

o

6

FIGURE 9.6 Yield Spreads as a Function 0.05, p 0.06, o 0.02, ro 0.2, a 0.2, b =

=

=

=

=

i

i

i

I

'

I

8

10

12

14

16

18

of Time-to-Maturity: -0.25

20

Maturity

(yrs)

to

=

0.8,

«»

=

correlated, the corporate interest rates and the value of the firm's assets are negatively spread is a single peaked function (first decreasing, then increasing) of the volatility of the firm. Figure 9.8 indicates that neither Merton's result nor Decamps's proposition extends to our setting (see, for instance, Panels A and C). coefficient p on the level of Figure 9.9 describes the impact of the correlation For positive values of the corporate spreads as a function of the firm's asset volatility. monotonically while for negative relationship increasing, correlation coefficient p, the is

Yield spreads

Yield spreads

(bps)

(bps)

1000

700

800 600

Panel A : lo

=

600

_

"v

-

1.1

=

500

0 3 a

400

=

o

=

400

himbf A : lo -

o2

-

0.1

qa= 0.9 le

300 200

200

100

2

6

8

10

12

14

16

18

20

0.20

0.15

0.25

Asset volatility

0.30

(bps) 450

go= 0

400

1000

a

=

0 3

350 300

800

00

2

4

6

8

10

12

14

16

18

20

·

qa= 0.9 le

00

MatuSrty

Yield spreads

0.05

0.10

0.20

0.15

0.25

0.30

Yield spreads

(bps)

(bps)

140

200

120

qa=0

-

o,=0.3

100

to

0 10

Yield spreads

1200

0.4

0.05

(yg

(bps)

:

-

00

Mategy

Yield spreads

Panet C

·

-

0

=

0

°=

-

150

-

80

·

qa=0.91,

-

60

°v=02

-

40

Panel C

:

to

=

0.4

100

=

no

-

lo

-

50

·

200

FIGURE 9.7 Yield Spreads f2 0.8, a 0.2, b 0.06, « =

=

=

=

0

2

i

'

I

4

6

8

10

12

14

16

of Time-to-Maturity: as a Function 0.05, p = -0.25 0.02, ro

18

20

Maturity

go

O

(yrs) =

0.914,

fi

=

=

FIGURE 9.8 0.8, a

=

0.2, 6

0.05

O

0.10

0.15

0.20

Yield Spreads as a Function of Asset -0.25 0.02, ro 0.05, p 0.06, «

=

=

=

0.25

Volatility:

0.30

Asset volatility

T

10,

=

f,

=

f2

=

=

values

of the same coefficient this is no longer true. As a matter of fact, corporate is measured by the quantity Ï(t, T) which is the volatility of the ratio A,/P(t, T). As already mentioned above, this last quantity is the forward price of assets, namely the relevant corporate zeroasset for pricing the corporate But Ï(t, T) is a monotonically increasing function coupon bond under the Q-economy. of the asset volatility Ja if p is positive, while it is first decreasing, then increasing, if p is negative. volatility

'underlying'

9.4.4

The Interest

Rate Elasticity

of Corporate

Bonds

model has interesting implications for both portfolio and asset-liability management (ALM). Indeed, bond portfolios and the balance sheets of many institutions movements in the are interest rate sensitive. To protect these latter against unexpected term structure of interest rates, the investor needs to evaluate his risk exposure accurately.

This valuation

220

OPTIONS,

AND EXOTIC

FUTURES

MiiRIVATIVES

Yield spreads

PRICING

CORPORATE

BONDS

gli

Let r¡n denote the interest elasticity

(bps)

1 8 Do

340 r¡n 330 -

1.1

320

By applying obtams:

0.5

=

p

Panel A : \p

.

p

300

0

=

=

-,

I

'

'

'

0.05

0.10

0.15

0.20

025

0.30

Itô's lemma to the closed-form

o

=

N(-d

lo

)- -

N(-ds)

-

(1

)

n(di)

250

-

150

100

(1 - fa)

N(-di)

d3)

- N(

Ï(0, T)

n(d3)

qan(ds)

qan(d4)

Ï(0, T)

Ï(0, T)

Ï(0,

=

p

-

From the expression for yo, several polar cases can be recovered. The first deals with the situation where there is no violation of the absolute priority rule and where the safety 1). Under these assurnptions, the interest rate covenant is fully protective (i.e. a elasticity of the corporate bond is given by Vasicek's formula for the interest rate elasticity of a riskless zero-coupon bond:

0.5

-

,

=

o_o

=

p'

0

0.05

-o.5

'

'

'

'

0.10

0.15

0.20

0.25

0.30

Asset volatuity

r¡o

Yield sppsreads

=

r¡,

=

-

B(T)

(60)

In the second case, no safety covenant is attached to the corporate zero-coupon 0). There is, however, still no deviation from the strict priority rule (fi a As a result, r¡o collapses to Merton's formula extended by Decamps (1994): =

2oo 150 1oo

so

p=0.5

fi

=

f2

9.9 =

0.8,

r¡s=

·

_0 5 I

0.05

0.10

I

I

'

0.15

0.20

0.25

Yield Spreads as a Function of Asset Volatility: 0.25 0.06, « 0.02, ro 0.05, p 0.2, b a =

=

=

=

-B(T)+

+B(T) a

-

o0 FIGURE

bond (i.e. =

f2

=

1.0).

·

p=o.o

PanelO: ¼=04

(59)

T)

-

50 0

Do, the following expression

2n(ds)

N(-d3)

3oo

-

solution

with

Asset volatility

(bps)

Panel B : Io ¯ 0.8

(57)

-

Do Bro

0.0

Yield spreads

200

of the corporate bond:

measure

=

0.30

T

=

Asset volatility

10, go

=

0.914,

Interest rate elasticity and duration measures are now commonplace. Nevertheless, most of them are quite restrictive and apply only under a specific set of assumptions. Corporate default, for instance, is rarely taken into account. This is unfortunate and produces biased estimates of the true elasticity of a corporate bond: see Ambarish and Subrahmanyam (1989)and Chance (1990). In that respect, this model corrects these deficiencies. Moreover the simple structure ofour corporate bond pricing formula makes it easy to compute the relevant elasticity measure.

Do

.N(-d,)

(6!)

A careful inspection of this last expression enables a better understanding of the more complex expression for yo. Indeed, the last formula is basically composed of three terms which can be disentangled as follows. As already mentioned above, the first term corresponds to the interest rate elasticity of a riskless zero-coupon bond. The second term (withmbrackets) is equivalent to the interest rate elasticity gap between the firm's assets and the default free zero-coupon bond. This is so because the ratio pos/o measures the interest rate elasticity of the firm's assets. At this point it is worth mentioning the crucial role that the correlation coefficient p plays in the determination of the overall interest rate elasticity of the corporate bond. Other things being equal, a negative p entails a higher interest rate elasticity. The third term can be defined as a probability adjusted ratio. The general expression for yo is obviously more complex. The full Ro term reflects the impacts of both the safety covenant and the violations of the absolute priority rule. When it comes to immunization or related techniques, practitioners use the duration tool (especiallyMacaulay duration) more frequently than the interest rate elasticity itself Duration-matching methods, for instance, are quite popular. The accuracy of these methods, however, crucially depends on the correct measurement of duration. To provide results that are closer to current managerial practices, we now define the effective .

-

.

.

222

FUTURES

OPTIONS,

AND EXOTIC

DERIVATIVES

PRI ING CORPORATE

Duration

duration of the corporate zero-coupon bond. This duration Ao is defined as the maturity of the default-free bond which exhibits the same interest rate elasticity as the risky corporate zero-coupon bond, namely =

Qo

-

a

p(0,

(yrs)

=

-

-

6

A p)

(62)

-

=

5

PMWBA : qa Oh =

4 3 -

1

(63)

o0

90 T

(years) Ao

(years)

=

«

Ao as a Function of Time-to-Maturity: -0.25 0.02, ro 0.05, p =

90 Ao

0.2,

=

fi

=

| 5 10 I5 20 Panel I 5 IO IS 20

=

,

.

8

10

12

14

.

•

16

18

(years)

=

0.9/o Ao - T T

go Ao

(years)

=

Panel

C: lo - 0.4

I 5 10 l5 20

I.00 4.90 8.64 i i.i i |2.63

(yrs)

11- I2 - 1ß

-

12

-

10

Panel 8 : go

=

0.9 le

8

-

i

6 4

(2

=

·

-

4

0

=

t2 0.6

-

10

12

14

16

18

20

Maturity

(yrs)

Duration

0

(yrs) =

20

Ao - T T

2

1.0

·

2-0.8

176.0% -10.6% -21.0% -27.5% -34.3%

4.20 4.14 6.83 9.2I i I.02

320.0% -17.2% -31.7% -38.6% -44.9%

3.73 3.79 4.80 5.4| 5.75

273.0% -24.2% -52.0% -63.9% -71.3%

Panel C : go

-

le

10

-

2=0.6

5 -

0

0

I

I

2

4

I

6

'

8

I

·

·

·

'

10

12

14

16

18

20

0.8

2.02 4.43 7.57 10.23 12.23

Maturity

20

I.I

2.76 4.47 7.90 I0.87 13.14 B: 12

,

6

16 14

15

PanelA:Io

=

,

4

=

=

10 Ao - T T

Ja

,

2

ouration (yrs)

2

=

=

¯

2

In (1 + aq,,)

Table 9.3 and Figure 9.10 give some numerical insights into the behavior of the duration of the corporate zero-coupon bond. Table 9.3 is quite informative about the influence of the early default ratio on the effective duration. Indeed, other things being equal, the closer qa gets to zero, the smaller is the effective duration. This result makes sense if one remembers that a safety covenant is equivalent for bondholders to a long position on a put option. If this long position is progressively lifted, a gradual decrease in the duration of the corporate bond is rather intuitive. Table 9.3 also delivers that the effective duration is usually smaller than the maturity, i.e. the Macaulay duration. But this result is not general and does not carry over to short-term bonds. In Panel A, the duration of a one year to

=

1.o 0.8 0.6

=

=

-

a

TABLE 9.3 Duration 0.06, 0.8, a 0.2, b

t f, y - (2 fi I2 =

7

In a Vasicek framework, this can be rewritten as Ai,

223

BONDS

102.0% -I l.4% -24.3% -31.8% -38.9%

l.82 4.27 6.77 8.84 10.40

82.0% -I4.6% -32.3% -4l.l% -48.0%

l.76 4.20 5.77 6.49 6.78

76.0% -|6.0% -42.3% -56.7% -66.\%

0.0% -2.0% -13.6% -25.9% -36.9%

l.00 4.90 8.45 10.42 i I.46

0.0% -2.0% 15.5% -30.5% 42.7%

l.0 4.91 8.35 9.74 9.91

0.0% -1.8% -16.5% -35.1% -50

FIGURE 9.10 Duration as a Function -0.25 0.06, « 0.02, ro 0.05, p =

=

of Time-to-Maturity:

lo

=

Maturity

(yrs) 0.8, a

=

0.2, b

=

=

1.1 and qa maturity corporate bond, for instance, is 3.73 years (when lo 0). This In a constant interest rate environment, Leland result may seem quite counter-intuitive. (1994a,1994b) obtained the result that effective duration is always less than Macaulay duration. This is not true here. This point is confirmed by Figure 9.10. In the three panels, effective duration is greater than Macaulay duration for short-term bonds. Moreover, this holds true whatever the intensity of the deviations from the strict priority rule. In the specific case where qa 1.0 (Panel C), effective duration is equal to lo and i f Macaulay duration (45° line). But in this scenario the risky bond becomes a riskless bond. =

=

=

=

=

224

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

This lengthening of the effective duration finds its roots in the interest rate elasticity gap between the firm's assets and the default-free bond. This gap is one of the determinants of the overall interest rate elasticity of the corporate bond. For short maturities, the quantity B( T) is small. Moreover, Figure 9.10 displays results in the case of a negative correlation coefficient p. Other things being equal, this entails a stronger clasticity r¡n which in turn translates into a higher duration. For a positive correlation coefficient, Leland's result would be recovered. Finally it is worth pointing out that more stringent deviations from the strict priority rule contribute to shorten the effective duration.

•

•

Ÿricing

IRSurance

•

Linked SOnds

SUMMARY The purpose of this chapter is to show how the contingent-claims framework is appropriate for analyzing the effect of default risk on corporate bonds. Black and Scholes (1973)and Merton (1974) have modeled corporate liabilities as options on the total value of the firm. Since their seminal papers, this approach has been extensively used to price more complex securities. More recently, contributions have proposed modeling frameworks where new issues are taken into account: interest rate uncertainty, the bankruptcy-triggering mechanism and deviations from the strict priority rule. The Longstaff and Schwartz (1995)model introduced default risk and interest rate risk. Financial distress occurs when the value of assets reaches a barrier. Briys and de Varenne (1997)develop a corporate bond valuation model which avoids some of the shortcomings of the previous models. The bankruptcy-triggering mechanism is directly related to the pay-off received by bondholders when early bankruptcy is forced. Because it accounts for stochastic interest rates, default risk and deviations from the absolute priority rule, their model is capable of producing quite diverse shapes for the term structure of corporate spreads. Interest rate elasticity and duration measures have also been derived. Among other things, effective duration has been shown to be significantly different from the traditional Macaulay duration.

POINTS • • • • • •

• • •

OUTLINE

This chapter is organized as follows:

Genoese 1. Section 10.1 starts this chapter by telling the story of a thirteenth-century merchant who protected his business by issuing what could be called the ancestor of insurance linked bonds. This transaction is indeed very similar to the type of transaction that nowadays involves insurance companies and capital market players. Some of these modern transactions are then briefly described. Finally, this section draws the analogy between insurance linked bonds (alsocalled nature-linked bonds) and junk bonds. 2. Section 10.2 describes the basic features of an insurance linked bond. It shows how such a bond is created and what the basic transactions involved are. 3. In section 10.3, the pricing model is built and the various assumptions underlying it are presented. A closed-form solution is derived under the assumption of constant interest rates. This section delivers an in-depth analysis of the spread over Treasury of an insurance linked bond and of its duration. Implications of the pricing model on the validity of traditional performance statistics (suchas mean return and standard deviation of return) currently used in the analysis of insurance linked bonds are also analyzed.

FOR DISCUSSION

What are the underlying assumptions of the Black-Scholes model? What is a put-to-default? Wliat is the deviation from the strict priority rule? Give some examples of deviations from the strict priority rule Why is the contingent-claims framework so useful in modeling default risk? What does default' mean? What are the three weaknesses of the traditional approach? Describe the Longstaff and Schwartz (1995) model What is the main defect in their model? Describe the Briys and de Varenne (1997) model· 'carly

•

CHAPTER

INTRODUCTION Insurance risk or nature risk is considered by many as an emerging asset class. Indeed, the record number of natural catastrophes topped by the significant number of man-made catastrophes has raised concern about the available capacity in the insurance and reinsurance markets. While a S10 billion fluctuation is easily absorbed by the capital markets, the same is not true for the insurance industry. These facts have led to the idea that insurance or nature risks can also be directly priced and exchanged on capital based on markets. The Chicago Board of Trade has introduced derivatives contracts insurance indices. More recently, investment banks have structured private placements involving bonds with coupons and/or principal exposed to insurance or nature risks.

226

OPTIONS,

FUTURES

AND EXOTIC

DERæ

The purpose of this chapter is to examine such bonds. More specifically, it addresses the issues of the pricing and interest rate risk of insurance linked bonds. These questions are important since insurance linked bonds are often presented as overperforming financial instruments. It is claimed, for instance, that they outperform Treasury bonds while exposing the investor to a lower risk level. This last claim is carefully investigated in this chapter. It is shown that it is rather dubious, given the embedded non-stationarity of insurance linked bond returns The purpose of the present chapter is twofold. First, it proposes a simple pricing model for insurance linked bonds and derives spread over Treasury and duration measures for such bonds. Surprisingly enough, the whole issue of duration measurement has been neglected in most publications dealing with insurance linked bonds. Such an omission is rather unfortunate. Indeed, we show hereafter that an insurance linked bond is quite sensitive to interest rate movements. Second, based on the simple pricing model, it reexamines the case for outstanding performance of insurance linked bonds. Among other things, it shows that the methodological critique addressed by Ambarish and Subrahmanyam (1989)to performance studies of junk bonds applies also to insurance linked bonds-

10.1

NATURAL HAZARDS, INSURANCE INSURANCE LINKED BONDS

RISKS AND

Quirksof Mother Nature have always been threatening mankind. In ancient times, peasants whose sole means of subsistence were agricultural had to resort to some naïve pantheon animated by benevolent saints and gods. In Burgundy, a French province famous for its magnificent wines, wine growers prayed to Saint Médard and Sainte Barbe for favourable weather. Saint Médard was venerated on the grounds of his ability to make rain while Sainte Barbe was invoked to chase away lightning. This celestial type of coverage was usually rounded off by some more terrestrial type of hedging. For instance, sea navigators would not only pray to Saint Guénolé but also arrange some contracts to mitigate their exposure to sea perils. Indeed, in 1298, a famous Genoese merchant, Benedetto Zaccaria, structured an interesting deal with two Genoese bankers, Baliano Grilli and Enrico Suppa (see Favier, 1987). Zaccaria had to ship 30 tons of alum from Aigues-Mortes in France to Bruges in Belgium. The only way to deliver such a heavy quantity of alum was to freight a vessel Zaccaria, like his contemporary fellow merchants, knew that sea navigation was a very risky venture. The odds that the vessel would never reach its final destination were fairly high. Sea hazards and unexpected storms could simply send the vessel and its valuable freight to the bottom of the ocean. Even if the seas were quiet, there was always the danger of greedy buccaneers and corsairs preying on merchant shipping. In both cases, the freight would be lost and the merchant would eventually face bankruptcy. Zaccaria was well aware of these contingencies. With the help of Grilli and Suppa, he designed a contract to shift the risk away from his business. The deal was the following. Zaccaria sold the alum to Suppa and Grilli on the spot at an agreed price. The alum was then loaded on the vessel that headed to Bruges. If the vessel safely reached Bruges, Zaccaria had to purchase the alum back from Suppa and Grilli. He then sold it to its local client. The repurchase price was obviously much higher than the price of the initial transaction. If things went wrong, say because of a storm, and the alum went lost during i

i

PRICING

INSURANCE

127

LINKED BONDS

the maritime trip, Zaccaria did not owe anything to Suppa and Grilli. In other words, Suppa and Grilli granted Zaccaria an option to default on the repurchase transaction provided the feared event occurred. A modern economist would simply say that the three risk-sharing rule. Viewed from Genoese businessmen had come up with a non-linear risk-sharing option rule is convex: he was long an to default. Zaccaria's standpoint, the with rule reconciled risk-sharing this 1298, be ingenious Although dating back to can Zaccaria risk-bearing. Indeed, faced of optimal theory two results of the recent some risk types of risk, namely a businessman risk and an idiosyncratic risk. The idiosyncratic the time, of risk. At the businessman is due to the sea hazards and is usually independent this risk was not easily hedgeable. In such a setting and with some additional restrictions, Franke, Stapleton and Subrahmanyam (1995)and Briys and Viala (1996)have shown that The Genoese contract also fits well in rule is indeed non-linear. the optimal risk-sharing marketplace. More and more investment the current developments that affect the fmancial loss the whose directly by pattern of natural hazards returns are determined instruments specifically, or some other contingencies are either traded or privately placed. More natural hazards risks and various insurance risks are securitized in order to place them with investors on the marketplace in the form of securities. Other things being equal, this is exactly what Zaccaria's loan was all about. Grilli and Suppa extended a natural hazardlinked loan through which the risk of sea perils was securitized. Nowadays, more sophisticated versions of Zaccaria's loan are issued by investment banks such as J.P. Morgan, Salomon Brothers, AIG Combined Risks, Swiss Re Financial Products or Merrill Lynch. For instance, Nationwide Mutual Insurance Company issued $400 million in 30-year secured bonds at 2% above long-term Treasury bond rates. Using the proceeds, the company bought Treasury bonds to use as collateral. If a catastrophe occurs in the next 10 years, Nationwide will substitute company surplus notes and use the Treasury securities to pay losses. The risk for investors is the possibility that Nationwide Other bonds where the would default on the surplus notes following a major catastrophe. risk principal is itself partially or totally at are also structured. These are usually zerocandidates potential for securitization cover quite a large coupon bonds. Hazards that are airline crashes, and so on. According earthquakes, hurricanes, including floods, spectrum advantages': 'In addition offer investors such bonds Re Sigma Swiss (1996b), to to providing an above-average yield potential, such instruments are attractive since their performance is not correlated with any other financial risks. They therefore promise an outstanding diversification effect.' To the reader familiar with the CAPM theory, this is a free lunch'. Indeed, if insurance linked bonds have argument may sound like no systematic risk, their return should equal the risk-free return. It is interesting to observe that insurance linked bonds (also called insurance linked assets) are often compared to corporate bonds. Indeed, investment banks use credit implied by the spreads on corporate bonds as a benchmark to evaluate the richness is that insurancespreads on insurance-linked bonds. The output of such comparisons conclusion, characteristics. The common linked bonds offer outstanding risk-return based upon historical analysis, is that an investment in an insurance linked bond, after adjusting for risk, tends to outperform an investment in, say, Treasury bonds. For instance, according to AlG Combined Risks, if an investment in natural catastrophe linked bonds with principal fully at risk had been available over the last 25 years, its return would have been four times that of Treasury bonds with a lower volatility. The same kind of argument was applied in the 1980s to investment in junk bonds: see Altman (1987) and Blume and 'decisive

'there

OPTIONS,

228

FUTURES

AND EXOTIC

DERIVATIVES

performance fact and the lack of correlation risks led Swiss Re Sigma (1996b)and many possibility shifts the so-called additional efScient frontiers upward, within both national and international modeling frameworks'. The same shift was documented with the introduction of emerging market assets. Finally, it should be emphasized that insurance linked securities make it possible for investors to play the insurance game without having to carry the other risks associated with an investment in shares of an insurance company.

Keim (1987, 1989). This risk-adjusted between natural hazards and financial investment houses to conclude that 'this

10.2

THE STRUCTURE BONDS

.

OF INSURANCE

.

.

LINKED

PRICING

INSURANCE

LINKED BONDS

229

reinsurance markets. Indeed, according to Swiss Re Sigma, the potential losses from catastrophe risks worldwide exceed the cover capacity ofthe insurance industry. The capital reinsurance), as the argument goes, no and the surplus of the insurance industry (including longer offer a sufficient cushion to absorb the losses caused by natural hazards. The same kind of concern was expressed by Kleindorfer and Kunreuther (1996)and Cummins et al. (1996).In a nutshell, the insurance industry feels that it can no longer provide coverage against natural hazards without facing the risk of insolvencies or significant equity losses. Compared to the daily fluctuations on the financial markets worldwide, these potential Seventy basis points of fluctuations on the US financial losses seem less worrying. markets is equivalent to a fiuctuation of $US 133 billion on average (Swiss Re Sigma, 1996a). This a lot more than the size of the Kobe losses. Still, a 70 basis point daily fluctuation is something which is the common rule rather than the exception on financial markets. Canter, Cole and Sandor (1996)noted that a $50 billion catastrophe were to occur in the U.S., approximately 20% of the capital of the primary and reinsurance industries would be wiped out'. The same point was made by Cutler and Zeckhauser $10 billion is large relative to insurance (1996).They argue, for instance, that markets, it is tiny relative to asset markets as a whole'. This prosaic comparison has recently revived the idea that insurance risks can also be priced and exchanged outside the traditional insurance and reinsurance markets, namely on the financial markets. The first proposal emanated from the Chicago Board of Trade where insurance futures contracts and options are now traded. The next step is now towards securitization and private placements. Numerous institutions including investcompanies and direct insurers are currently developing new ment banks, reinsurance classes of assets, the so-called insurance linked assets. These assets are very close in spirit and in action to the 1298 Genoese loan described in the introduction. The most common ones are insurance linked bonds. A typical private placement can be disentangled as follows (see Lehman Brothers (1997)).Suppose there is a corporation that wants to be insured against some nature risk in some geographical area for two years. This corporation cannot find suitable insurance coverage. It turns to an investment bank which suggests the following sequence of transactions. The corporation pays an insurance premium at the same time that some investors bring cash to a so-called Special Purpose Vehiclei (SPV). The insurance premium and the cash payment are bundled together to purchase some Treasury bonds or strips. For the sake of simplicity, we assume that the Treasury instruments purchased are zero-coupon bonds. After the purchase of the strips, the SPV issues two-year naturelinked zero-coupon bonds yielding a spread over Treasury. These bonds are delivered to the investors in exchange for their initial cash payment. As a result, investors hold a nature-linked investment. They accept to lose part of the principal of the bond if some event (saytoo much wind or too much hail) occurs in the designated geographical area during the first two years. The two-year period is called the exposure period. If the event occurs during the first two years, the corporation receives its indemnity from the SPV The SPV is able to cash out this indemnity because it keeps the fraction of the principal of the nature-linked bond that the investors have agreed to give up on the event Cash flows are occurrence. The risk has been passed on to the financial market. 'if

In his masterly piece on the history of climate since the year 1000, Le Roy Ladurie that 1316 had been very wet, with some rather unexpected consequences. Inhabitants of Tournai, a wealthy town of Belgium, were forced to drink their execrable local wine because French vineyards had been ruined by heavy rainfall! On a more of nature's and weather's serious note, it is clear that the financial consequences misbehavior can be very significant. For instance, strong hailstorms have caused heavy damage to vehicles, mobile homes, windows, and so on. Hailstones can be surprisingly heavy (morethan a kilogram) and hence dangerous. In 1986, a violent hailstorm killed 100 people and devastated more than 80 000 real estate properties in central China, In Bangladesh, a sudden hailstorm on 14 April 1986 killed 92 people (seeGauthier (1995)). The spectrum of hazards generated by Mother Nature is wide. In the United States, in 10 major populaaccording to Simpson (1997),over 1983-88 temperature vanations tion centers caused the cost of energy consumed in space heating and cooling to vary by variation was $9 billion over the whole an average of S3.6 billion per year. The total market of the for heating and cooling energy. $75 billion US country. This represents 12% northeast the United States can have weatheroil in and distributors heating Natural gas driven sales that exceed 90% of their sales during the winter season (see Simpson (1997)).In the United Kingdom, subsidence risk came second only to gales and storms in terms of costs to insurers over the last decade. Although buildings are constructed to receive support from the ground, shrinkage of clays caused by prolonged drought presents a significant subsidence peril to these buildings. According to Swiss Re Sigma (1996a),the insurance industry has been significantly hit in the 1990s by a record number of natural catastrophes. In 1995, they caused insured losses of $US 12.4 billion, more than half of which were accounted for by four single hazards costing some billion dollars each: the Kobe earthquake, hurricane Opal, a hailstorm in Texas and winter storms combined with floods in Northern Europe. The Kobe earthquake was by far the most severe disaster: total damages were estimated at $US 82.4 billion and insured losses at SUS 2.5 billion. Losses in excess of $US 1 billion are much $US more frequent. Since 1988, every year has seen at least one disaster costing more than also 1988. on encountered often Storms this less size were before 1 billion. Disasters of are the rise. They are the most costly cause of damages for the insurance industry. Over the period 1989 -95, they entailed losses amounting to nearly $US 10 billion. topped by the significant number of manThese record numbers of natural catastrophes capacity m the insurance and made catastrophes have raised concern about the available

(1983)reported

'while

'bankruptey-remote'.

SPV is necess to make the nature-Unked bond designated event only. This is our assumption in what follows.

Indeed, the bond has to be contingent

on the

FUTURES

OPTIONS,

230

AND EXOTIC

DERIVATIVES

RICING INSURANCE

to the state of nature where they are the most badly needed. Howeve f occurs during the exposure period, the investors receive the Treasury yield topped by the proceeds from the insurance premium. bond looks like a junk bond. A junk bond offers To put it in a nutshell, a nature-linked an interesting spread over Treasury investments because its holder grants the shareholders of the issuing company an option to walk away (which is usually deep in-the-money). A nature-linked bond is a junk' bond where the investor is short an option to insure the corporation in case things go wrong. As should be obvious by now, the whole problem is to make sure that the level of the insurance premium is high enough to warrant an attractive spread over Treasury.

reallocated nothing

'natural

10.3

A SIMPLE PRICING MODEL OF INSURANCE/ NATURE LINKED BONDS

10.3.1

The Valuation

Model

model for valuing nature-linked In this section we develop a simple continuous-time non-catastrophic. risks words, we restricted other that In nature is to bonds. The scope are rather than the Poisson type of uncertainty. focus on the Wiener type of uncertainty Although a more realistic pricing setting would incorporate at the same time both interest rate risk and natural hazards or insurance risk, we concentrate in the following on the simpler case of constant interest rates.2 In other words, the only source of uncertainty stems from natural hazards or some other insurance risk. This uncertainty is usually index. captured by a suitable index. The return on the nature-linked bond is pegged to this index like published those official such institutions either by This index as an can be industry statistical agent, Sigma, Property Claims Services (PCS hereafter), an insurance 3 index or a customized index a service of Swiss Reinsurance Company, a meteorological covered Simpson reficcting (orcausing) the contingency to be (1997)). (see,for instance, period. If this A trigger level is defined which remains active during the exposure of the bond is allowed to default. trigger is hit by the index the issuer Hereafter, the index and the trigger level are labeled I(t) and K(as oftime t) respectively. index. The bond is a zero-coupon The bond payoff is contingent upon the value taken by the is characterized by E It maturity Tand value of face bond an exposure period maturing at time T' s T. If the index does not exceed4 the trigger level K during the exposure period, .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-

.

.

.

.

.

the investor is sure to receive the face value E If, however, nature decides otherwise (thesocalled event'), the investor loses part of the face value of the bond and receives coef6cient applied to the bond. In practice, only (1 a)F where a denotes the write-down the period between the end of the exposure period and the maturity of the bond is used for , accurately assessing the value taken by the mdex during the exposure period. 'qualifying

.

.

.

.

.

of 2This assumption is less drastic than it appears at first glance. Indeed, as can be seen from Appendix 10.A, even in the case the stochastic interest rates (i.e. a Vasicek term structurek we maintain a zero correlation assumption between the process for interest rate and that for the loss index. To avoid moral hazard problems, it is clear that the index shall not be subject to manipulations 4We assume that the index may bit the trigger from below We could have looked at the sciated case where the trigger is hit from above. A typical case would be a rain index tracking the risk of a low rainfall level. "This is easily done for a weather-related index. 11 is more dif6cult for a loss-related index.

LINKED BONDS

231

stochastic process. In the We assume that the index is governed by a continuous-time data is reasonable. Indeed, one of case of meteorological data, the assumption of the primary tasks of meteorological offices is to release reliable on-line data on temperature (such as degree-days measured by the National Weather Service in the USA), windspeed (the famous Beaufort scale6), river water level or rainfall. As far as insurance loss-related indices are concerned, the assurnption of an instantaneous claim process is quite common in the literature (see, for instance, Shimko (1991, 1992), Cummins and Geman (1995)).The index I(t) is assumed to be driven by a geometric Brownian motion: 'on-line'

dl -I

=

µ dt + o dz

(1)

where µ denotes the instantaneous expected change in the index and a its instantaneous standard deviation. The uncertainty of nature is captured by the increments of a standard Wiener process z(t). The drift µ is an interesting quantity in itself. It is usually the object of intense discussion in scientific circles. If one assumes that the index I(t) is a weatherlinked index, µ can be related to the famous El Niño or the global warming effect' currently under discussion at the Kyoto Conference. We denote by T,,, the first passage time of I(t) through the constant trigger level K: II,K

=

{inf(t >

T; I(t)

0; 0 < ts

=

K)}

(2)

.

by a payoff that is pegged to aggregate bonds are characterized Some nature-linked This is the case, for instance, for hurricane bonds whose return is linked to aggregate damage caused by the occurrence of hurricanes. A new stochastic process has thus to be defined. If I(s) denotes the instantaneous claim level, aggregate claims are I(s) ds. The first passage time is then defined with respect to A(t). We given by A(t) could also consider an event in the spirit of Parisian options. In this case, the event is described as one where the index is above the trigger and stays there for a given amount of time. For example, if the index measures the temperature level the corporation issuing the bond will try to hedge a persistent drought. To keep things simplc and tractable, this chapter is based on a parsimonious frarnework. . imprecise) Indeed, a proxy (admittedly for the two previous definitions is to choose a high value for K, whatever If the index reaches that level (say,temperature or means. windspeed), significant losses are to be expected. In January 1943, Spearfish, Dakota, was hit by the chinook ('snow-eater'),a Rocky Mountains foehnwind whose blow can have devastating effects if it is too strong. The chinook blew so strongly that the temperature rose from -20° to +7°C in less than two minutes! House, building and shop windows were blown out in a wide area, causing significant damage to individuals and local businesses. As stated above, the interest rate r is assumed to be constant over the lifetime of the option. Financial markets are assurned to be complete and frictionless. Trading takes place continuously. Under this hypothesis, Harrison and Kreps (1979)have shown that there is a unique probability measure Q-the risk-neutral probability-under which the continuously values.

.

.

.

=

.

.

.

'high'

.

.

.

6The Beaufort scale ranges from 0 to 17. A value between 0 and I corresponds to a windspeed weather vane. For values between 12 and 17, the wind speed is above l 18 km/h. Severe damage for values of 10 and above 'Scientists may even disagree on the true magnitude of µ. [n the case of an insurable risk, this Company bond quite fat (compared may find the spread over Treasury implied by a nature-linked as a direct reinsurer) and decide to invest in it

to 5 km h, namely a still constructions usually occurs

of0 to

may explain why an insurance to what it would have charged

OPTIONS,

232

FUTURES

AND EXOTIC

DERIVATIVES

The fact that markets are considered complete is discounted loss index is a Q-martingale. not such an heroic assumption as it sounds. Indeed, the Chicago Board of Trade has introduced derivative assets, including options and futures, based on natural catastrophe implies that fmancial assets loss indices: see, for instance, D'Arcy and France (1993).This perfectly correlated to the changes m the mdex I(t)." exist whose returns are (almost) To clarify the valuation procedure, we first look at the cash flows which bondholders are entitled to under the various scenarios. They have a claim on the following potential cash flows: .

.

.

.

.

.

.

.

•

No trigger during the exposure period: Under this scenario, bondholders receive the linked zero-coupon bond, namely F.1 , where full face value of the insurance la is function for the event B. the indicator Under this second scenario, bondholders Trigger hit during the exposure period: .

.

•

receive

an amount equal to a)F.1rer,

0 of the insurance linked zero-coupon bond is thus given by The price as of time t value of future expected cash flows under the risk-neutral probability Q: =

the discounted

=

LINKED BONDS

EU[exp(-rT)(F.1,

=

e +

(1 -

a)F.17,

s,)]

exp(-rT)F

-

EU[exp(-rT)aF.1,

(4)

is easy to interpret. It simply says that holding an insurance linked bond investing in a portfolio which is long a default-free zero-coupon bond of face short a conditional binary option to default. formulation, the probability distribution of the first passage time of I(t) through barrier K appears explicitly as the key factor driving the pricing process: Do= exp(-rT)F(l

-

aQr,

c)

(5)

Qr, is the one-sided risk-neutral first passage time distribution of the index f(t) through the barrier K. of the risk-neutral The explicit computation first passage time distribution (see Appendix 10.B) yields the following closed-form solution for the insurance linked bond:

where

,

Do

=

Fexp (-rT)

1- a

N(di)

nhenomenn

lo2)T'

In(Io K)+(r-

di=

naturnt

(r - ja2)F

In this pricing formula, N(.) denotes the normal distribution function. The term withm coefficient a, is simply the one-sided risk-neutral first the brackets, after the write-down Ice" K. passage time distribution. This closed-form solution depends on the ratio E, vulnerability' reminiscent is of It This ratio can be viewed as a measure. the quasi-leverage ratio of Merton (1974).Indeed, the underlymg stochastic process (the nature index here or the corporate assets in Merton's case) is redefmed through the change of numeraire (i.e. under the new probability measure) and compared to the trigger level (K here or the face value of the corporate debt in Merton's case). In what follows, it will be shown that this ratio plays a sigmficant role. When a is equal to zero, the price of the insurance linked bond collapses to that of an otherwise equivalent default-free zero-coupon bond, P(0, T). The same holds true when the exposure period is nil, namely when T' is equal to zero. For an mfinite trigger point K, the insurance linked bond tends to its default-free value. Some other properties of the pricing formula are also noteworthy. The first yields the obvious result that the price ofthe insurance linked bond tends to P(0, T)(1 - a) when lo tends to K. As a result, when the principal ofthe 1, the bond price tends to zero, other things being equal. The bond is fully at risk, i.e. a second interesting property deals with the impact of the index volatility on the price of the insurance linked bond. When the index volatility goes to infinity, the bond value tends to .

.

.

=

'pseudo-nature

·

-

-

=

Do

2

(7)

of

such

as

the Re:nifort

scale

are not tra
this

P(0, T)

=

Ïo 1- a K

(9)

Io/K is the value of the risk-neutral first passage time distribution for an infinite level. This quantity is less than l because even though the volatility is infinite there are still some states of nature with strictly positive probability of occurrence where the bond will not default. volatility

10.3.2

The Valuation

case

of Insurance

Linked Spreads

The pricing model derived in the previous section quantifies the discount that applies to an insurance linked bond when it is compared to an otherwise equivalent default-free bond. This discount can be translated into a spread value. More specifically, a whole-term structure of insurance linked spreads can be derived. For the sake of simplicity and without loss of generality, we let F 1 and consider the spread values as of time t 0. nbol d whose maturity We denote by Yothe yield of an insurance Un is T: =

=

(10) Using the

of

K) -

(6)

N(d2)

with

"It is true that streimths

=

where

7)]

This expression amounts to value Fand In a third the constant

ln(lo

d

Another equivalent formulation for the price of the insurance linked bond is possible. Indeed, it can be rewritten as Do

233

.

(1 -

Do

PRICING INSURANCE

a risk

oremium

has to be

closed-form

pricing formula for Do yields

Yo=r--In

1 7

1-a

N(d)÷N(da)

la -K

(ll)

The insurance linked spread So is defined as the difference between the yield Yo and the yield of an otherwise riskless bond. The insurance linked spread is thus zero-coupon

234

OPTIOilIS, FUTURES

AND EXOTIC

VATIVES

1 2r/o3

So

=

-

ln

1

-

N(di) + N(d2)

a

(12)

235

LINKED BONDS

PRICING INSURANCE

When the quasi-exposure ratio is less than l, the term structure of insurance linked spreads is humped as shown in Figure 10.2. When the bond maturity and the exposure period increase, the spread first deteriorates. Indeed, a longer time period can carry bad news, namely a shift from a insolvent' position. Afterwards, it decreases again for solvent' position to a the same reason as before. 'technically

Other things being equal, the insurance linked spread is a decreasing function of the bond maturity T when T' is strictly smaller than T. Indeed, the ratio T'/ T improves when T increases for a fixed T'. In other words, the relative time exposure decreases. The spread is an increasing function of the percentage a at risk. Indeed, the more the mvestor may lose upon default, the higher the required spread. The spread decreases when the trigger point K increases. In other words, the higher the trigger level, the smaller the first passage time probability. As a consequence, the insurance linked bond trades at a price which gets closer to its default-free value. The index volatility affects positively the insurance linked spread. When the volatility goes to infinity, the first passage time probability tends to the value lo/K. In this case, the quoted spread is equal to So When T'

=

-

T

ln

1- a

(13)

K

,

T, the influence of the exposure period T' on the insurance linked spread is Indeed, the yield spread can either rise or fall when the exposure period is lengthened. This result may seem rather counter-intuitive and requires some further ratio Er is at work. The nature-vulnerability' explanations. Here again, the of change the yield spread the length of the period depends on in to exposure response a whether the quasi-exposure is smaller or greater than 1. When the ratio is greater than 1, the bond is unsafe' and its spread is a decreasing function of time as shown in Figure 10.1. In the risk-neutral economy, the bond is bankrupt' since the index is expected to cross the barrier. The longer the time to maturity, however, the greater the chance that the bondholder will receive good news. The bond is somehow given more chance to recover. The same kind of behavior affects corporate bond spreads when the quasi debt-to-firm-value ratio is greater than l=

ambiguous.

'pseudo

'technically

'technically

10.3.3

The Duration

This valuation model has interesting implications for both portfolio and immunization issues. Indeed bond portfolios are exposed to shifts in interest rates. To protect these portfolios against unexpected movements in the term structure of interest rates, the investor needs to evaluate his risk exposure accurately. Interest rate elasticity and duration measures restrictive and only apply are now commonplace. Nevertheless, most of them are quite rarely taken into is assumptions. default, for instance, specific of under a Corporate set of a of elasticity estimates unfortunate produces and true biased This the is account. corporate bond: see Ambarish and Subrahmanyam (1989),Chance (1990)and Briys and deVarenne(1997).lnterestinglyenough,theissueoftheinterestrateelasticityofmsurance linked bonds is rarely analyzed. The various documents-see, for instance, Sigma (1996b) with msurance linked bonds insist on their lack of and Goldman Sachs (1996)Mealing correlation with traditional assets. They argue that mtroducmg msurance risks into a portfobo can improve returns while reducing risk. This absence of any reference to any duration concept is obviously surprising. After all, the premier feature of an insurance linked bond is precisely its bond nature. In the followmg, we derive an interest rate elasticity measure for insurance linked bonds and examine its basic properties, Let denote the interest rate elasticity measure of the msurance linked bond:

go

Do

The insurance linked bond price is given by the following expression:

0.075

-

0.070 3

-

-

-

0,065

2

(14)

or

Spread

Spread 4

'technically

¯

0.060 0.055

-

-

-

1-

0.050 0

FIGURE «=0.5,

2

10.1 Spread as a 1, T'= T r= 0.I,«=

4

Function

6

of

8

10

Time-to-Maturity:

Maturity

lo = 99, K= 100,

0

2

FIGURE 10.2 Spread as a a=0.5, r= 0.I, a= 1, T'= T

4

Function

6 of

8

Time-to-Maturity:

10

Maturity.

lo = 35,

K*140,

236

OPTIONS,

Do

=

exp(-rT)F(1

FUTURES

-

aQr,

Applying the elasticity computation to this last expression

ANSWEKOTIC DERIVATIVES c)

pluCINg

4NSURANCE

yieldS

5.14

5.08

When the fraction a of the principal at risk is nil, the duration of the insurance linked bond is equal to T which is the Macaulay duration of an otherwise equivalent default-free bond. This no longer holds true when the bond is exposed to insurance risks. zero-coupon Indeed, the magnitude of the duration of the insurance linked bond depends primarily on the sign of the partial derivative of the risk-neutral first passage time distribution with respect to the interest rate. This partial derivative is equal to

DQr,ar

or

_

N'(di) a

- N'(d2)

lo -K

1 2r

o'

BONDS

0.1

0.2

237

Duration

(15)

-

UNKED

5.06

-

-

-

5.04 5.02

-

o

0.3

0.4

0.5

Interest rate

i-2,/o2

2 - N(d2)- a2

lo -

K

10

ln K

(17)

where N'(.) denotes the standard normal density function. It can be shown that the above partial derivative is positive. The intuition is that a greater drift increases the odds that the trigger is hit in the future. This in turn implies that the duration of an insurance linked bond is always greater than its Macaulay duration, namely the maturity of its underlying default-free bond. This result may seem surprising at first result that the duration of a corporate bond is usually smaller than Indeed, it is a well-known its Macaulay duration. A corporate zero-coupon bond can be disentangled into a long position on an otherwise equivalent default-free bond and a short position on a put-todefault. The strike of this put is given by the face value of the bond. If the corporate assets fall below that level, the bond defaults. The short position on the put-to-default contributes to shortening the effective duration of the corporate bond. Here, the situation is different· The trigger point is not given by the bond itself. It is defined as an exogenous barrier whose magnitude depends on the chosen insurance index (forinstance, the AIG Combined Risks CatBonds are characterized by a trigger at $7.5 bn for the USA and $3.0 bn for Europe). The return on the bond is inversely related to the value of this insurance index. Thus, an insurance linked bond behaves like an inverse floater. It is well known that inverse floaters in interest rates. Jorion (1995)gave the example of a fiveare quite sensitive to movements whose interest rate exposure is roughly that of a ten-year indexed Libor inverse floater year fixed rate note. The case of insurance linked bonds is, however, more subtle. Indeed, the principal of insurance linked bonds does not contractually move inversely with movements in a benchmark interest rate. Moreover, the selling point is precisely that interest rates are usually not correlated to insurance risks such as natural hazards. Insurance linked assets are sold as pure insurance play. However, this pure insurance play should not be misunderstood. Indeed, the movements in interest rates have an effect on the risk-neutral first passage time distribution. The overall interest rate sensitivity of an insurance linked bond is driven at the same time by both the interest rate sensitivity of the equivalent default-free bond and that of the first passage time distribution. The two effects reinforce each other and lead to a duration which is higher than the Macaulay duration. Figure 10.3 illustrates this effect for an insurance linked bond for which the insurance index is far from the trigger point. In Figure 10.3 the parameters are such that the bond is in a safe position, Even though the duration of the insurance linked bond is close to its Macaulay duration, it is always above it.

FIGURE 10.3 T=5,T'=2,a=I

Duration

as a Function

of Interest

Rate:

lo = 10, K= 100, a= 0.5,

Figure 10.4 documents the case where the bond is in a very vulnerable position. When the bond is much exposed to the trigger risk, the duration of the insurance linked bond can greatly depart from its Macaulay level. Indeed, for an interest rate level of 15%, the effective duration is roughly 8.5 years. Black and Scholes (1973)modeled corporate bonds as a long position on a riskiess bond and a short position on a put to default. Chance (1990)has shown that in this case the duration of the risky bond is always less than that of the riskless bond. This result makes sense. In the risk-neutral economy, the put to default has positive duration: the higher the riskless rate (i.e.the drift of corporate assets), the smaller the probability that the corporate bond will eventually default. In our setting, the nature risk exposure is captured by an index potentially hitting a given strike from below. The higher the riskless rate, the higher the probability that the binary option to default will end up in-the-money. As a result, catbond investors should be aware of especially when the bond is their potential increased exposure to interest rate movements, in a vulnerable position. In other words, when it rains, it may pour! Duration

10.00

-

9.so 9.00 6 8.00

-

¯ -

0

FAGURE 10.4 Duration = 2, a = I

T= 5, T'

0.1

0.2

as a Function

0.3

0.5

0.4

of interest

Rate:

jnterest

rate

lo = 95, K = 100, o

0.5,

OPTIONS,

238

FUTURES

AND EXOTIC

DERIVATIVES

These results cast some doubt on the validity of the comparison between an investment in insurance linked bonds and an investment in Treasury bonds. Indeed, to be convincing such a comparison requires that the respective performances be measured at the same risk level. The standard deviation of time-series returns is often used as a risk measure. Guy Carpenter and Co. (1995)claimed that a 5% investment in insurance linked assets increases the expected return of a 60/40 equity bond portfolio by 1.25% while decreasing its standard deviation by 0.25%. These nguresshould be taken with extreme care First, the relevance of such statistics is rather dubious for assets whose returns ar truncated. Indeed, the distribution of return on an insurance linked bond is highly skewed. Once the trigger is hit, the game is over. Second, the very fact that the respective durations of insurance of the two investments are divergent indicates that any outperformance linked bonds has to be interpreted with much care.

I 0.3.4

Time-Series

Properties

We have seen that the insurance risk exposure has important implications for the measurement of the duration of insurance linked bonds. An insurance linked bond exposes its holder to a leverage effect in the same way an inverse floater does. To put it in a nutshell, the investor has thus to be aware that an insurance linked bond is at the same time both an insurance play and an interest rate play. When contemplating the prospect of investing in an insurance linked bond, the investor should also be aware that the past performance of insurance linked bonds is no indication of their future performance. This is obviously true of any asset. However, the caveat is even stronger in the case of insurance linked bonds One has to be extremely careful about the interpretation of comparative time-series sample statisticsofrealizedreturnsoninsurancelinkedbondsandTreasurybonds.Thesamepoint that Ambarish and Subrahmanyam (1989) raised about corporate bonds applies to time-series sample means and insurance linked bonds. These authors showed that standard deviations (of returns on corporate bonds) would generally be poor estimators of the ex ante expected return and the standard deviations of return'. This finding cast some doubt on the relevance of studies such as those of Blume and Keim (1989), for instance. The reason for this result is that a corporate bond can be decomposed into a portfolio which is long an otherwise equivalent default-free bond and short a put option to default. The time-series sample returns of a corporate bond are thus the time-series sample returns of a portfolio. However, the returns are not extracted from the same portfolio at each point in time. Indeed, the allocation of the portfolio between the default-free bond and the put option to default varies as time goes by. For example, the corporate bond can move from an AAA position (lowweight of the put option to default) to a junk position (heavyweight of the put option to default). This in turn implies that the returns are drawn from an 'the

heterogeneoussample,namelyfromahighlynon-stationarydistributionofreturns. It is fairly easy to show that the same caveat should be borne in mind as far as We have shown previously that the price of an insurance linked bonds are concerned insurance linked bond is given by Do

=

exp(-rT)F

An insurance linked bond can be viewed

-aexp(-rT)FQ, as a portfolio

c

of two assets:

(18) a long position on

PRICING

INSURANCE

LINKED BONDS

239

an otherwise equivalent default-free bond and a short position on a binary option. The expected return on an insurance linked bond can be written as E(Ro,)

BeE(Rp)

=

-

BoE(Ro)

(19)

where E(Ro,) E(Ro) E(Ro) 8, the the =

=

=

=

=

the expected return on the msurance linked bond, the expected return on the equivalent riskless bond, return' on the binary option, the portfolio weight of the riskless bond, portfolio weight of the binary option. 'expected

in terms of the risk-neutral first passage any loss of generality that the face value F is equal to l and that the percentage principal at risk a is equal to l. The portfolio weights of the two components are thus respectively given by the following expressions:

It is fairly easy to express the portfolio time distribution. Let us assume without

weights

8" 1

(20)

Go

(21)

=

Q Ûr « 1 - Ûrer

It is then obvious that these portfolio weights evolve stochastically over time. Their respective values track the movements of the first passage time probability distribution. Consequently, in the same way that the expected return on a corporate bond is highly non-stationary, the expected return on an insurance linked bond is also highly nonby stationary. The same holds true for the standard deviation of return which is will be confronted with significant an identical problem. This implies that the investor difficulties when it comes to interpreting statistics on the supposed overperformance of insurance linked bonds. In any case, it is misleading to draw conclusions on the superiority of insurance linked bonds over Treasury bonds as long as the embedded nonstationarity problem has not been cured. At most, any claimed superiority should invite the investor to investigate carefully the sample path over which the research has been run. 'plagued'

EUMMARY 'a

Brealey and Myers (1995)defined finance as fun game to play, but hard to win'. There is no doubt that insurance linked assets are fascinating instruments. They enable investors to introduce a new asset class in their portfolios. They demonstrate the endless ability of financial markets to engineer instruments which suitably reallocate cash flows across states of nature. The recent increase in natural disasters and the surplus shortage of the industry explain why they attract much attention. However, insurance and reinsurance when careful dealing with these insurance linked assets. Most recent should be one studies are quite misleading and may lead the investor to conclude that insurance linked assets are some kind of Holy Grail. The simple pricing model derived in this chapter and that shows that insurance linked bonds can be highly sensitive to interest movements

240

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

These caveats clearly do not question their time-series returns are highly non-stationary. the usefulness of insurance or nature linked assets. Extensions are of course possible. We mention briefly two of them. First, it would be an obvious improvement to incorporate stochastic interest rates as indicated in Appendix 10.A. Second, a better model for catastrophic events should either consider an aggregate index or contain a jump process on top of the Wiener process introduced in this chapter. This would provide a more suitable model of catastrophic events.

POINTS • • •

• •

FOR DISCUSSION

Generalization to jump Processes, Stochastic Volatilities and Information .

What is the rationale behind the emergence of insurance linked assets? Why would an insurance company issue a catastrophe bond? What should one think of the empirical studies documenting an overperformance insurance linked assets? What are the specific duration properties of a catastrophe bond? Why are the returns on an insurance linked bond highly non-stationary?

10.A: EXTENSION RATES

COStS

This chapter is organized as follows:

The case of stochastic

interest rates can be handled in the same way as Longstaff and Schwartz the issue of the valuation of corporate bonds under stochastic interest rates. The only difference is that the first passage time probability distribution is related to a stochastic process which may hit the trigger from below and not from above. Although the closed-form expression for the price of the insurance linked bond in this case turns out be more complex, the basic results remain valid. Indeed, the issue of the correlation between the interest rate snovements and the loss index variations can be overlooked. The key tenet of insurance linked assets proponents is precisely to say that financial risks and insurance risks are uncorrelated

APPENDIX 10.B: COMPUTATION PASSAGE TIME DISTRIBUTION the Q-probability,

stochastic I,

The condition

=

OF THE FIRST

process governing the loss index implies that lo exp((r

OUTLINE

TO STOCHASTIC

(1994)tackled

Under the

.

of

CHAPTER APPENDIX INTEREST

•

•

FUrther

-

a2/2)t

+ oz,)

(23)

I, a K implies that 1 z* a - In

K

(24)

1. Section 11.1 presents briefly the main results in Merton's and Cox and Ross's models. 2. Section 11.2 presents Hull and White's (1987)model, which is one of the simplest models of option pricing with stochastic volatilities. 3. Section 1 l.3 develops the main results of Stem and Stein's (1991)model. 4. In section 1l.4, a generalization of derivative asset pricing models to a context of markets within complete stochastic volatilities and incomplete is presented. In particular the main results in Heston's (1993a, 1993b) model and HPS's (1992)model are reviewed. It is important to note that an incomplete market is simply a market in which there is no unique equivalent martingale measure or risk-neutral probability for the asset price. This is exactly the opposite of the defmition of a complete market given in the first chapters of this book. Note that all our previous results have been obtained in the context of complete markets, as stated at the beginning of this book. In such a setting, any source of risk is tradable, namely hedgeable. 5. Section 11.5 reviews the model for the valuation of options with information costs and shows how it might explain the smile effect. 6. Section l l.6 is devoted to the theory of option pricing biases and in particular to the effect'. strike price bias, which is commonly known as the 7. Finally, section 11.7 presents recent evidence regarding the effect'. 8. The appendix. characterizes the Poisson process. .

'smile

'smile

where z* is linked to previous Wiener process Z, by the following equality:

This new Wiener process is a Wiener process with drift. The drift is given by the term within brackets. The variance of this new process is equal to I. The final step is to compute the first passage tirneb fstri ti<mMehrdihis p8r2oc sHav the ti9m8e window [0, T]. This first passage time distribution

INTRODUCTION

Since the pathbreaking contribution by B-S (1973)was published, many papers have tried to relax its most stringent assumptions. In particular, the introduction of a stochastic

OPTIONS,

242

FUTURES AND EXOTIC DERIVATIVES

has been considered by many authors. Much of the known bias repo empirical studies based on the B-S formula has something to do with this assumption. The biases reported in Rubinstein (1985,1994) for stock options, Melino and Turnbull (1990)and Knoch (1992) for options on other underlying assets, seem to be more pronounced for foreign currency options. These biases are not surprising since the model with known mean and assumes a log-normal distribution for the underlying asset price

JUMP PROCESSES,

STOCHASTIC

volatility

variance

The fact that asset prices do not move continuously but rather jump from time to time led Cox and Ross (1976) to price options for alternative stochastic processes. In the same vein, Merton (1976)used a combination of a jump process and a diffusion process. Scott (1987,1991), Hull and White (1987),Wiggins (1987),Stein and Stein (1991), Hoffman, Platen and Schweizer (1992),hereafter HPS, and Heston (1993a, 1993b), others, have generalized the B-S model to allow for a stochastic volatility. among However, it should be noted that most of these papers do not present closed-form solutions and require the use of numerical techniques to obtain option prices. This should come as no surprise. Indeed, a stochastic volatility or stochastic asset price jumps are probability non-hedgeable sources of risk. As a result, there is no unique risk-neutral measure to price the option. The required market completeness fails to hold. The valuation of stock and index options, in the context of Merton's (1987)model of capital market equilibrium with incomplete information, allows the derivation of a simple analytic formula for the valuation of these options in an economy in which information is not costless. The simple analytic formula given in Bellalah and Jacquillat (1995) explains some of the biases reported in the literature, and in particular the smile effect. Our concern here is to present the main results in these models and to show how they can explain some of the biases reported when testing the B-S model.

I I.I I 1.1.1

AND THE CONSTANT THE jUMP-DIFFUSION ELASTICITY OF VARIANCE (CEV) MODEL The jump-Diffusion

Model

a combination of a jump process and a diffusion process, namely a mixed process. By constraining the jumps in such a way that they are distributed log. normally, Merton (1976)presented the following formula for the pricing of European call options: Merton

(1976)used

r[A(1 + h)T]"

C=

ni '

rN(d2)] - Ke-,

r -

o'2

Ah +

In(S/K)

+

(r'+)¤'2)

ln (S/K)

+

(r' - ja'2)T

=

=

o'

T

243

h)

n In (1 +

a2 +

-

(4) (5)

o

where

Ais the rate at which jumps occur, h is the average jump size measured as a proportional inestaschada#S¾ek a is the variance in the distribution of jumps. Table 11.1 gives examples of call values for such a introduction to a Poisson process, see Appendix 1l.A. TABLEII.! h=2%,r=

Call 10%

with

Values

T=0./ = 5%

T=0.5 = 5%

1

\0.979

15.304

20.012

3.049

8.306

13.308

a= IO% K = 110

0.279

3.775

a= 25% K = 90

I I.139

a 25% K = 100 a=25% y ,,o

1 10%

a=

K 90 10% a K 100

T= I = 5%

pace,

jump-4i minaproces for an

jump-Diffusion

Process

T=0./ = 10%

1

for

T=0.5 = 10%

1

10.899

oj =IO%

i

'

T= I = 10%

14.439

18.657

1.842 I

5.928

10.404

8.229

0.020

1.244

4.379

16.108

21.187

I l.142

\6.\24

3.669

9.606

15.008

3.678

9.63

15.0413

0.597

5.\50

10.198

0.606

5.177

10.236

=

=

=

=

«=50%

!2.223

2l.44

28.659

!2.927

K = 90 = a 50%

6.785

16.27

23.944

3.067

12.l69

19.949

a=50% K = I lo

The Constant

Elasticity

of Variance

The family of constant elasticity of variance following stochastic differential equation:

with

d2

=

SMILE EFFECT

2l.2|O9

2l.46

28.675

6.791

16.289

23.962

3.073

12.183

19.968

(CEV) Diffusion

Process

(1)

o

d

r'

11.1.2 [SN(di)

VOLATILITIES,

(2) (3)

dS

=

diffusion processes

µSdt + öS"/2 dW

is described by the

(6)

where µ, 6 and å > 0 and W is a Wiener process. In the above expression, öS" is the instantaneous variance of the stock price where 8 is the elasticity of this variance with respect to S. In this equation, the instantaneous

OPTIONS,

244

FUTURES

AND EXOTIC

DERIVATIVES

of the return U2 is given by a2 _ 2Sf" 2). When 0<2, this varianc decreasing function of the asset price. 2, the instantaneous variance of returns is ö2 and the process reduces to that When 6 used in B-S (1973).The variance is independent of the asset price level. The formula presented in Cox and Ross (1976)for the valuation of a call at any instant of time is:

variance

=

C

=

S

- Ke

G

g

jUMP PROCESSES,

STOCHASTIC

VOLATILITIES,

Using the additional assumption

4

S, - K

__

(7)

245

risk does not entail any risk This convenient assumption implies that the volatility premium. With these additional assumptions, it is possible to get a unique option price, computed as the expectation of the discounted terminal pay-off under a risk-neutral probability measure. The terminal boundary condition at the maturity date Tis

G

g

SMILE EFFECT

(14)

that the variance

v

is not influenced by the stock price

and setting:

where

S'

Tí2

2re

S'

=

m)

'' -

v

U

E(2-0)

=

e"x'"

l'(m)

9)

the gamma density function

G(x, m)

=

r=g(y,

ds

=

(15)

(ll)

S,, a))

c(t,

(10)

m)dy

when

the volatility

is

uncorrelated

with the stock

pTlCC IS

' ,

T

the value of a European call option

2r(e-cr<2 or _

ö2(2 - 8)

=

,

(8)

ö2(2 - 0)

we E'

">[e

cos(t,

=

S,,

v

,

/S,, o2)

T)dF(v

(16)

where ces is the usual B-S option price corresponding to the variance, v conditional distribution of v* under the risk-neutral probability measure,

,

and F is the given S, and

Û2

and K is the strike price, r is the riskless interest rate and Tis the time to expiration. This model can be easily simulated. In fact, given a daily variance of return and a value of 8, ö can be chosen to satisfy U2 __ 2S(0-2).

'The value of cas(t, S,, v*,) is given by CBS(t,

THE HULL AND WHITE

where

=

dv,

=

and

WI,

(12) (13)

µ (S,, a,, t)S, dt + a,S, dW

(o,, t)v, dt + ö (a,, t)v,

S, stands for the stock price at time t, v,

of returns

v*,

-

Ke"'N(d2)

T)T

d2

=

MODEL

Hull and White (1988)considered the following model for the dynamics of the stock price and its volatility, which is defined as a separate stochastic variable: dS,

r) -- S,N(di)

ln(S/K)+(r

di

I I.2

S,,

and W) are two Brownian

=

o motions

dW 2

(17) *

di -

=

(18)

It is important to note that the moments of F can not be determined in general, except in v*, is special cases when y and are constants, and an integral over log-normal variables. In this very particular situation, using a Taylor expansion of (16), Hull and White solution: following the (1988)gave

o

is the instantaneous variance correlated

with

a correlation

c(t,

S,, o')

=

SN(d,)

-

Ke "N(d2)

+ ¼SvitN(d,)(d,da

- 1) 2a

coefficient

p. In this context, the pricing of a call option implies the construction of a riskless portfolio containing the option, the stock and a second call option with the same exercise price but a different time to maturity. Let c(t, S,) be the value of the first call option which depends on time t and the stock price S,. This approach, which was also used by Johnson and Shanno (1987),Scott (1987, 1991) and Wiggins (1987),yields a partial differential equation for the option price. However, the solution to that equation is not unique unless the price function for the second call is known. To get a unique solution, Hull and White (1987)made the additional assumptions that the two Wiener processes are independent and that the variance has no systematic risk-

+ ¼SN(di)[(did2

+

(o6 3h3}

- 3)(did2 - 1) -

[e3"- (9 +

18h)e" +

(d (8

+ d

24h

\

e" - h

h2

2)

)] 18h' +

6/Ÿ)]

NJ

with h e2t. Table 11.2 gives examples of Hull and White model pricing for e 0.5 and 2.0. Table l 1.3 compares Hull and White call prices with B-S prices for different parameter values: S 100, r 10%. =

=

=

=

OPTIONS,

246

TABLE

I I.2

Volatility

AND EXOTIC

DERIVATIVES

Model Pricing

Hull and White K

=

0./0

0.5

0.50

/.00

2.0

=

e 0./0

0.50

0.00

18.57 9.98 3.68

10.90 1.00 0.00

14.39 5.19 I.I6

20.53 0.00 195.97

10.90 I.l8 0.00

14.82 7.79 3.68

20.20 13.87 8.80

10.90 l.19 0.00

15.16 5.60 6.97

0.00 0.00 477.38

I I.I2 3.65 0.9I

18.79 13.66 9.89

26.07 21.27 17.24

11.\4 3.67 0.83

14.89 !3.07 15.94

0.00 ISI.!! 816.43

90 \00 I 10

10.90 l.00

25%

90 100 I \0 90 100 110

jUMP PROCESSES,

STOCHASTIC

YOLATILITIES,

Volatility

Comparison

between

.

.

do

-ö(o

=

\0%

90 100 I 10

25%

50%

I I.3

0.5

e

=

P

l

/.00

0.00 0.00 -0.14

0.00 0.04 0.35 -

0.00 0.19 -0.52

-0.l5

0.94

1.96 -33.61 191.78

0.19 0.l8 -0.88

0.47 -0.l3 -l.26

0.00 0.09 -0.41

0.53 -2.00 2.41

-170.96 -139.3\ 467.32

0.39

0.07 -l.02 -2.05

0.11 0.22 -0.75

-3.52 -1.10 4.56

-37\.2l 158.82 797.\4

90 100 I10

0.\0 0.20 -0.67

STEIN AND STEIN'S

-0.51

-l.49

0.l0

0.50

0.00 0.00

0.00 0.27

l.00

(y2+þ

e"""'"'drj

(22)

e "'So(Pe

=

using

where the integral I(.) is calculated The European call price is Co(P, t)

' =

[P

e ,

.

.

.

Usmg this model

.

implies

equations

pg

-

K]S(P,

'")

(3) to (8)in Stein

and

(23) $tein(1991).

t|õ, r, x, 0)dP

(24)

.

values for the parameters ö, K and 8. the choice of reasonable values are 8 studies by Stein (1989) show that reasonable 0.3.

Results of the empirical ö 16 and x 0.4. Simulations of the Stein and Stein (1991)model show that stochastic volatility exerts an upward influence on all option prices. Also, the stochastic volatility impact is more important for out-of-the-money or in-the-money options than for at-the-money options, in the context of this model exhibit a U-shape as the strike i.e. the implied volatilities price is varied. Hence, an at-the-money option has the lowest implied volatility, and this volatility rises in either direction as the strike price moves. =

=

=

MODEL

stock price distributions when prices follow diffusion varying volatility with parameter. They obtained interesting results regarding processes a option pricing with stochastic volatilities and the relationship between this parameter and the nature of fat tails in stock price distributions Their model is more general than that of Hull and White for the following two reasons First, Hull and White used a Taylor series expansion to solve explicitly for the option price, about the point where the volatility is non-stochastic, i.e. ö 0. Second, it is not

Stein and Stein

(21)

0, the solution is

drift, µ

2.0

0.50

0.00 0.07 -0.38

«dW2

=

When the stock has a non-zero

0.l0

90 100 110

-8)dt+

This process has been used by many researchers in modehng and studymg the empirical properties of volatility. From equations (20) and (21) it is possible to get an explicit closed-form solution for the stock price distribution. Setting Po equal to 1 and denoting by Sc(P, t) the time t stock price distribution when the stock price drift µ 0, the solution given by Stein and Stein (1991)is

S(P, t) =

e

(20)

BS and HW Call Values Time to Moturity

K

247

where ö and a stand respectively for the expected instantaneous return and variance and Wi is a Wiener process. Let the dynamics of the volatility be governed by an Ornstein-Uhlenbeck process with a tendency to revert back to a long-run average level 8 as follows:

Sc(P, t)=

TABLE 11.3

SMILE EFFECT

dP=µedt+opdW,

I.00

14.39 4.95 0.00

10%

y

clear that this expansion provides a good approximation to option prices when the value of ö is far from zero. Using Stein and Stein's notations, let the dynamics of the stock price P be represented by the following familiar equation:

Time to Maturity e

50%

FUTURES

(1991)studied

=

I 1.4

GENERALIZATION VOLATILITIES

TO STOCHASTIC

Stein (1989) and Stein and Stein (1991) built their models on the assumption that volatility is uncorrelated with the spot asset. They obtained solutions that look like an average of B-S values over different paths of volatility. The absence of this correlation implies that the model can not capture important skewness effects. Heston (1993a, 1993b) used a new technique to derive an analytical solution for a

FUTURES

OPTIONS,

248

AND EXOTIC

DERIVAT VES

context. His model, which accou ts for interest rates, can be used in the pricing of bond volatility

European call option in a stochastic

stochastic volatilities and stochastic options and foreign currency options. Hoffman, Platen and Schweizer (1992), hereafter HPS, considered a very g e neral diffusion model for asset prices in which volatilitics are stochastic and past-dependent. We present the main results in both models-

Heston's

i 1.4.1

Model

Heston (1993a, 1993b) used a new technique to derive a closed-form solution for a European call option in a stochastic volatility context. His model allows for arbitrary correlation between the spot asset returns and the volatility. Also, he introduced stochastic interest rates and applied the model to the pricing of bond options and foreign currency options. The dynamics of the spot asset at time t are described by the following diffusion equation:

dS(t)

SdW

µS dt +

=

(25)

The volatility dynamics are governed by the Ornstein-Uhlenbeck and Stein (1991):

d

di + ö dW

-ß

=

Using Itö's lemma and standard arbitrage arguments, Heston the price of a European call is given by v, t)

c(S,

which

where pi and p2 are probabilities p;(x, v, T, ln(K))

l

=

l

Re

=

process used by Stein

(26) (1993a,1993b) showed that

Spi - KP(t, T)p2 (27) the following formula: calculated using be can e

I"**)

(x, v,

T,

¢) d¢, -

y

=

1, 2

(28)

with f;(x, v, t; ¢) given by equations (16)and (17)in Stein and Stein the asset price S and the exercise price K The probabilities in equation (28)multiplying in (27)must be calculated to get the option price. Following Merton (1973a)and Ingersoll (1989),Heston incorporated stochastic interest rates in his option pricing model and applied it to options on bonds and options on foreign currencies. This is possible if one modifies equation (25) as follows, to allow time dependence in the volatility of the spot asset, as(t): dS(t)

=

µsSdt + os(t)

Sd W

(29)

This equation is satisfied by discount bond prices in the model of Cox, Ingersoll and Ross (1985a,1985b). The dynamics of the bond price P(t, T) at time t, for a maturity date T are specified by the following equation: dP(t,

T)= µ,P(t,

T)dt+ap(i)

P(t, T)dW

(30)

It is convenient to note that variances of the spot asset and the bond are given by the same variable v(t). In this context, the valuation equation is given by equation (22) in Heston

(1993a).

jUMP PROCESSES,

STOCHASTIC

VOLATILITIES,

SMILE EFFECT

249

The model can also be applied to options on foreign currencies. In fact, when S(t) stands for the dollar price of a foreign currency, and the dynamics of the foreign price of F(t, T), are given by the following equation: a foreign discount bond, dF(t,

T)

µ,F(t,

=

T)dt+ap(t)

F(t, T)dW

2

(31)

then equation (26) in Heston (1993a)can be used for the pricing of European currency options. The model can be used for the valuation of stock options, bond options and currency options. It is interesting since it links most biases in option prices to the dynamics of the spot price and the distribution of spot returns. With a proper choice of its parameters, the stochastic volatility model seems to be flexible and promising in the description of option prices. However, the main drawback of this model is the number of parameters to be estimated.

I 1.4.2

Platen

Hoffman,

and

Schweizer's

Model

Hoffman, Platen and Schweizer (HPS) (1992) considered a very general diffusion model for asset prices allowing the description of stochastic and past-dependent volatilities. Since their model implied an incomplete market, they were unable to derive an analytical solution. They used stochastic numerical methods for the study of option prices and hedging strategies. HPS (1992)provided an approach to option pricing which allows the specification of general patterns of volatility behavior. The approach combines the use of a highin an incomplete dimensional Markovian model with stochastic numerical methods market. The assumption of an incomplete market implies that there is no unique equivastock price lent martingale measure or risk-neutral probability for the underlying dynamics. When the volatility is both stochastic and path-dependent, the following multidimensional process was used by HPS (1992): dX' '

for i

=

a'(t,

X,) dt +

b'

WI (t, X,) d

(32)

1 to n. 0 to m and j In this formulation, the riskless assset is represented by X° and the dynamics of the underlying asset (the stock) are given by X I. The other components of X can be used to model other assets. For example, they could model the stochastic volatility and its dependence on the past. It may be noted that such a model goes back to Merton (1971, 1973a, 1992). In this formulation, an option or a contingent claim is a random variable of the form c(X,). Within this general formulation, there is no unique equivalent martingale measure for the stock price process. This leads to the problem of pricing in incomplete markets. To overcome this problem, and to be able to compute option prices, HPS (1992) supposed that there is a investor who should use the minimal equivalent martingale measure. This allowed them to decompose the space of all those assets compatible with the given stock in a direct sum of two subspaces: purely traded assets and totally nontradable asssets. =

=

'small'

OPTIONS,

250

FUTURES

AND EXOTIC

DERIYATIVES

and generality of the model give rise to stochastic differential equations The nexibility that do not have explicit solutions. HPS used numerical methods to compute option prices and simulate the performance of hedging strategies under various possible scenarios. Such a model is from a practical perspective since it is rather difficult to implement by market participants. The main drawback is the number of parameters to be estimated. 'poor'

I 1.5

THE THEORY

OF VOLATILITY

SMILES

When the underlying asset dynamics are believed to be stochastic, Hull and White (1987),Stein and Stein (1991)and Heston (1993a,1993b), among others, proved that the implied volatility may vary with the option's strike price. Simulations of their models show that a plot of theoretical implied against strike prices displays a U-shaped curve, effect'. known as the Although there have been several models to explain the strike price bias and the smile effect, little empirical work has been done in the area except by Xu and Taylor (1994)and 'smile

the references therein. .

The Smile Effect in Stock and Index Options

I 1.5.1

. .

.

.

.

.

for stock price distributions when the volatihty They applied the results obtained is driven by an Ornstein- Uhlenbeck process (ARl). and to the analysis of the from the model to option pricing with stochastic volatilities relationships between stochastic volatilities and the nature of fat tails m stock price .

.

.

.

.

.

.

.

.

.

.

distributions. .

.

.

.

.

'mixing

.

I I.5.2

.

.

.

.

.

.

The Smile Effect for Bond and Currency

VOLATILITIES,

SMILE EFFECT

25l

When examining the effect of stochastic volatilities on option prices in comparison to the B-S model, Heston used a B-S model with a volatility parameter that matches the variance of the spot return over the option life. The parameters used correspond roughly to those estimated by Knoch (1992)for yen and Deutschmark currency options. The model links the biases of the B-S model to the dynamics of the spot price and its distribution. Ileston showed that B-S prices are virtually identical to his model prices for at-the-money options, explaining some of the empirical support for the B-S model. Also correlation between volatility and the spot price is necessary in explaining skewness that options. affects the pricing of in-the-money options relative to out-of-the-money In fact, a positive correlation coefficient results in high variance which spreads the right tail of the probability density and results in a thin left tail in the distribution. This option prices and decreases in-the-money option prices increases out-of-the-money relative to the B-S model with a comparable volatility. A negative correlation coefficient has the opposite effect, i.e. it decreases out-of-themoney option prices relative to in-the-money option prices with respect to B-S prices. Hence, without this correlation, stochastic volatility does not change skewness and options relative to affects only the kurtosis. This latter affects prices of near-the-money far-from-the-money options. The model seems to impart not only the strike price bias but also other biases reported when testing the B-S model. Heston (1993a,1993b) presented an option pricing model consistent with empirical biases reported in the B-S formula. The model, based on a log-gamma process, explains in particular the smile effect with respect to the distribution's skewness. Model simulations show that option prices are similar to those of B-S for at-the-money options. options and lower However, Heston's model assigns higher prices to out-of-the-money prices to in-the-money options when compared to B-S prices. The difference in prices is economically sigmficant. The strike biases are similar to the skewness-related biases m models where stochastic volatility is correlated with stock returns. .

.

parameters from data In order to simulate their model, they estimated the necessary used in Stem (1989)concernmg mdividual stocks and the S&Pl00 index. When comparing option prices, they calculated option values using their model and a B-S model based on the implied volatility associated with their model' Their results showed that a stochastic volatility exerts an upward influence on all option options, prices, and that stochastic volatility is more important for away-from-the-money corresponding that is, that implied volatilities to new option prices exhibit a U-shape as volatility options and is lowest for at-the-money the strike price is changed. The implied rises as the strike price moves in either direction. options, the distribution' they use explains the UFor away-from-the-money volatilities. They find that the overall impact on option prices is shape in implied economically sigmficant, especially for out-of-the-money options. For some parameter values, their model prices were l1.7% higher than the B-S model's, and even larger effects are observed with cheaper options. .

STOCHASTIC

.

a model

(1991) presented

Stem and Stem

UMP PROCESSES,

I I.6

.

.

.

.

.

.

.

OPTION VALUATION WITH INFORMATION COSTS: THE BELLALAH-)ACQUILLAT MODEL

.

Smee the acquisition of information and its dissemination are central activities in finance and especially in capital markets, Merton (1987)developed a model of capital market information equilibrium with mcomplete to provide some msights into the behavior of .

.

security prices. He also studied the equilibrium structure of asset prices and its connection with empirical anomalies in financial markets in the same context. Such a model might help to understand why the B-S model (Black and Scholes, 1973) volatilities leads to theoretical prices which are systematically biased and why implied differ from one strike price to another for the same underlying asset. .

Options I I.6.I

In order to explain strike biases in the B-S model, Heston (1993a,1993b) proposed an between volatility and spot asset option pricing model allowing for arbitrary correlation follows an Ornstein-Uhlenbeck process as in Stein and Stein returns when volatility and applied bond options His model to currency options. can be (1991).

The Model

Financial models based on complete information might be inadequate to capture the complexity of rationality in action. Some factors and constraints, such as entry into the dealer business, are not costless and may influence the short-run behavior of security

FUTURES

OPTIONS,

252

AND EXOTIC

DERIVATIVES

jUMP PROCESSES,

STOCHASTIC

VOLATILITIES,

SMILE EFFECT

253

prices. Hence, most models developed in financial economics do not explicitly provide a functional role for the complicated and dynamic system of dealers, market makers and

money, at-the-money and out-of-the-money options negotiated on the following stocks: Michelin, Peugeot, Saint Gobain, Elf Aquitaine, Paribas, Compagnie du Midi, Thomson

traders-

and Lafarge. The study covers the period June 1986 to March 1988. refers to an option whose strike price is closest to the closing The term stock price. The pricing model requires the use of the following parameters: the stock price, strike price, time to maturity, interest rate, volatility and information costs. While the first three parameters are observed in the market, the others must be estimated. The risk-free rate used is the continuously compounded equivalent yield on French treasury bills having a maturity closest to the option's expiration date. Historical data corresponding to the period of three months before the option is quoted is motivated also by a The use of historical volatility are used to estimate volatility. who with makers and told historical volatility is traders that market us discussion some often used as a benchmark for trading options. While most of the parameters are easily observed or estimated, the only difficulty the options and their underlying concerns the estimation of information costs regarding used. different approaches could be least two assets. At study of the equilibrium For stocks (assets),it is possible to use a cross-sectional is theoretical in nature, structure of asset returns as in Merton. However, this approach using the same methodological approach employed in empirical studies of equilibrium of asset returns. Since no empirical work has been done to estimate information costs in option markets and since the main purpose of this section is to calibrate the model to the market, one way to do this is to infer information costs from observed market prices. The information costs inferred from the model are expressed in the form of a discount rate which is added the riskless rate when discounting the cash fiows of an option in a context of mcomplete mformation. The estimates of information costs show that they vary from one day to another and A priori, results show that information costs according to the stock used in calculations. are associated with the underlying asset and the option's maturity date. The sample shows that information costs are between 0.001 and 0.12 for Michelin and Elf Aquitaine, 0.01 and 0.16 for Peugeot, 0.06 and 0.16 for Saint Gobain, 0.09 and 0.3 for Paribas and Compagnie du Midi, 0.02 and 0.05 for Thomson, and 0.05 and 0.17 for Lafarge (exceptfor some extreme values of 0.38). In order to compare model prices to market prices, we estimate a weighted average cost of incomplete information using the following regression:

Besides, the treatment of information and its associated costs play a central role in capital markets. If an investor does not know about a trading opportunity, he will not act to implement an appropriate strategy to benefit from it. However, the investor must determine if potential gains are sufficient to warrant the costs of implementing the strategy. If profitable implementation of a strategy justifies the costs of its implementation, these costs include time and expenses required to create a database to support the strategy and to build models. This argument applies in varying degrees to the adoption in practice of new structural models of evaluation, i.e. option pricing modelsFrom the model of Merton (1987),it appears that taking into account the effect of incomplete information on the equilibrium price of an asset is similar to applying an additional discount rate to this asset's future cash flows. In fact, the expected return on the asset is given by the appropriate discount rate that must be applied to its future cash flows. Then, relying on Merton's model to derive a valuation formula should lead to more accurate theoretical prices since information costs are included for stocks and options. We denote by C(S, t) the value of the option as a function of the underlying asset and time. Let T be the maturity date of the call and E be its strike price. The boundary condition at maturity is

C(S, T)

=

if S else

S - E 0

E

>

(33)

lo

Using standard methods for the price of a European call, the solution is found to be

C(S, T)

=

N(di)

Se

-

Ee

'

""N(d2)

(34)

with

ln(S/E) b'y

I 1.6.2

Empirical

di

=

+

(r

Ü2

+ L)T ,

d2

=

di - o

v/Ï

'at-the-money'

(35)

Tests

The model presented above is simulated here with reference to the B-S model and tested with regard to stock options traded on the MONEP, the Marché des Options Négociables de la Bourse de Paris. The focus of this section is to calibrate the model to the observed prices in the option market. For the specificities of this market, see for example Bellalah and Prigent (1997a,1997b). Stocks and options prices were given by the database of AFFI, Association Française de Finance, and missing data were collected from the journals La Tríbune and Les Echos· We have selected closing prices for stocks and the mid-quoted bid-ask spread for options Options are quoted for different strike prices and maturities. Most options are traded for the nearest maturities of three and six months. Options with nine months to maturity are rarely traded. The database englobes in-the-

C,,,

=

as, +ß,,,M,,,

+

E¡¡,

(36)

where

l'

S,, - Xe X,,e

"

"

(37)

In the above regression as, is an implied weighted average cost for each option j on asset i on day t. Some results are reported in Tables I 1.4-11.6, from Bellalah and Jacquillat

(1995). To get an idea of the amount of mispricing, and using the weighted average cost of incomplete information and the historical volatility, we calculate each day the difference

254

OPTIONS,

TABLE I I.4 Estimates age Costs of Incomplete Peugeot

AND EXOTIC

of Implied Weighted Averfor Options on Information

Dote

N Options

a'

2-01-88 2-02-88 2-03-88 2-04-88 2-05-88 2-08-88 2-09-88 2-10-88 2-I I-88 2-12-88

63 53 32

0.03 0.03 0.01

20 26 24 2I 25

0.03

R

ß

0.52

0.50 0.10

0.83

-0.88

0.97 -i.08

56 64

TABLE I l.5 Estimates Costs of incomplete Paribas

age

Date

N Options

2-0 1-88 2-02-88 2-03-88 2-04-88 2-05-88 2-08-88 2-09-88 2- I0-88 2- I I 2-12-88

45 I4 IS 15 I2 5I 27 i5 48

-88

FUTURES

74

-0.89

0.01 0.02 0.09 0.\0 0.09 0.08

0.58 0.42 0.71

0.70 0.62

0.93 0.82 0.52 0.89 0.78 0.69 0.75

of Implied Weighted AverInformation for Options on a'

0.72 0.42

0.07 0.08 0.\7 0.09 0.2I 0. I2 0.13 0. I l 0. I 3 0.l8

R2

ß'

-0.88

.08

- I -0.89 0.89 0.72 0.7 I 0.80 0.62

0.80 0.83 0.97 0.93 0.82 0.72

0.69 0.78 0.69 0.74

DERililATIVES

jUMP PROCESSES,

of Implied

Averfor Options on

Weighted

Information

Date

N Options

a'

ß'

R2

2-01-88 2-02-88 2-03-88 2-04-88 2-05-88 2-08-88 2-09-88 2-10-88 2-Il-88 2- I2-88

43 46 21 33 29 40 53 IS 52 29

0.\3 0.03 0.0 I 0.07 0.0 I 0.08 0.13 0.05 0.06 0.04

0.52 0.56 -0.88 0.73 -0.69 0.48 0.62 0.76 0.60 0.62

0.67 0.54 0.87 0.90 0.72 0.49 0.84 0.78 0.7\ 0.40

VOLATILITIES,

SMILE EFFECT

255

between theoretical or model prices (TP) and market prices (MP), expressed in French Francs. The difference is designated YCI. Also, a ratio RCI is calculated by dividing YCI by TP. When the difference is positive, the model price is greater than the market price and vice versa. expressed by RCl lies between 0.01 and 0.2 Results show that the relative mispricing 0.06 for Peugeot, 0.02 and 0.14 for Saint 0.01 Elf Aquitaine, and and Michelin for Gobain, 0.029 and 0.13 for Paribas and Compagnie du Midi, 0.01 and 0.11 for Thomson, and 0.03 and 0.15 for Lafarge (exceptfor some extreme values around 0.2). To quantify the magnitude of mispricing (under-or overevaluation of the option price with respect to the market price), the means and standard deviations of YCI, RCI and Ml were calculated. Results are reported in Tables 11.7 and 11.8, from Bellalah and .

Jacquillat (1995). Subscript l means that options are out-of-the-money.

Subscript 2 refers to at- and inthe-money options. It seems that the model values correctly at- and in-the-money options and misprices out-of-the-inoney. This result is interesting since the B-S model often overvalues out-ofthe-money calls and undervalues in-the-money calls. If information costs are correctly estimated, then calculations of implied volatilities from our model would not show different volatilities for different strike prices. This

TABLE

I I.7

Magnitude

of Mispricing

for Out-of-the-Money

Options

Options

MOY YCII

STD YCII

MOYRCl|

STD RClI

MOY MI\

STD MI i

Paribas Thomson

2.746 40 -0.284 98 5.746 20 5.905 60 1.78063 4. IO9 42 5.\4294 4.46758

3.595 42 2.824 63 8.0 18 44 6.842 65 2.28472 2.52 103 4.81947 3.58582

8.88 I 26 0. I l 6 30 0.66 I 80 0.042 20 0.23979 0.540 35 0.27562 0.67030

4.679 05 0.690 67 0.790 37 0.072 32 0.28931 0.434 13 0.30343 1.52086

-0.200 78 -0. I52 43 -0. I98 57 -0.094 62 -0.08478 -0.102 33 -0.0547\ -0.22227

0.109 2 I 0.|38 I0 0. i08 62 0.04 I 63 0.05334 0.078 80 0.02257 0.13122

Lafarge

Midi Michelin Elf Gobain

TABLE I I.6 Estimates age Costs of incomplete Lafarge

STOCHASTIC

Peugeot

TABLE

I I.8

Magnitude

of Mispricing

for At- and In-the-Money

Options

Options

MOY YCl2

STD YCl2

MOYRCl2

STD RCl2

MOY M/2

STD MI2

Paribas Thomson Lafarge Midi Michelin

3.4 18 70 -0.64 I 22 3.128 59 -0.569 63 2.74482 2.I l842 5.23294 6.577 58

4.095 72 3.464 43 |2.072 844 9.094 69 2.48362 2.155213 5.18147 |4.556 92

0. I3 I 26 -0.020 40 -0.005 02 0.006 94 0.I25 13 0.140 IS 0.14562 0.084 30

0. I83 68 0.|5 I 85 0.222 88 0.035 92 0.I2654 0.10949 0.14937 0. I 59 I l

0.09| 57 0.089 42 0.064 8 i 0.233 02 0.09768 0.l5494 0.02984 0.061 24

0.074 57 0.069 I 3 0.060 02 0. I62 63 0.07422 0.080 23 0.03235 0.050 32

Elf

Gobain Peugeot

256

OPTIONS,

FUTURES

would

AND EXOTIC

mean that an appropriate choice of information costs appearance of the U-shaped curve, known as the smile of volatilitics.

I l.7

VOLATILITY

SMILES: EMPIRICAL

could

STOCHASTIC

257

jUMP PROCESSES,

eliminate

biases. These biases are shown to be more pronounced for foreign exchange options than for stock options. The extensions of the B-S model include constant elasticity of variance processes and jump-diffusion processes to explain some of the option biases. The extension of this model to incomplete markets takes into account the stochastic character of volatilities. strike price bias. However, the This allows in part the explanation of the well-known major drawback of these models is the need to estimate several non-observable parameters which limit their implementation. On top of this, they lose the convenient

the

EVIDENCE

volatility The expected of an underlying asset can be inferred from the option price. Implied volatilities can be taken for different times to maturity and different exercise with rows ordered by prices. Hence, using option prices, a matrix of implied volatilities the exercise price and columns ordered by time to maturity can be constructed. The rows of the implied volatility matrix may provide information about the term structure of expected future volatility. Stein (1989) and Franks and Schwartz (1988) studied the term structure of implied volatilities using only two times to maturity. The time-series studies done by Day and Lewis (in press) used one implied volatility per day and ignored the term structure effects, since time to maturity varies from day to

day Stein (1989)used two daily time series on implied volatilities for the S&Pl00 index options over the period 1983-87. Based on the assumption that the volatility is meanreverting, he concluded that long-maturity options tend to overreact to changes in the implied volatility of short-maturity options. This conclusion has been disputed by other authors who showed that overreaction depends on the model used to represent changes in the volatility. Xu and Taylor (1994)presented and illustrated methods for estimating this term matrix corresponding structure from one row of the implied volatility to nearest-theand its time-series money options. They modeled the term structure of expected volatility properties. They used spot currency options on the British pound, Deutschmark, Japanese yen and Swiss Franc quoted against the US dollar in their empirical work. The study concerned the period from January 1985 to November 1989. The main results appeared in Xu and Taylor (1994). When examining the relation between short- and long-term implied volatilities for the European option exchange index and Philips stock, the principal result in Heynen, Kemma and Vorst's study is that the major determinant for the specification of the term structure of implied volatility relations is the level of the unconditional volatility. This latter seems correctly specified in the case of the EGARCH(l, 1) stock return volatility model. Xu and Taylor's study presented the following results. First, implied volatilities vary significantly across maturities. Second, the direction of the term structure of implied volatilities changes (up or down) nearly once every two or three months. Third, the variations m expectations regarding long-term volatility are significant, although the process of adjustment of expectations is slower than that regarding short-term volatility

uniqueness of pricing of complete markets. This chapter presents recent developments along these lines, especially the models of Merton (1976),Cox and Ross (1976),Hull and White (1988),Stein and Stein (1991), Heston (1993a,1993b) and Hoffman, Platen and Schweizer (1992).From a theoretical point of view, these models are rather interesting. They point out the difficulty of pricing assets in an incomplete market. From a practical point of view, the use of these models is very limited given the burden of parameter estimation to implement them. effect' is reviewed and some of the findings The recent evidence regarding the volatility models. Finally it is noteworthy to are explained with respect to stochastic mention that other models have the potential to explain the smile effect without the use of stochastic volatilities. 'smile

POINTS • • • • • • • • •

FOR DISCUSSION

What are the specificities of the jump-diffusion model? What are the specificities of the constant elasticity of variance model? What are the deficiencies of Hull and White's model? How can we proceed to price options in a stochastic volatility economy? What are the basic assumptions behind Stein and Stein's model? What are the basic assumptions behind Heston's model? What are the basic assumptions behind HPS's model? What are the theoretical reasons for the existence of volatility smiles? What is the empirical evidence regarding volatility smiles?

If (N,),x

is a Poisson process with an intensity 1, then for all t > 0 the random variable N, satisfies: P(N,

=

n)

SUMMARY Also, when s Many authors, well-known

'

e

=

(38)

(At)"

In particular Var(N,)

its

PROCESS

\ I.A: THE POISSON

APPENDIX

E(N,)

The B-S model is the simplest model in the theory of option pnemg however, have tried to relax some of its assumptions in order to explain

SMILE EFFECT

VOLATILITIES,

DERIVATIVES

>

0,

E(sN,

s

O

=

E(N))

=

-

li (E(N,))2

=

At

(39) (40)

l2 The Lattice Approach and the Binomial Model

CHAPTER

OUTLINE

This chapter is organized as follows: 1. In section 12.1, a short survey of numerical techniques in derivative asset valuation is presented and the lattice approach is applied to the valuation of European and American equity and futures options. The treatment of discrete and continuous dividends within the lattice approach is presented. Simulation results are provided. The approach is also extended and applied to the valuation of options in the French market by taking into account the specificities of the Paris Bourse, and in particular, the mechanism. 2. Section 12.2 deals with the pricing of interest rate derivative assets in the context of lattice approaches, First, the Ho and Lee interest rate model is presented and applied to the valuation of derivative assets whose values depend on interest rates. Some of its deficiencies are analyzed. Second, Hull and White's interest rate model is presented. 'report'

INTRODUCTION with the valuation of derivative assets using the binomial or the lattice approach. The valuation of options in a discrete-time setting is more pedagogical than in a continuous time setting. Ironically enough, however, the more complex approach, namely the Black and Scholes one, was discovered before the simple binomial approach. efficient, Even if the discrete-time approach is not always computationally option valuation with the lattice approach is very flexible. It can handle many situations where for the valuation of no analytical solutions are possible. This is the case, for example, stock options and index options when there are several discrete cash payouts made by the underlying asset, such as cash dividends on a stock. The model proposed by Cox, Ross and Rubinstein (1979), hereafter CRR, is one of the It is framed in the context most successful models dealing with derivative asset valuation. of a lattice approach. The model was first proposed for the valuation of stock options. It

This chapter deals

OPTIONS,

260

FUTURES

AND EXOTIC

DERIVATIVES

valuation of several derivative assets with complex pay-offs and was then extended to the different underlying assets. The lattice approach has also been used by several authors to model the dynamics of the term structure of interest rates and to value bonds and bond options. There have been using a one-factor many attempts and approaches to describe yield curve movements model. The approach presented by Ho and Lee (1986), in the form of a binomial tree for discount bonds, provides an exact fit to the current term structure of interest rates. Their model is convenient since it takes market data such as the current term structure of interest rates as given. In that respect, it is close to a binomial stock option pricing approach where the current stock price is taken as an input to the model. Unlike most interest rate contingent claims models, this model uses full information on the current term structure. In fact, using an ingenious discrete-time approach for pricing bonds and interest rate contingent claims, Ho and Lee succeeded in incorporating all information about the yield curve in their model. An alternative to the Ho and Lee model was proposed by Black, Derman and Toy (1990)and Hull and White (1993)among others. Black, Derman and Toy (1990) used a binomial tree to construct a one-factor model of of all discount bond yields as well as the the short rate that fits the current volatilities Hull and White (1993)presented a general rates. of interest structure current term numerical procedure involving the use of trinomial trees for constructing one-factor models where the short rate is Markovian and the models are consistent with initial snarket data. Their procedure is efficient and provides a convenient way of implementing models already suggested in the literature. In that respect, their contribution is more a numerical one than a financial theoretical one. This chapter covers binomial and more general lattice approaches for the pricing of equity and interest rate dependent claims.

12.1

LATTICE APPROACHES

12.1.1

A Survey

THE LATTICE

AND THE BINOMIAL

MODEL

26 I

Since then, the CRR approach has been extended and extensively used for the valuation of many contingent claims and options. Rendleman and Barter (1980)applied the CRR methodology to the pricing of options on debt instruments. Trigeorgis (1993) applied the methodology to the valuation of investments with multiple real options and to the pricing of managerial flexibility implicit in investment opportunities. Hull and White (1988, 1993) provided a modified binomial lattice model for the valuation of stock options and interest rate options. Boyle (1986)proposed a trinomial option pricing model in which the stock price can at a given time period. In another move either upwards or downwards or stay unchanged paper, Boyle (1988)showed how a five-jump, three-dimensional lattice can be used for the valuation of options on two underlying assets. Omberg (1988)studied a family of discrete-time jump processes and applied a GaussHermite quadrature technique to derive prices of options on options or compound options. Yisong (1993)modified Boyle's approach by presenting a general methodology that lattice approach. He proposed a modified can be applied to any multi-dimensional approach to the selection of lattice parameters including probabilities and jumps using additional restrictions. Sandmann (1993)developed a model for the pricing of European options under the assumption of a stochastic interest rate in a discrete-time setting. He used a combination of the binomial model for a stock with a binomial model for the spot interest rate. Rubinstein (1994)developed a new method for inferring risk-neutral probabilities or option prices from observed market prices. These probabilities were used to infer a binomial tree by implementing a simple backward recursive procedure. However, the approach is restricted to European options, and future research remains to be done regarding the pricing of American options.

12.1.2

for the pricing of Various numerical procedures have been proposed by many researchers derivative securities. These procedures include the lattice approach, finite difference schemes and the Monte-Carlo method, among others. When pricing European options, equivalent to these procedures are, under certain specific conditions, asymptotically closed-form solutions à la Black and Scholes. However, they may be more practical for the pricing of American options and complex derivative securities. In fact, these approaches can handle more complex situations such as cash dividend payouts. this chapter, In we are interested in the lattice approach pioneered by CRR (1979)· These authors proposed a binomial model in a discrete-time setting for the valuation of framework, their approach was based on the construction options. Using the risk-neutral argument, of a binomial lattice for stock prices. They applied the risk-neutral valuation pioneered by B-S, which simply means that one can value the option at hand as if With an appropriate choice of the binomial parameters, they investors were risk-neutral. result established of their model to that of B-S. a convergence

APPROACH

The Model for Options

on a Spot Asset without

any

Payouts

The lattice approach can be introduced by first looking at a stock option whose underlying asset does not pay any dividend. Let T be the option's maturity date which is divided into N reasonably small intervals of length At, so that T NAt. During each time interval, the stock price moves either upwards from S to uS or downwards from S to dS. This movement in the stock price is binomial with a probability p attached to an upward jump and a probability (1 p) to a downward movement. The binomial process can be represented as shown in Figure 12.1. The parameters u, d and p are functions of the mean and variance of the rates of return on S during the interval At. Indeed, intuition has it that the larger the spread between u and d, the greater is the variance of the return on S. world, it is possible to show that model parameters must satisfy the In a risk-neutral following relations: =

u

=

e

(1)

OPTIONS,

262

FUTUftEGdilMIN

1|Il Ti AIMllM¥ATIVES

THE LATTICE

Su

AND THE BINOMIAL

APPROACH

At time T, the call's value

P

is given by the

MODEL

263

maximum

between its intrinsic

value

and

zero: C,

=

max

[0, S, -

K]

2

-

or 1-p

Sd

gg.t

=max[0,SudN-j

C,

At the same date, the put's pay-off is given by

Price

gefetmAsset

d

p

e-c,m

=

=

a

Pr

(2)

e'*'

(3) (4)

=

u - d

The stock value at each node is given by Su d' for the value of S is observed, at instant t + At it takes takes on three values, and at instant t + iAt it takes stock price dynamics over three periods. The value of a call option, C,_, (or a put, P(i, j)), starting from date T, where the option terminal backward through the tree.

j

on two values, at instant t + 2At it i values. Figure 12.2 represents the

-

.

at time /

iAt can be calculated by pay-off is known, and proceeding

=

[0, K

- Sr]

max

[0, K

SuddN-j

-

Since in a risk-neutral world, the option's value is given by its expected pay-off discounted at an appropriate risk-free interest rate for the length of time considered, it is possible to calculate the option value at each node using the following equation: Cy for 0

=

'

'[pC,

e

+

(1 -

p)C,

,]

(7)

i GN - 1 and 0 sje also for a put. i. This relationship If the option is of the American type, then its price must at least be equal to its intrinsic value. For an American call, the condition applies

<

C

;

=

i

max {Suid'

-

K, e '*'[pC,aya

+

(1 -

;)}

p)C

must be used. In the case when there is no cash dividend, the American is always equal to the right-hand term of the max expression. For an American put, the equivalent condition is P,,7

su2

max

or

from 0 to i. Since at time t

varying

=

/

max {K - Suid'

i,

'*'

[pP,o

+(1

(8)

call option price

;]}

(9) - p)P, In the case of an American put, early exercise is always possible. Indeed, if the stock price were driven very close to zero because of a highly probable bankruptcy, it would certainly make sense to the option before maturity. Condition (9) takes this into account by picking the greater of the two cash flows. =

e

yo

'kill'

su

Su

Sd

Sd

12.1.3

Sd"

Sd3

The Model for Futures

Options

Merton (1973a), Black (1976) and Barone-Adesi and Whaley (1987), among others, showed that futures contracts, stock index options and currency options may be assimilated to options on a stock that pays a continuous dividend. In a risk-neutral economy, the expected return on an asset paying a continuous dividend yield b is (r b) so that referring to Figure 12.1 we can write

-

e

In addition, t

Figure

12.2

Dynamics

At

t+2At

all the preceding

'"^'

=

Price over

t+3At

Three

Periods

For a futures, contract,

r

=

-

equations are used, except a=L

of the Asset

pu + (I

b and a

=

l.

74,

p)d for

(10) (3) which

must be rewritten

as

(ll)

12.!.4

i

AND EXOTIC

FUTURES

OPTIONS,

264

DERIVATIVES

THE LATTICE

APPROM N A94D THE BtNOMIAL

MODEL

265

Discrete Dividends

Model with Dividends

A Known Dividend Yield

For stock options, the underlying asset pays a discrete dividend rather than a dividend yield. When there is a discrete dividend payment, the tree does not recombine (Figure

The lattice or binomial approach can be easily modified when the underlying asset pays a known dividend yield. date, stock prices are given by At an instant t + ¡At, prior to the ex-dividend

12.4).

Su d'

for j

=

0, 1,

.

.

.,

i

(12) u S- D

At an mstant t + iAt, just after the ex-dividend date, stock prices are given by us '

i

S(1 - ð,)u d'

for j

0, 1,

=

i

...,

S

(13)

'

S More generally, if is the total dividend yield corresponding then stock prices are given by

o

i

S(1-ö)ud

to all ex-dividend

D

dS d S-D

dates,

(14)

forj=0,1,...,i

Figure

12.4

Ðynamics

of the Asset Price with

Discrete

Dividends

This allows the construction of the binomial tree for stock prices (Figure 1.23) The same recursive procedure applies for the determination of option prices. Assume that there is only one ex-dividend date r, between an instant l)At. If i < K, the nodes on the tree at time t + iAt correspond to stock prices:

KAt and

(K

su4(1-6) Su

sj(1

-

(1 -

6)

(1 -

I

for j

0, 1, 2,

=

SJ(1 8) -

6)

...,

(Suid'

i

-

D)u

and

(Su d'

I

-

D)d

(15)

i

If i K + l, and D stands for the dividend amount, the nodes stock prices: =

8) Su

Suid'

on the tree correspond

for j=

0, 1, 2,

...,

i

to

(16)

S(1 -6) Sd

S Sd

(1 -

s (1 - 8)

(1 - 6)

6) Sd

(1-

Sd (1 6) 4

6)

Sd (1 -8)

Figure

12.3

Dynamics

of the Asset

Price with

a Continuous

Dividend

Yield

1) rather than (i + 2) nodes. so that there are 2(i At an instant (i + n)At, there are (n l)(i l) rather than (i + n + 1) nodes. As in Hull and White (1990), Hull (1993) and Bellalah (1995b), it is possible to design trees where the number of nodes at time iAt is always (i + l) by assuming that the stock price has two components: a stochastic component and a part corresponding to the present V31ue of all future dividends during the option's life. Denote by r the ex-dividend date such that: KAt ers

(K + 1)At

(17)

266

AND4][gOTIC

OPTIONS,FUTURES

and write the stochastic part S *

S (t ,

|S(l')

)

if t'>r else

S(t') - De*¯©

.

S*(t')u

'

d' d

De

'"

of S*, arafall the .,

50.3914 10*

52.4258 4*

i,

if At>r if t' Gr

'^6

2. 42 1*

Examples

40.0000

42.0344 5*

=

31.7515

33.7859 6* 30.3403

=

=

=

3&B379

37.6893

In these examples we use the following parameters for the valuation of a European and an American put option on a stock paying a dividend of 2.05 in three and a half months: i 40, S 0.4. 0.1, N 5, T 45, r 42, K I month, « 5 months, At The first step is the calculation of the parameters u, d, m and p. Since u e" e"^', p 1 u, a d (a - d)/(u d) and q (1 - p), we get u 1.1224, d 1.0084, p 0.5073 and q 0.8909, a 0.4927. Using these parameters, the dynamics of the underlying asset can be generated as shown in Figure 12.5. The values of the underlying asset in nodes 1* to 13* are generated as follows: =

44.8960

46.9O6

37.6556 3*

=

56.5593

Rßt07

46.9136

(S*(t')u

=

267

63.4822 9*

-

=

MODEL

AND THE BINOMIAL

71.2525 12*

=

12.1.5

APPROACH

at time t' as

t is the dividend date. In this context, the volatility a of S is replaced by o*, the volatility other parameters, p, u and d are conserved. 0, 1, 2, At time (t + iA t), stock prices can be written as, for j . where

LATTICE

DERIVATIVES

=

,

=

=

=

=

=

=

=

-

25 2039

=

=

*

S(0, 0)

S*(u°d°) + De

"

=

2*

S(1, 1)

S*(u'd°) + De

"i

=

2

3'

S(1, 0)

=

S*(u°d')

"

2

4*

S(2, 2)

=

S*(u2d")+

De

l'

2

5*

S(2, 1)

=

S*(uld')+

De

""

2

6*

S(2, 0)

=

S*(u°d2) + De

l'

2

7*

S(3, 3)= S*(u'd°)+

""

2

8*

S(3, 0)

l

=

S*(u"d')

+ De

De De-r(T-3)

+

22.4554 13*

I :

Vigure it&

The European

Signadmissofthe

Price:

Asset

Date

f dividend

Example

S*(u°d")

11*

S(4, 0)

=

12*

S(5, 5)

=

S*(u'd°)

13*

S(5, 0)

=

S*(u°d")

Put Price with Dividends

12

Figure 12.6 shows the process used to generate nodes 14* and 15*: When x> r, we do not discount the dividends: 9* 10*

S(4. 4) - S*(u'd°) S(4, 3)

=

S*(u'd')

14*

P(5, 0)

=

15*

P(4, 0)

=

max

[0, K

- S(5, 0)]

[pP(5, l) + qP(5, 0)]

The European put price with dividends is equal to 5.8190.

a

OFTIONS,

268

FUTURES

XOT1C

AND

DERIVATIVES

THE LAT I gbAPPRO

10144

ID HE BINOMIAI.. MODEL

0

0.0000 0

.0.0248

0.0000

0.0248

0.0508

1.1294

269

1.2186

0.0508 16*

0.1040

2.2861

3.1876

4.6266

5,3610

58190

5.633Ï

6.0633 9.3621

8.6183

8.6274

5.0000 17*

8.9441

13.2485

12.5047 16.7111

15.9673

19.7961

22.5446 14*

12.6

Put Price with

European

The American

22;5446

Figure

Dimitiends

Put Price with Dividends

a

-

-

For the put price, P(i, j)

=

max

{[pP(i

For example, the values 16*

17

P(4, 4)

P(4, 3)

+ 1, j + 1) + qP(i + 1, j)] a; max

at nodes

- S(i, j)]}

16* and 17* (Figure 12.7) are calculated

=

max

max [0; 45 {0.0508;

=

max

(0.0508;0)

max

max [0; 45 {4.6266;

=

[0, K

=

-

(20)

as follows:

50.3914]}

0.0508 -

40]}

=

max

(4.6266; 5)

=

5

The price of the American put when there are dividends is 6.0633. The difference between the American price and the European price corresponds to the early exercise premium.

12.7

American Put Price

The Model for American

Options

in the French

with

Dividends

distributions. This model is fully developed in Bellalah (1995b). The stocks are traded in the RM market, the Marché à Règlement Mensuel, and present 13 cash payments per year in the form of report, déport and dividends. When there is a déport, the stock price is reduced by the amount of déport in the same way as for dividends. Therefore, it can be handled as a dividend. When there is a report, the stock price rises by the amount of report. These cash distributions can induce early exercise of American options. Since 10 September 1987 American options on stocks and stock indices have been negotiated on the Paris options market, called the MONEP, Marché des Options Négociables de la Bourse de Paris. The market is organized under the authority of the Conseil des Bourses des Valeurs. In April 1992, 27 classes of stock options were traded. and Eurotunnel Each option contract implies 100 stocks except for Euro-Disneyland where lots of 500 shares are used. The underlying RM market quoted 240 stocks in 1992. Since 1991, long-term options have been traded on the CAC40 index. .

The RM Market 12.1.6

1$.7111

16.ËW7

19.4226 15*

Figure

9.3621

8.9887

12.8Ì51

12.1375

0.1040

2.4685

3.3657

and the Report

Mechanism

Market

The RM

of the basic lattice approach to the valuation of This subsection provides an extension American options traded in the Paris Bourse when there are many discrete cash

market can be regarded as a forward market since the buyer (the seller) of a stock at an instant t pays (receives)and takes delivery of (delivers)the asset at a specified date t,, known as the liquidation. The liquidation day corresponds to the sixth day

OPTIONS,

270

FUTURES

AND EXOTIC

DERIVATIVES

preceding the last trading day of the month. The buyer assumes a long position and agrees to buy the stock at a specified future date for a specified price, which is the stock price at instant t. The seller assumes a short position and agrees to sell the stock on the same date for a specified quoted price at t. On the liquidation day, the buyer pays and takes delivery of the stock and the seller receives the payment and delivers the stock. If the buyer (seller)decides to maintain his long (short)position, he can defer the maturity of the initial transaction to the next liquidation date, and so on. The Franc amount of the long net position often exceeds the short net position (nearly 10 times). Hence, if the short investor has no difnculty in obtaining the stocks from the long one, the long investor may not have enough money to defer his or her position for the next month. There is another group of investors who accept delay on the long positions by buying the stocks for one month from the long positions and selling them the next month. This operation is realized at a price called the report. For some stocks, it may happen that the short position exceeds the long position. In this case, there is a shortage of capital to defer the long net position and stocks are necessary for the delay of short positions. In this case, there is a déport. Each stock traded in the RM market gives the right to the long or short investor to pay or receive at liquidation a cash amount called the report and sometimes the déport. Also, each stock provides unknown cash incomes 13 times per year corresponding to 12 reports or déports and a dividend. For the CAC40 stock index, there are 520 unknown cash income distributions a year. The Modified

Lattice

Options traded in the Paris Bourse can be priced by an extension ofthe basic lattice approach pioneered by CRR and extended by Hull and White. In fact, if we suppose the existence of only an ex-report date t', the following condition must be added in the lattice approach: '^6

S(t) - R, e S(t)

if ts t' if t > T'

(21)

kAt < t' s (k + l)At, where R, stands for the amount of report or déport. This amount is positive for a report and negative for a déport. asset values can Hence, at each day of liquidation, at an instant t + iAt, the underlying be written as with

(S*(t)uíd'i-Roe

S'(t)uid'

i

'^6

ifAtr'

if

forj=0,1,...,i

APPROACH

TABLE

AND THE BINOMIAL

12.!

Paribas

MODEL

27]

Call Values, june Maturity Date

(K,o)

24.4

27.4

30.4

33.4

36.4

260 280 300 320 340 360 380

|63.0 143.0 123.0 103.0 83.0 63.0 43.2

163.0 143.0 123.0 103.0 83.0 63.0 43.6

I63.0 143.0 123.0 103.0 83.0 63.! 44.0

I63.0 143.0 123.0 103.0 83.0 63.5 45.2

400

25.2

26.1

27.2

420 440 460 480

I l.7 4.3 1.3 0.3

13.I 5.6 2.I 0.7

14.6 6.9 2.9 f

I63.0 143.0 I23.0 103.0 83.0 63.2 44.5 28.3 16.1

ABLE 12.2

Paribas

.!

Call Values,

8.3 3.9 l.7

September

29.5 17.6 9.7 5.0 2.4

Matur-

(K, o)

24.7

27.7

30.7

33.7

36.7

300

123.0 103.0 83.1 64.5 48.3 34.6 23.6 15.0 8.8 4.6

123.0 103.0

123.0 103.1

83.5 65.8 50.2 37.0 26.2 17.7 11.2 6.4

84.2 67.2

123.0 103.4 85.2 68.9 54.4 41.9 31.5 23.0 I6.0 10.5

l23.I 104.1 86.5 70.5 56.6 44,4 34.2 25,6 |8.6 12.8

320 340 360 380 400 420 440 460 480

52.2 39.4

28.9 20.3 I3.6 8.4

(22)

When using this model, at each liquidation date, the amount of report or déport is given by the product between the asset value and the interest rate corresponding to the valuation period. For example, the one-month report is estimated by the value of the asset price times the one-month

THE LATTICE

interest rate

Examples Tables 12.1--12.6 give simulation results of the model presented here for Amdaican calls and puts traded on the Paris Bourse for the Paribas stock.

TABLE 12.3 ity Date

Paribas Call Values,

December

Matur-

(K, a,)

24.3

27.3

30.3

33.3

36.3

360 380 400 420 00

73.2 57.8 44.5 33.2

75.2 60.3 47.5 36.5

77.3 63.0 50.5 39.9

79.6 65.7 53.6 43.2

82.0 68.5 56.7 46.6

272

OPTIONS,

TABLE

'

~

12.4

Paribas

FUTURES

Put Values,

AND EXOTIC

June Maturity

Date

(K, ax)

24.4

27.4

30.4

33.4

36.4

260 280 300 320 340 360 380 400 420 440 460 480

0.0 0.0 0.0 0.I 0.2 0.8 2.8 7.8 \7.5 3 I.9 49.6 68.8

0.0 0.0 0.1 0.2 0.4 1.3 3.8 9.3 19.0 33.0 50.2 69.0

0.0 0. I 0.I 0.3 0.7 l.9 4.9 I0.7 20.5 34.2 50.9 69.3

0.1 0. I 0.2 0.5 l.I 2.6 6.0 I2.2

0.I 0.2 0.3 0.7 1.6 3.4 7,2 I3.7 23.6 36.9 52.7 70.4

TABLE 12.5 ity Date

Paribas

Put Values,

22.1 35.5 5 I.8 69.8

September

DERIVATIYES

Matur-

(K, o)

24.7

27.7

30.7

33.7

36.7

300 320 340 360 380 400 420 440 460 480

0.3 0.6 1.5 3.5 7.3 13.8 23.2 36.0 5\.6 69.4

0.5 f.! 2.4 4.9 9.3 16.2 25.8 38.2 53.2 70.2

0.9 1.7 3.4 6.5 l1.4 18.6 28.4 40.6 55.0 71.3

l.4 2.5 4.7 8.2 |3.6 2l.I 31.0 43.0 57.0 72.8

2.0 3.5 6.0 10.0 I5.8 23.7 33.6 45.5 59.I 74.4

THE LATTICE

APPROACH

AND THE BINOMIAL

MODEL

273

The following dates and amounts are retained for reports

24 05/1995: 3.08 FF 24 06/1995: 24 07/1995: 24 08/1995: 24 09/1995: 24 10/1995: 24 l 1/1995: 24/12/1995:

3.08 2.99 2.98 2.99 2.82 2.82 2.82

FF FF FF FF FF FF FF

Comparisons of these option values with those obtained from the CRR or Hull and White models show significant differences varying from 2% to 30%. For more details concerning options and dividends, the reader is referred to Bellalah (1998).

12.2

THE LATTICE APPROACH DERIVATIVE ASSETS

12.2.1

The Ho and Lee Model for Interest

FOR INTEREST

RATE

Rates and Bond Prices

(1986),let P'"(.) be the equilibrium price of a discount bond of T for state i. Thus, P'"'(.) is a discount function that completely describes the term structure of interest rates and satisfies the following two conditions: P"'(0) 1 Following Ho and Lee

maturity

'

(23) (24)

=

lim Pl">(T)

0

=

In order to describe the binomial lattice, let us denote at the initial time: P() P (.) 1 (25) At time l, the discount function is specified by two possible functions P\"(0) and corresponding respectively to an upstate and a downstate. Hence, at time n, the binomial process is specified by a discount function P)"'(.) which can move upward to a function se "(.) for i 0 to n. (.) and downwards to a function P)"' Ho and Lee (1986)defined two functions caHed perturbation functions, h(T) and h'(T), such that in the upstate =

=

PL"(0)

TABLE Date

12.6

(K, ox) 360 380 400 420 440 46052.6

Paribas

Put Values,

December

Maturity

24.3

27.3

30.3

33.3

36.3

6.0 10.I 16.7 26.0 38.0

8.\ |2.8 19.8 29.2 40.9 54.8

IO.3 15.5 22.9 32.4 44.0 57.2

|2.8 18.4 26.0

15.2 21.2 29.2 39.0 50.3 62.8

35.7 47. I

59.9

4+,

=

Pí' "(T)

pt")(y '

=

r

p"+"(T)=

(26)

P"'(T

1) "'(1)

h*(T)

(27)

h*(0) with h(0) 1. These functions specify the deviations of the discount functions from the implied forward functions. To insure the absence of arbitrage profits when forming portfolios of discount bonds, they define an implied binomial probability x independent of time Tand the initial discount function P( T) as =

December.

h(T)

and in the downstate

P

Simulations were run for 11 May 1995, when the stock price quoted was 423 FF, the dividend was 18 FF on 5 June 1995, and the avoir fiscal (tax credit) was 6 FF. The interest rate is taken to be 8.75% for the maturity date of June, 8.5% for September and 8% for

1)

=

OPTIONS,

274

x)h*(T)

xh(T)

AND EXOTIC DERIVATIVES

FtJTURES

for

1

=

n, i>0

(28)

THE LATTICE

Example

APPROACH

AND THE BINOMIAL

MODEL

275

2

and "(T l)]P"'(l) (29) - l)+ (1 -X)P value bond discounted by This equation shows that the bond price equals the the prevailing one-period rate. Hence, x may be interpreted as the implied risk-neutral probability. Assuming that the discount function evolves from one state to another dependmg only assuming that a on the number of upward and downward movements is equivalent to downward movement followed by an upward movement is equivalent to an upward movement followed by a downward movement, This is the path independence condition which gives rise to the following values of h and h*: P)"'(T)

"(T

[xP

=

'expected'

.

.

h(T)

1/[x + (I - x)ör]

=

for T

>

0

(30)

Consider the following parameters the Ho and Lee model:

The current

x +

Poo(3)

TABLE

(l

-

x)ö r

=

=

=

funtion: Poo(3)

=

TABLE

The Term

12.7

T Poo(K) Pio(K) Pii(K) P20(K) P2,(K) P22(K) Pao(K) P3|(K) P,,(K) Ps3(K) P4o(K) P4 (K)

I | 1 I I I | 1 I I I i

1

=

Poo(4)

0.98260 0.97923 0.98514 0.97570 0.98|64 0.98756 0.97235 0.97822 0.984\2 0.99006 0.969I5 0.97500 0 98 0.99276

Poo(2)

0.9826, =

0.9651

=

Poo(5)

0.9296,

=

0.9119

e (g)

in the Ho and Lee Model, Example

Structure

/

0

( P (K)

Poo(l)

0.9474,

12.8

Poo(K) Rio(K) Po(K) P2o(K) P2i(K) Pa(K) Pao(K) P3;(K) P32(K) Ps(K) P4e(K) P4,(K) P42(K) P43(K)

.

1,

0.5, t

=

365 days.

(Table 12.8) is specified

term structure Poo(0)

Consider the following parameters for the determination of discount functions in 0.5, t 365 days. 4, ó 0.994, Jr the Ho and Lee model: n The current term structure (Table 12.7) is specified by the following discount =

=

by the following discount

.

f

Poo(0)

0.994, x

=

function:

This model is convenient since it uses all information available in the current term structure of interest rates. Besides, only two parameters need to be estimated.

=

4, ö

=

=

1,

Poo(l)

0.941,

The Term

=

Poo(4)

Structure

Poo(2)

0.982, =

0.921,

=

Poo(5)

0.961 =

0.911

in the Ho and Lee Model,

Example

2

T0/2345

h*(T)=

Example

=

n

for the determination of discount functions in

2 6 0.95837 0.96997 0.95l64 0.963I6 0.97482 0.94520 0.95664 0.96823 0.97995

3 0.94740

0.93752 0.95460 0.92780 0.94477 0.96198

4 0.929 0.91687 0.9392\

I 1 I

I I 1 I I i I i I I I i

0.98200 0.97567 0.98155 0.97320 0.97917 0.98508 0.9699 I 0.97576 0.98165 0.98758 0.97720 0.983 I 3

0.96 l00 0.95248 0.96604 0.94680 0.95832 0.96992 0.95069 0.96220 0.97385 0.98564

0.94 l00

0.92941 0.94634

0.92100 0.91653 0.93886

0.9 I 100

0.93090 0.94786 0.96513

0.98907 0.99504 i

I

5

12.2.2

The Ho and Lee Model for Contingent

Consider the valuation of

a contingent

Claims

claim C, with a maturity date Tand pay-off

such that C(T, i)

=

f(i)

for 0

6

i GT

f(i) (31)

Let C(n, i) stand for the contingent claim value at time n and state i. The contingent claim has a minimal value L and a maximum value U such that at time n and state i, L(n, i), C(n, i) and U(n, i) satisfy the following relationship: L(n, i)

The holder of this

contingent

<

C(n, i)

<

U(n, i)

(32)

< T. claim receives X(n, i) at time n and state i for lsn This contingent claim may refer to an interest rate futures option, interest rate futures, a bond option and so on. Ho and Lee (1986) showed that a portfolio comprising the

OPTIONS,

276

contingent claim and discount bonds, preventing following equation: C(n, i)

=

[x{C(n +(1

4

FUTURES

riskless

AND EXOTIC

arbitrage

profits, implies the

+ 1, i + 1) + X(n + 1, i + 1)}

- X){C(n+

1, i)

X(n+

DERIVATIVES

=

max[L(T

min(C*(T - 1, i),

parameters: tion is

(33)

1, i)}]P"'(1)

n

AND THE BINOMIAL

4, ö

=

Poo(0)

=

=

Poo(3)

0.99, x

=

Poo(l)

1,

0.957,

=

0.5, At

=

MODEL

Poo(4)

=

The current discount func-

3 months.

0.992,

=

277

Poo(2)

0.939,

=

Poo(5)

0.975 =

0.921

Poo(l) is the price of a discount bond paying off one dollar in three months. The option values at the maturity date are calculated in Figure 12.9 using

where

"'(1)

C(T - l, i)

APPROACH

*

P denotes the one-period discount bond price at the point (n, i). This equation gives the arbitrage-free price ofthe asset, one period before expiration, C*(T - 1, i). The market price must be where

THE LATTICE

P

=

[980-

max

1000Pe(l),

0]

for j

0, 1, 2, 3, 4

=

Using the backward procedure at node I, the option price is calculated as

- 1, i), U(T - 1, i))]

(34)

Using the recursive procedure as in the binomial model for stock prices, it is possible to 0, and hence the initial price. obtam the asset value at n .

=

[0.5(0)+ 0.5(9.173)]O.9761

=

4.477

At node J, the option price is

[0.5(4.477)+ Hence, the option

value

0.5(13.556)]0.9716

8.761

=

is 3.196.

Examples The application of this model to the pricing of any type of interest contingent claim requires the estimation of the probability x and the spread ö between the two perturbation functions. First, the discount function at the time of pricing must be estimated. A procedure like that of McCulloch (1975)or Litzenberger and Rolfo (1984)can be used. Then the parameters x and ö are estimated using a non-linear procedure like that in Ho and Lee (1986) or Whaley (1986). The estimation approach uses observed contingent claim prices and a pricing model in order to infer the parameters in the same way as we calculate an implied volatility for stock options. of a one-year European put option on a three-month Consider the valuation treasury bill. The face value of the bill is $1000 and the option's strike price is 980. Figure 12.8 shows the lattice for the bond price with respect to the following

0

1.085 2.197

0

3.196

4.477

5.358 8.761

9.173 13.556

18.882

Figure

12.9

The Lattice

for the Put Price: n

=

4, ó

=

0.99, x

=

0.5, K

=

980

0.9958 0.9914 0.9905 0.9860

0.9878 0.9814

I 2.2.3 0.9806

0.9761

0.9779 0.9716

0.97os

0.9664 0.9611

Deficiency

in the Ho and Lee Model

It is worth noting that the Ho and Lee model presents some deficiency. In fact, the of the entire discount function are not sufficient to constraints imposed on movements eliminate negative interest rates. This deficiency has been recognized by Heath, Jarrow and Morton (1992), Pedersen, Shiu and Thorlacius (1998),Ritchken and Boenawen (1990),among others. These authors generalized the Ho and Lee model and provided the model. necessary adjustments in order to obtain an economically meaningful Ritchken and Boenawen (1990) showed that the restrictions imposed by Ho and Lee probability imply that bonds are priced by a model characterized by a evolution of the and spread', However, the rate ö. discount parameter x an 'risk-neutral'

Figure 12.8 The

Lattimagr

Bond Prices

'interest

278

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

function, namely the default-free zero-coupon bond prices, may not be bounded in the interval (0, 1]. To show this, they developed an example where X 0.4 and ö 0.8. They generated prices of pure discount bonds at all the vertices and found that some prices exceed 1. This indicates the presence of negative interest rates in the lattice. In fact, if we modify their ö from 0.8 to 0.9 and generate bond prices as in Figure 12.10, it is clear that some bond prices exceed 1. To avoid negative interest rates, the Gl constraint P must be added for each time period in the lattice. The figure uses the parameters ð 0.9, x 0.5 and n 4 and the current yield curve for the first five periods: =

=

">(1)

=

=

ro(l) ro(2)

=

=

ro(3)

9 P

(3)

=

09

=

p(l)

=

e,""

8.6178%,

p(2)

=

e-ro(2)2

7.696%,

p(3)

=

e

9.531%,

7.2321%,

p(4)

=

e

to(5)

=

7.2321%,

p(5)

=

e

P P P P

(1) (2) (3) (4)

= = = =

P P P

0.97457 0.96486 0.9527 0.92531

P P P

P

(1) (2) (3) (4)

=

0.87712

=

0.78154

= =

(1) (2) (3)

=

=

1.04214 1.08026

^

(1) (2) (3)

= =

=

P P

P P

(1) 0.84413

P P

(2) (3)

(1) (2)

=

1.091135

=

1.16448

(1) (2)

0.98202 0.94322

=

=

(1) (2)

= =

0.88381 0.76401

=

0.70875 0.58366

=

µ[0, r, t]dt + od W(t)

The short interest rate r corresponds to the continuously compounded yield on a discount bond maturing at date At. When the tree is constructed, the values of r are equally spaced and have the form To + jAt. Note that ro is the current interest rate value and j is an integer which may be positive or negative. Also, the values of At are equally Spaced with a positive integer i. As shown in Hull and White (1990),the variables At and and Ar must be chosen in such a way that Ar lies in an interval between 20\ ÃÏ. We use the following notation:

iAt and r, ro + jAr with i a 2, a node on the tree for the values of t the yield at time zero on a discount bond maturing at time iAt, the drift rate of r at node (i,j) with r, ro + jAr, respectively to the upper, 1, 2, 3, the probabilities corresponding for k middle and lower branches emanating from node (i, j).

12.2.4

12.10

Deficiency

=

=

If the tree constructed up to time nAt(n > 0) is consistent with the observed R(i) and the interest rate r at time iA t applies to the interval between iAt and (i + 1)A t, then the tree reflects the values of R(i) for i sn + 1. Note that between times nAt and (n + 1)At, the value of 8(nAt) must be chosen in such a way that the tree is consistent with R(n + 2). Once the value of 0(nAt) is known, then it is possible to calculate the drift rates µs for r at time nAt using ,

µ[8(nAt,

=

Trinomial

ro +

jar,

nAt)]

(36)

The three nodes from node (n, j) are (n + 1, k + 1), (n + 1, k) and (n + 1, k - 1), where must be chosen in such a way that r, (the value of the interest rate reached by the middle branch) is very close to the expected value of the interest rate, r, + µ,.;A t. Hull and White (1993)gave the following probabilities:

P (1) = 0.79543 P(2)=0.61885

2At

of the Ho and Lee Model

The Hull and White

=

=

Pi(n, j) Figure

(35)

k

= =

279

In this model, the volatility a is a known constant and the functional form of µ is known. The value of 8, is unknown.

µ, P

0.6945 0.60709

dr

R(i) µ(i, j) P

0.93792 0.87501 0.80063

/

MODEL

AND THE BINOMIAL

provides a convenient way of implementing models already suggested in the literature. It should be noted in passing that a one-period trinomial tree is somehow equivalent to a standard two-period binomial tree. For example, they studied the case where the process for r(t) has the general form considered by Hull and White (1990),also called the extended Vasicek model:

(i, j)

P P P P P

.•

1.09826

=

APPROACH

aN/2

'

ro(4)4

=

.6

5

'

ro(4)

0.79383

)

=

THE LATTICE

procedure involving the use of Hull and White (1993) presented a general numerical trinomial trees for constructing one-factor models which are consistent with initial market data where the short rate folows a Markovian process. Their procedure is efficient and

PJn

)

--

2Ar2

2Ar2

P2(n, j)

Model

42

=

=

1

=

2Ar2

y + 2Ar

a2At

y2

Ar

Ar2

+

2Ar2

2Ar

(37) (38) (39)

with 4

=

2n At +

(j- k)Ar

(40)

280

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

-

This general procedure for fitting a one-factor model of the short rate to the initial yield curve using a trinomial interest rate tree can be used to test the effect of a wide range of assumptions about the interest rate process on the prices of interest rate derivatives However, research remains to be done to get better models for interest rates.

e

y

Numerical Methods for American Option Pricing

UMMARY This chapter presents the basic concepts and techmques underlying the pricing of derivative assets within the context of binomial models and lattice approaches. The lattice approach is applied to the valuation of European and American equity options when the underlying asset is traded in a spot or in a futures market. The approach is extended to the valuation of options traded on the Paris Bourse by taking into account several cash distributions to the underlying assets, i.e. report, déport and dividends. It is convenient to note that lattice approaches can be easily implemented and adapted to different derivative asset pay-offs. The approach is more pedagogical than the continuous time approach. However, it takes some time to offer accurate option prices, which is obviously not a major handicap when there are no closed-form or analytic solutions. The basic models for the valuation of interest rate dependent claims and derivative assets are presented in the same context. First, the Ho and Lee model is detailed. Second, it is applied to the valuation of derivative assets. Third, Hull and White's trinomial model .

D

is

•

FOR DISCUSSION

• • •

•

•

OUTLINE

-

This chapter

is

orgamzed as follows:

1. Section 13.1 presents a numerical solution to the valuation of an American on a dividend paying stock. 2. Section 13.2 develops a numerical solution for the pricing of an American on a dividend paying stock. 3. Section 13.3 presents a numerical solution to the valuation of an American bond with many embedded call and put options. 4. Appendix 13.A gives a detailed algorithm for the valuation of American there are several dividends. 5. Appendix 13.B gives a detailed algorithm for the valuation of American .

.

What are the key valuation parameters in the lattice approach for pricing stock tions? o How is an option priced in the lattice framework? What modifications are necessary to the standard lattice approach when applying it to American options? What are the effects of cash distributions on the stock price? What are the specificities of the Paris RM market? What are the specificities of the Ho and Lee approach in the description of the term structure of interest rates9 What are the specificities of the Ho and Lee approach for the valuation of interest rate dependent contingent claims? What are the deficiencies in the Ho and Lee model? .

•

.

presented.

POINTS •

.

CHAPTER

.

.

.

call option

.

put option convertible

.

calls when

.

puts in the same context. 6. Appendix 13.C provides a detailed algorithm for the valuation of American convertible bonds with several call and put provisions in the presence of dividends and coupon payments. .

|NTRODUCTION Financial

often resort to numerical methods. They particularly use finite that must be satisfied by the to solve partial differential equations prices of derivative securities. Indeed, these methods are a powerful tool in option pricing when there are no closed-form solutions. The finite difference method consists in discretizing the partial differential pricing equation and the boundary conditions using a forward or a backward difference approximation scheme. The resulting system is then solved iteratively. This gives the derivative asset price at each instant of time as a function of different levels of the underlying asset price.

difference

economists methods

OPTIONS,

282

13.1 13.1.1

FUTURES

TO AMERICAN APPLICATION DIVIDEND PAYING STOCKS The Schwartz

AND EXOTI

$4GigillMCAL f4ETNODS

ERIVATIVES

CALLS ON

82C DC + rS -OS OS2

DC

+

-8t

- rC

=

0

'

=

C(0, t)

)

max

=

[0, S -

max

K, C(S - d, T

)]

when

where d stands for the dividend amount, and T and T* refer to the instants just before and just after the underlying asset goes ex-dividend The first condition gives the call pay-off at the maturity date. The second condition shows that the call is worthless when the underlying asset is zero. The third condition value when the shows that the call value cannot be less than its immediate exercise underlying asset goes ex-dividend. This condition characterizes the existence ofa certain level of the underlying asset for which the option value (withdividends) is equal to its value upon exercise. This is the critical underlying asset price corresponding to the situation where the option intrinsic value is above C(S, T*). More generally, this situation requires the use of the following condition on the option's derivative with respect to the underlying asset: .

hm

=

1

This condition must be satisfied for a sufficiently high level of the underlying

82 C

If we replace these partial derivatives equation, we get

Consider a subdivision of the state variable into h equally spaced units of theenderlying asset and the time variable into k units of time, or S, Ty

=

=

ih

for i

jk

for j=

Hence, the option price u(S, T) can be written C(S, T) The partial derivative point by the difference

with

respect

with

their values

=

a,

=

as

C(S;, T;)

=

(6)

C(ih, jk)

to time, (DC 8t), can be approximated

at the

(i, j)

C(i, j - 1)

=

(10)

hi c,

Boundary condition

(rik- ja2i2k

(11)

1 + rk + a2i2k

(I2)

=

=

(2) for the American

(3), C(0,

t)

=

0, is approximated

for j

0

C(i, j*)

=

values

C(i - d/h,

f

ih

by

if i a k/h else

0

=

(13)

call is approximated ih - k

=

2k

-jrik -jo2

=

(14)

by

0, 1,

...,

by

C(i - d/h, j for C(i - d/h, j

of the underlying asset, condition =

h

(15)

m

) for

j

- k

C(i, j) - C(n - I, j)

0 to m

in the B-S partial differential

with

For sufficiently high

0 to n

=

:

2 C(i, j) + C(i - 1, j)

-

At a dividend date, condition (4) is approximated

Solution

The Numerical

(8)

point by

(i, j)

a,C(i - l, j) + b,C(i, j) + c,C(i + 1, j)

Boundary condition

asset.

at the

C(i + 1, j)

C(0, j) 13.1.2

the (i, j)

2h

(5)

as

s

at

C(i + 1, j) - C(i - 1, j)

C(i, 0)

ßC(S, r)

1)

'

(2) (3) (4)

,

0

=

[0, S -

K]

V41ue

283

k

The term 82C/DS2 can be approximated

He used the following boundary conditions to solve for the American call there are dividends: C(S, 0)

SC OS

113

PRICING

to the asset price can be approxunated

respect

applies between divadend dates:

OPTION

C(i, j) - C(i, j-

_

E

The partial derivative with point by the difference

Schwartz (1977)assumed that the B-S (1973)equation

C(S, T

DC

y

Model

la2S2 2

FOR AMERICAN

for j=

)> )E

ih - k ih - k

(16)

(5) is approximated by 0,

...,

m

(17)

with n + l Hence, for each value of j there is a system of n - 1 linear equations unknowns. Using conditions (3) and (5) gives a system with n + 1 equations and n + l unknowns. This system can be solved by inverting the matrix to give all possible values of the option price at each instant j. Appendix 13.A presents a detailed algorithm corresponding to this modet

284

13.2

FUTURES

OPTIONS,

The Brennan

and Schwartz

DERIVATIVES

NUMERICAL

solution for the valuation Brennan and Schwartz (1977b) presented of American put options when there are discrete distributions to the underlying asset. The valuation of the put is given by the solution to the B-S partial differential equation under the following conditions:

P(S, t)

>

max

max P(S, t)

[0, K - S] [0, K - S]

(18) (19) (20) (21)

0

>

P(S, t) GK SP(S,

.

ini

-

t)

0

=

BS

(22)

to the put value at the maturity date which is simply the greater of zero and the intrinsic value. Condition (19)shows that the American put value must be greater than its exercise value at each instant. Conditions (20) and (21) give respectively the minimum and the maximum price for a put option. Condition (22)results from conditions (20)and (21)and the convexity of the option's price. On a dividend date, the following condition must be satisfied: Condition

P(n - 1, f)

.

-

-

P(S, t )=max{K-S,

t*)}

P(S-D,,

Solution

of the state variable (the stock) space into h small, equally spaced units and the time variable (time)into k small units. Also, we will use a new time variable, T - t, instead of t, the calendar time. The discretization of the asset price r and the time to maturity can be written as

Consider the discretization =

S, ry

ih

for i

jk

for j

=

=

(28)

-¡rik-ja2i2k

(29)

The put price P(S, r) is approximated by P(S,, r Approximating the option partial derivatives differential equation gives the following system:

=

0 to

n

=

0 to

m

(24) (25)

) P(ih, jk). by their values

-

P(n, j)

=

for j

0

by

1 to m

=

(30)

=

for K - ih condition. satisfy this must the underlying asset for which this ich, corresponding asset price, Se date, condition (23) is approximated P(i, j)

and the solution The value of gives the critical At a dividend

>

=

.

_

i

=

0 to n

inequality becomes a strict equality to an optimal exercise policy. by

P(i - d/h, j for P(i - d/h, j

P(i - d/h, j K ih

) for

-

(31)

)>

)<

K - ih K - ih

(32)

.

This system can be solved by inverting the matrix to give all possible values of the option price at each instant j as a function of the values an instant before. Appendix 13.B presents a detailed algorithm corresponding to this model.

13.3

APPLICATION

13.3.1

The Specificities

TO CONVERTIBLE of

ConYertible

BONDS

Bonds

bond is a security paying periodic coupons. It is more complex than the a dual option. It gives the right to the bondholder to convert the bond into common stocks and gives the issuing firm the right to call the bond for early redemption. Following Brennan and Schwartz (1977c),we use the following notation: The convertible

warrant

The Numerical

=

1 + rk + a2:2k

(23)

This condition shows that just before the stock goes ex-dividend (instantt ) the put value must be equal to the greater of the intrinsic value and the put price when the stock is exdividend (instantt*). This problem has no analytical solution and numerical methods must be used.

I 3.2.2

c,

285

System (26)and condition (22) represent a set of n linear equations with n + l unknowns 0 to n. The use of condition (18) allows the solution of P(i, j) as a u(i, j) for i of P(i, function j - l). Condition (19)is approximated by

(18) corresponds .

=

PRICING

=

=

a numerical

=

b,

OPTION

1 to m. 1 to n - 1 and j for i The boundary condition (22)for each value of j is approximated

Model

P(S, T)

FOR AMERICAN

METHODS

PUTS ON

APPLICATION TO AMERICAN DIVIDEND PAYING STOCKS

13.2.1

AND EXOTIC

and

V(t): W(V, t): CP(t): B(V, t): D(t):

involves

the the the the the

value of the firm's securities, market value of a convertible bond with par value $1000, call price at time t at which the bonds may be called for redemption, value of an otherwise identical bond with no conversion provision dividend payment to the common stocks.

market

=

and replacing

in the

Suppose that there are Nc convertible bonds and No shares before conversion. We denote by q(t) the number of shares into which a bond can be converted at time t, I the coupon I/Nc the periodic coupon payment per bond. payments at each payment date, and i Since each convertible bond can be converted into q(t) shares, the conversion value, =

a;P(i

-

1, j) + b,P(i, j) + c,P(i+

1, j)

=

P(i

j -

1)

(26)

C(V, t), is given by

with =jrik

a,

-

(<>2i2k

(27)

C(V, T)

=

q(t)V(t)/[No

+ Neq(t)]

=

z(t)V(t)

(33)

AND EXOTIC

OPTIONS,?UTURES

286

T1VES

with z(t)

q(t) [No + Neq(t)]

=

(34)

Since an optimal conversion strategy implies that the value of the unconverted bond is at value, the following arbitrage condition must be satisfied: least equal to the conversion W( V, t)

C( V, t)

>

(35)

The bondholder has the choice at each call date to receive the call price CP(t) or the conversion value, C( V, t). Therefore, the value of the called bond VIC( V, t) must satisfy the following condition: VIC(V, t)

C(V, t)]

max [CP(t),

=

t*, when the bond becomes callable Moreover, at time t not called until after a certain period), its value must satisfy =

W(V, t*)

if C(V, t*)

C(V, t*)

=

(36)

(becausein practice bonds are >

CP(t*)

(37)

and at any time during the call period, the bond's value cannot exceed the call price, W(V, t)

I 3.3.2

The Valuation

CP(t)

<

i.e.

(38)

2

Equation

2

82W -

BW -

BW

+ rV

2

-kW

0

=

W(0, t) W(V, z*(t)

0

=

t)
is the maximum value of z(r) for r

(t,

in

W( V, t) > C( V, t) z(t)V, z(T)V W(V, t)

V/Ne,

VG

W(V,

lini W( V, t

)

=

)

OPTION

PRICING

287

callable.

=

t)

The Numerical

<

[W( V

Solution

Since there is no closed-form solution to the B-S differential equation under the above boundary conditions, numerical methods are useful in solving such problems. T - t, instead of the calendar time t, the discretization of the Using a time variable r underlying asset price and the time to maturity is =

V=ih r

(43)

z(t) V

(44)

z(t)

D, t*), z(t

) V]

W( V - I, t*) + i min W( V [ - I, t ) + i, CP(t =

)]

(51)

Otom

=

1) W(i, j -

which is

(52)

with

(45)

b,

(46)

c,

(47) (48) (49)

(50)

forj=

to the B-S partial differential equation

=

a,

CP(t)

fori=0ton

- 1, j) + b(i)W(i, j) + c(i)W(i + 1, j)

a(i)W(i

1000/z(T)

jk

=

The convertible bond is the solution approximated by

1000Ne

=

)

13.3.3

T);

=

EVE

BW(V, t)

max

W( V, t W( V, t

FOR AMERICAN

Condition (40)shows that the total value of the bonds is less than the firm's value. This is value is equal to that of its stocks and its bonds. so because the firm's Condition (41)indicates that the bond value is zero when the firm is worthless. Condition (42) shows that the convertible bond value is less than the value of an equivalent straight bond and the maximum number of shares in which it can be converted. Condition (43)illustrates the option offered to the bondholder for conversion. Condition (44)corresponds to the pay-off of the convertible bond at the maturity date. Condition (45)shows the constraint on the call price during the call period. It indicates that the convertible bond price can not exceed the call price, otherwise the issuer will call back the bonds. Condition (46)is a high-contact condition which applies for a sufficiently high value of V Condition (47)must be applied at each dividend date where the instants just before and just after are denoted respectively t and t*. Condition (48) must be applied at each coupon date when the bond is not currently callable. The instants just before and just after are denoted by t and t*. Condition (49) must be applied at each coupon date when the bond is currently

1000

>

1000, 1000Ne

=

(39) (40) (41) (42)

Ne W( V, t) EV

where

METHODS

solution for the valuation of Brennan and Schwartz (1977c)presented a numerical American convertible bonds when there are discrete distributions to the underlying asset and call and put provisions.

The convertible bond can be valued using the B-S partial differential equation under the following appropriate boundary conditions: Io2

NUMERICAL

for i

(rik («2i2k 1 + rk + a2i2k

=

-irik- ja2i2k

=

1 to n and j 1 to m. For a given j, equations with n + 1 unknowns, when i varies from 0 to n. Condition (4l) is written as =

=

W(0, j)

=

0

for j

=

(52)-(55)give n -

0 to m

(53) (54) (55) I equations

(56)

OPTIONS,PAPfURES

288

The discretization of condition

for zhi a P for ps

P

=

V

.

hi

where P samds for the par value of the convertible Dendmon(46) is approximated by

'

(57) (58) (59)

bond.

i, j)) h (W(n, y) - W(n -

Using conditions

P/z

<

for hi GP

hi

=

=

(60)

z

the values of W(i, j) can be determined m a recursive manner from W(i, j - 1) since all the W(i, 0) are given for all values of i. At a call date, condition (46) is replaced by (45), or

(56)-(60),

W(i, j)

<

CPU)

(61)

Smce the bond can be called before the maturity date, the value of W(i, j) for a certain i greater than a given value q, given by

On a dividend date, condition

(47) is approximated

W(i - D|h, j,,) zih M/(i, j,,)

zih

=

.

./c)

approximated .

WO, and condition

(49)is approximated

A detailed algorithm

13.3.4

is

by

(63)

by

.

W(i - I h, je)

=

I

(64)

by

W(i - I/ h, je) CP(jc)

W(i, jc)

not defined

(62)

if W(i - D h, j,,) > zV if W(i - D h, j,,) < zih

=

On a coupon date, condition (48) is

is

CP(j)/zh

=

q

I

given m Appendix

for W(i - I h, ic) + I < CP(jc) for W(i - I h, je) + I> CPUc)

(65)

13.C

Simulations

Using the algorithm

in Appendix 13 C, the convertible bond pnce is iiníulated using the

following data: Par value of the bond: 40, Semi-annual coupon: 1.1, Quarterlydividend: 1.1 Firm variance rate: 0.0012 per month, Risk-free rate: 0.0057 per month. The bond is not callable for five years; it is callable at 43 (plus accrued interest) for the next five years, at 42 for the next five years and at 41 for the last five years. Using 200 iterations, h 200 240 months and a time step of one V 2.5, T - t month, Table 13.1 gives the prices for different levels of the underlying asset V =

=

NUMERICAL

TABLE

(44) gives z(t) V= zhi

W(i, 0)

AND EXOTICHIINtiVATIVES

=

13.1

METHODS

Prices

0.000000 µ(4) = 4.740 000 u(8) = 7.990000 u(l2) = 11.230000 u(l6) = 14.480000 u(20) = 17.940000 u(24) = 22.730000 u(28) = 29.400 000 u(32) = 34.520000 u(36)=36.080000 u(40) = 36.214000 u(44) = 36.2 I 5 870 u(48) = 36.2 I5 875 u(52) = 36.2 I5 875 u(56) = 36.2 I5 875 u(60) = 36.2I5876 u(64) = 36.215880 u(68) = 36.2l6000 u(72) = 36.217 100 u(76) = 36.223 600 u(80) = 36.250 000 u(84) = 36.320 000 u(88) = 36.480000 u(92) = 36.720 000 u(96) = 36.990 000 u(l00) = 37.280000 u(IO4) = 37.580000 u(108) = 37.920000 u(I 12) = 38.270 000 u(I 16) = 38.610000 u(120) = 38.910000 u(l24) = 39.\90000 u(l 28) = 39.490 000 u(132) = 39.830 000 u(l36) = 40.240000 u(l40) = 40.660000 u(144) = 4l.040000 u(l48) = 4l.350000 u(l52) = 41.640000 u(156) = 4l.920000 u(l60) = 42.200000 U(l64) = 42.490000 u(168) = 42.780000 u(l72) = 42.960000 u(l76) = 43.990000 u(180) = 44.990000 u(l84) = 46.000000 u(l88) = 47.000000 u(192) = 48.000000 u(196) = 49.000000 u(0)

=

FOR AMERICAN

for Different

OPTION

PRICING

289

Levels of the Underlying Asset

u(I) = 1.960000 u(5) = 5.560 000 u(9) = 8.800000 u(l3) = 12.040000 a(l7) = 15.300000 u(21) = 18.950000 u(25) = 24.290000 u(29) = 3 l.000 000 u(33) = 35.190000 u(37)=36.160000 a(4I) = 36.215000 u(45) = 36.2 I5 875 u(49) = 36.2 I5 875 u(53) = 36.2 I 5 875 u(57) = 36.2 I5 875 u(6I) = 36.215876 u(65) = 36.215890 u(69) = 36.2l6 100 u(73) = 36.217900 u(77) = 36.227 500 a(8 I) = 36.260 000 u(85) = 36.360 000 u(89) = 36.540000 u(93) = 36.790 000 u(97) = 37.070 000 u(l0l) = 37.350000 u(IO5) = 37.660000 u(\09) = 38.000000 u(1 I3) = 38.360 000 u(I 17) = 38.690000 u(121) = 38.980000 u(l25) = 39.260000 u(l 29) = 39.570 000 u(l 33) = 39.930 000 u(I37) = 40.340000 u(l4l) = 40.760000 u(145) = 41.120000 u(l49) = Al.420000 u(l53) = 4l.7\0000 u(157) = 41.990000 u(\6I) = 42.270000 u(l65) = 42.560000 u(169) = 42.790 000 u(I73) = 43.260000 u(I77) = 44.250000 u(181) = 45.250000 o(iss) = 46.250000 u(l89) = 47.250000 u(193) = 48.250 000 u(l97) = 49.250000

u(2) = 3.230000 u(6) = 6.370 000 u(IO) = 9,610000 u(l4) = 12.850000 u(l8) = 16.140000 u(22) = 20.070000 a(26) = 25.960000 u(30) = 32.420 000 u(34) = 35.640000 u(38)=36.190000 u(42) = 36.215800 u(46) = 36.2 I5 875 u(50) = 36.2 I 5 875 u(54) = 36.2 l 5 875 u(58) = 36.2 I5 876 u(62) = 36.215878 u(66) = 36.215916 u(70) = 36.216 300 u(74) = 36.219 100 u(78) = 36.233000 u(82) = 36.280 000 u(86) = 36.390 000 u(90) = 36.590 000 u(94) = 36.860 000 u(98) = 37. I40 000 u(lO2) = 37.430000 u(106) = 37.740000 u(I 10) = 38.090000 u(I I4) = 38.440 000 u(118) = 38.760000 u(l22) = 39.050000 u(I26) = 39.330000 u(130) = 39.650 000 u(l 34) = 40.030 000 u(l38) = 40.450000 u(l42) = 40.860000 u(l46) = 41.200000 u(Iso) = 4l.500000 u(154) = 41.780000 u(l58) = 42.060000 u(l62) = 42.340000 u(166) = 42.630000 u(l70) = 42.810000 u(l74) = 43.490000 u(l78) = 44.490000 u(182) = 45.490000 =46.500000 u(l86) u(190) = 47.500000 u(194) = 48.500000 u(l98) = 49.500000

V

u(3) = 3,910000 u(7) = 7.180 000 u(I I) = 10.420000 u(l5) = 13.660000 u(19) = \7.020000 u(23) = 2l.330 000 u(27) = 27.690000 u(3 I) = 33.600 000 u(35) = 35.920000 u(39)=36.2l0000 a(43) = 36.215800 u(47) = 36.2|5 875 u(5 I) = 36.2 I5 875 u(55) = 36.2 I5 875 u(59) = 36.2 15 876 u(63) = 36.2I5880 u(67) = 36.21595\ u(7l) = 36.216 600 u(75) = 36.220 900 u(79) = 36.240 600 u(83) = 36.300 000 u(87) = 36.440 000 u(91) = 36.660000 u(95) = 36.920 000 u(99) = 37.2 IO000 u(l03) = 37.500000 u(IO7) = 37.830000 u(I I I) = 38.180000 u(I I5) = 38.530 000 u(I l9) = 38.840000 u(123) = 39.120000 u(I27) = 39.4l0000 u(I 3 I) = 39.740 000 u(135) = 40. l 30 000 u(l39) = 40.560000 u(l43) = 40.950 000 u(\47) = 41.280000 u(l5l) = 41.570000 u(155) = 4l.850000 u(l59) = 42.130000 u(l63) = 42.4l0000 u(167) = 42.7l0000 u(l71) = 42.850 000 u(l75) = 43.750000 u(179) = 44.750000 u(l83) = 45.750000 u(187) = 46.750000 u(\91) = 47.750000 u(\95) = 48.750 000 u(199) = 49.750000

OPTIObts,FUTURES

290

AND EXOTIC

DERIVATIVES

NUMERICAL

METHODS

FOR AMERICAN

OPTION

PRICING

291

d(i)=C(i,j-1)

UMMARY End

This chapter introduces the reader to the application of finite difference methods to the pricing of American options. First, the method is applied to the valuation of American call options when there are severaldividends. Second, the method is applied to the valuation of American puts in the same context. Note that in both cases, there is no analytical solution in the literature. Third, the method is illustrated for the valuation of American convertible bonds when there are several dividend dates, coupon dates and implicit call and put options. In each case, a detailed algorithm is given in an appendix to illustrate the determination of the critical levels of the underlying asset price corresponding to an optimal premature exercise.

=-1,

a (n) For i=1 w(i)

• • • • •

For each level S,

(n, j)

ih-K

ton, < 0 then

C

For

=h

a

(i) ) (b(i)-wfi-1)a(i'f)

=g

(n)

(i)

=g

=0,

End

(for

i=0

-w(i)

u

(i+1,

of the critical

to n do H (i) =C (i, j) +K-ih

1)

End

end

i=1

i=0

For

to k-1,

v (i) C (i-k,

=0

is done. If j is ûtì2

end

end for

1)

< endif

(ih-K)

then for

end

vii)

Fori=0

i=0

tondoC(i,j)=v(i)

to n write

end C

=ih-K

for

(i, j) end for,

end

if

End.

End to the inversion of a tridiagonal matrix by the Gauss method.

APPENDIX

\ 3.B:

THE ALGORITHM FOR THE AMERICAN PUT WITH DIVIDENDS

Forj=1tom,

(1) b (1) c (1) d (1) For

(i) b (i)

For each level ih of the underlying asset, the terminal boundary condition is written as

=0

a

, =1+o2k+r

=-1/2

k

=C

(1, j-1)

i=2

to n-1

=1/2

a

=1+a2i2k+r

c(i)=-1/2

i=0 to n, P (i, 0) =K~ih if K-ih < 0 then P (i, 0)

For

U2k

r k-1/2

=0,

r i k-1/2

U2i2k

,

End

Forj=1tom,

.

i=0

to n)

When the asset price is zero, the put value is equal to the strike price.

ik r ik-1/2a2i2k

(for

.

P(0,j)=K,

End

if

then k=nat(d/h)

to n) For

to an

,

At a dividend date, the following treatment corresponds to the integer part ofa number.

is written as

,

asset price corresponding

to n do if h ( i) 50 then C (i, j ) =ih-K, k-k+1 end if if k=1 then S*= (i-1) h+h H (i-1) / H(i-1) -H(i) End (for i=1 to n)

asset, the terminal boundary condition

C(0,j)=0,

$$ae following system corresponds

1=0

For

When the asset price is zero, the option is worthless. Forj=1tom,

(i) / (b(i)

For i=k to n if elsev(i)=C(i-k,j)

(i, 0)

d (n)

-w(i-1)

=c

i=n-l down to l, C (i, j) i=0 to n write C (i, j)

For

k=0

THE ALGORITHM FOR THE AMERICAN CALL WITH DIVIDENDS

ih ofthe underlying

,

Here, the last elements in the system are calculated. We generate and print all the unknowns at each instant of time using the Gauss method for the inverted matrix.

C(i,0)=ih-K

Fori=0

if

=

=0

End C

Why are numerical methods using in asset pricing? What is a finite difference scheme? What is an implicit scheme? What is an explicit scheme? What are the main characteristics of convertablebonds?

13.A:

c (n)

A special treatment is done for the determination optimal early exercise.

FOR DISCUSSION

APPENDIX

=1,

g(i)=(d(i)-g(i-1)a(i))/

For

POINTS

b (n) to n do

whee

int(.)

292

OPTIONS,

AND EXOTIC

FUTURES

DERIVATIVES

APPENDIX

For each time step the system must be solved using, for example, the Gauss method. For j=1 tom, =1+a2k+r

OPTION

293

PRICING

THE ALGORITHM FOR CONVERTIBLE CALL AND PUT PRICES

I 3.C

=0,

(1) b (1) c (1) d (1) a

k

=-1/2

r k-1/2

a2k

To run the program, enter Vmax, the bond price P, the volatility a, the interest rate r, the number of months until the maturity date nm, the maximum number of steps for the underlying asset nV, the

.

(1, j-1)

=P

i=2 to n-1 do a ( i ) 1 / 2 r i k-1/ 2
For

FOR AMERICAN

METHODS

NUMERICAL

=

number of periods where z changes, i

k

=1+a2i2k+r

d (i) =P (i, j -1) for (i=2 to n-1) b (n) a (n) c (n) For i=1 to n do

npl, the number of periods where the call price changes,

np2,

the amounts of dividends dv, the dates of dividends dd, the coupon amounts Ic, the coupon dates de, the call price vector CP(k), and the length of the call period, d2(k).

Initialization

13.C.I

End

=-1,

=-1,

w(i)

(i) / (b(i)

=c

=0 ,

-w(i-1)

End

(i=1

for

to npl,

enter

d1 (i)

z (i) , For k=1 to np2,

enter

d2 (k)

i=1

For

(i) )

a

g(i)=(d(i)-g(i-1)a(i))

=0

d (n)

/ (b(i)-w(1-1)a(i))

,

of

length

the CP

(k)

(CP

period

(0)

:

i in monthss,

no call)

·

nvv nv nvt=nvv h=Vmax/nV

to n)

P(n,j)=g(n)

Fori=0tonV Vs=ih in the system are calculated.

Here, the last elements

all the unknowns at each instant of

We generate

if

llme.

Fori=n-1downto1, End Fori=0

P(i,j)=g(i)-w(i)

i=n-1down

(for

u(i+1,j),

For

For

k=0 For

is done for the determination of the critical asset pnce correspondmg

end

to early

write

j i=1

,

=

z

(1)

Vs else

if vs

P

/b(0) to nv do (i) / (b (i) -w (i-1) w (i) i=1

=c

For

-H

=g

a

i=nv-1

downto

(i)

=g

-w

O do

(i) u (i+1)

rl to nm (time

Sc

to n)

(i) ) (b(i)-w(i-1)a(i))

n)

W (i) EF d

(i) and if,

(i=1 to (nv)

for

W (nv)

,

( for

j )

)=Vs,

=d(0)

End

for

i=1 to n do thenP(i,j)=K-ih, if H(i)<0 k=k+1endif if k=1 then Sc= (i-1) h+h H (i-1) / H (i-1)

End

W ( i,

.

g(i)=(d(i)-g(i-1)a(i))/

to n do H (i) =P (i, j) -K+ih

i=0

.

w(0)=c(0)/b(0) g(0)

A separate treatment exercise.

( s ) >P then

Theproceduretobeusedtoinvertthematrixisasfollows

to 1) endfor

tonwriteP(i,j)

z (1) v thenW(1,

until

earch for the value of z(i) corresponding

maturity) to a given mon&,

tiefelteemg

treatment is done:

dt=0,imp1=0 For a dividend date, the following treatment corresponds to the integer part ofa number For

i=0 i=k

For else

For i=0 Fori=0

is done. If / is in Jl then k=int

(d/h)

where int(.)

Repeat

imp1=imp1+1 dt=dt+d1(impl)

=0

v(i) end for to k-1, to n if P (i-k, j) < (K-ih) P ( i-k, j ) end i f end v ( i) =

tondoP(i,j)=v(i) tonwriteP(i,j)

end end

for,

until

(jddt)

-

then f or for endif

v

(i)

zz=z

=K-ih

Tosearch End

for,

End

(impl)

for the call price CP(j) corresponding

dt=0,

imp2=0,

to a given month j, the following treatment

is done:

OPTIONS,

294

FUTURES

AND EXOTIC

DERIVATIVES

OPTION

PRICING

295

uu(k)>(k+1-rk)uu(k)+(rk-k)uu(k+1)

Repeat

for

(jddt)

until

end

then i=1 to nvt

W(i,

else

(imp2)

CPP=CP CPP=0

ih)

ifuu(k)>(z

imp2=imp2+1 dt=dt+d2 (imp2) If

FOR AMERICAN

METHODS

NUMERICAL

j)

=uu

thenW(i,j)=uukelseW(i,j)=z

(i) end if

end

ihend

for

if

The treatment for the coupons is as follows:

to nvv

W(i)=zhi, nv=nvv nvt=nvv

.

k=trunc(j/dc) If (dc k-j) for i=0 tonvuu(i)=W(i, j) end for for i=0 to nv (Ic/h) r k=ik=trunc (r k) ifk>0 then uu (k) (k+1-rk) uu (k) + (rk-k) if (uuk+IcGCCP) then or (CPP=0) =0

end

(i=1

for

to nvt

to nvv)

Else (CPP/z

nvt1=trunc for i=nvt

h) +1

to nvt1,

W(i)

nv=nvt,

end

nvt=nvt1,

=zh

i end,

=

The equation is discretized as follows: to nv-1

i=1

For

2

a(i)=1/2ri-1/2Ü2 =1+G2i'+r

b (i) c End

(i)

U2i2

r i-1/2

d(i)

=W(i,

for

(i=1 to

j-1)

=

0 is

c(0)=0,

The boundary condition when i

=

a(nV)=-1b(nV)=1,

d(0)=0

d(nV)=hzz

c(nV)=0,

solve (no call) (a, b, c, d, w, nv)

the

tri-diagonal

system

else Repeat nv=nv-1,a(nv)=0,b(nv)=1,c(nv)=0,d(nv)=CPP

tri-diagonal

the

solve Endif

system

j)5Cpp

(CPP=0)

For the dividend, the following treatrnent is done: k=trunc (j/dd) if (ddk-j=0) then Fori=0tonv,uu(i)=W(i,j)Endfor(i=0tonv) For

i=0

to nv,

r k=i-dd/h

k=trunc if k>0

(rk) ,

+Ic

if end for End if

(i=0

to nv)

End

(j=1

to nm)

End.

n Vis

CPP=0

untilW(nv-1,

(k)

nv-1)

b(0)=1,

a(0)=0,

=uu

end

=-1/2

The boundary condition when i

If

W (i, j) ElseW(i,j)=uu(i)

do

then

(a, b,

c,

d, w, nv)

for

else

W(i,

j)

=CPP

uu end

(k+1)

I4 Exchange, Forward Start and Chooser Options

CHAPTER

OUTLINE

This chapter is organized as follows: 1. In section 14.1, the option to exchange one risky asset for another is analyzed and valued. The concept of exchange options is applied to the analysis and valuation of the performance incentive fee contract, the margin account and the exchange offer. Also simulations of the contract's values are provided. 2. Section 14.2 identifies options with an uncertain strike price and presents a method for their valuation. Call and put values when the strike price is uncertain are also simulated. 3. In section 14.3 forward start options are analyzed. 4. In section 14.4 pay later options are studied and valued. 5. In section 14.5 simple and complex chooser options are analyzed using the results given for compound options.

INTRODUCTION The theory for pricing an option to exchange one risky asset for another was proposed by Margrabe (1978).This theory grew out of the B-S (1973)and Merton (1973a)models. The option to exchange one risky asset for another is implicit in some common financial arrangements. Examples of such contracts include the investment adviser's performance fee, the exchange offer, the general margin account and the standby commitments, among others. These contracts have the features of options to exchange one risky asset for another. This theory was used later by several authors for the pricing of more complex contingent claims. When the total value of a firm is given by the market value of its stocks and bonds, B-S (1973)shows that corporate stocks in a levered firm are regarded as a call with a strike price equal to the payment to be made to bondholders. However, since bonds are and consequently denominated in real terms, the payment to bondholders is uncertain, so is the call's strike Drice. Hence, it is not nossible to annlv in a straightforward wav the

298

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

arbitrage argument for the valuation of an option with an indexed strike price. Besides, to such options, it is necessary to infer how an asset which hedges against changes in the strike price should be valued Forward start options are options which give an answer to the following question: how much can one pay for the opportunity to decide after a known time in the future to get an at-the-money call with another time to maturity? This opportunity is not accompanied by an additional cost. Pay later options are options for which the premium is paid upon exercise. They offer some protection against sharp price movements without tying up capital. These options are contingent options since the buyer has the obligation to pay upon exercise, when the option is in-the-money regardless of the amount by which the underlying asset price exceeds the strike price. As noted by Turnbull and Wakeman (1991),these options present some hedging difficulties since the pay-off is discontinuous. For example, the delta is negative for an out-of the-money call when the option is close to maturity. Chooser options allow the holder, immediately after a predetermined elapsed time, to choose whether the option is to be a call or a put. There are two kinds of chooser options: simple and complex choosers. Complex choosers can be valued using the concept of an option on an option, or a compound option. An option on an option is an option for which the underlying asset is an option. Compound options take the form of a call on a call, a call on a put, a put on a call or a put on a put. The underlying call or put may be a standard or an exotic option Geske (1979a,1979b) presented the formula for a call on a call and Rubinstein (1991b) generalized this result to include a put on a call, a call on a put and a put on a put value

EXCHANGE,

START AND CHOOSER

FORWARD

Since the investment is nil, the investment must also be zero:

ci

EXCHANGE

14.1.1

Identification

OPTIONS

dSi -

dS2

c2

=

0

(6)

Using stochastic calculus for the option's return gives dc

=

ci dSi + c2

with c3 Sc St, where and (6)yields =

(52

+ c3 dt + ([cavjSi

subscripts

([c,v Si +

+ c22v S + 2c,2viv2pi2SiS2]dt

refer to partial derivatives.

Combining

c22v S + 2ci2viv2Ûl2SiS2

The solution to this equation subject to conditions

(3) and

MSa S2, t) - S,N(d,)

-

3

=

0

di

In

=

og

+

\S2 y

(v2T

(8)

(4) is

S2N(d2)

(O

d2

,

=

di - v

(10) =

=

=

For American-type

-

c(S2, Si, t) + S2 - Si

options, the following relationship

C(Si, S2, t) - C(S2, The model is used to calculate the call values

S,

OO

must be satisfied:

t) + S2 -- Si

(12)

reported in Table 14.1.

Following Margrabe, let Si and S2 be the prices of two assets 1 and 2, for which the price dynamics are given by =

S,(a, dt

v, d W,)

for i

=

l, 2

(1) (2)

dWi dW2 - pi,

TABLE

14.1

Exchange

Call Values

Option

Time to Moturity

where

d W, is a Wiener process and a, and v, are constants. Let c(Si, S2, t) be the value of a European option with a maturity date t*. The option gives the holder the right to receive the difference (Si when exercised, or nothing if - S2) not exercised. Hence, its pay-off is given by c(Si, S2, i

=

inax

(0, 5, - S2)

0

<

c(Si, S2, t)

«

=

(Oc/DS,) units

=

c - ci

Si - c2S2

=

0

5,

0./0

0.25

0.50

0.75

l.00

IO%

90 100 i10

10.39 3.34 0.56

| l.52 5.28 l.9I

13.17 7.45 3.82

14.56 9.12 5.39

15.77 |0.53 6.75

25%

90 100 I 10

10.92 4.35 \.20

12.70 6.87 3.28

15.05 9.70 5.95

16.96 I l.86 8.08

18.59 |3.68 9.90

50%

90 \00 110

12.73 6.90 3.3\

16.\0 10.89 7.I l

20.1 I \5.36 l l.60

23.24 18.75 15.07

25.87 2\.58 17.99

(4)

S,

It is possible to construct the following hedged portfolio, by selling c, of asset l and buying -c2 (Oc/8S2) units of asset 2, or

Volotility

(3)

This option is worth at most Si and has a positive value, so one can write

(5)

(5)

t* where N(.) stands for the cumulative normal density function and T - t. Note that when v2 0, this solution reduces to the B-S formula. v and v2 Margrabe (1978)demonstrated that the usual put-call parity theorem holds for the options to exchange one asset against another. In particular, he showed that the foBowing relationship applies for European options and their underlying assets:

and Valuation

dS,

equatimes

(7)

where

c(Si, S2, O

14.1

299

of the hedged position is nil and the return on the

value

dc -

OPTIONS

OPTI

300

14.1.2

,

FUTURES

AND EXOTIC

DERIVATIYES

EXCHANGE,

FORWARD

START AND CHOOSER

the reopen, It has a longer time to maturity equation (9).

Applications

The Performance

*

OPTIONS

over weekends

30 I

and can be valued

using

Fee

Incentive

Some portfolio managers are paid a performance Modigliani and Pogue (1975)as Fee

=

ö(ri -

incentive fee. This fee is modeled by r2)

(13)

where

Example

Consider, as in Margrabe, an investor who wants to finance the purchase of worth of shares in one fund by short-selling shares in another fund. Suppose the two funds are closed-end funds with the same risk, p;2 - L 0.05, In this context, a broker arranging this transaction assumes less risk v2 vi than making a margin loan since the investor's option is worthless. However, if the short sale is riskier for the broker, since the two operations of short pi2 would selling and purchasing be twice as risky as either operation alone

$100 000 =

is the rate of return on the managed portfoho, r2 is the rate of return on the standard permitting measurement ö is the number of dollars invested in the portfolio ri

ofthe performance,

=

-l,

=

This fee is valuable to the adviser if he can declare personal bankruptcy when the fee is the fee, the negative. By forming a corporation, handling his busmess and collecting investment adviser would have limited liability if the fee were negative. Hence the fee is equivalent to an option that can be calculated using the equation presented above. Consider the example given by Margrabe (1978).The manager receives 10 million dollars of any superior performance with respect to the standard and would pay 10% (ö 1 million) of any inferior performance. If the fee arrangement is for six months, the volatility is 5% for both the standard and the managed portfolio, i.e. Si S2 10 million, T 6, vi v2 0.05 and pi2 0, equation (9) shows that the option would be worth 690 dollars. This price must be paid by the manager to get the business. Simulations of the option values for different periods are given in Table 14.2.

2vi 2v2L This option would cost $1433 over a three-day weekend:

(v • •

.

.

•

=

=

When vi When vi When vi

=

=

0.05, T 0.05, T 0.05, T

=

v2 v2 va

=

=

=

=

=

=

4, p 2, p 1, p

=

=

=

-1, the option price is 1585. -1, the option price is l 124. -1, the option price is 796.

=

=

=

=

=

=

=

The Margin Account

Consider an investor buying on margin securities which are worth Si. Buying on margin means borrowing a fraction of the purchase from the broker and securing the loan by some securities. Let B stand for the principal amount and interest. When this sum is due, the investor pays it and takes his collateral or default. Hence, he has an option on th e collateral. If the collateral value is measured every T months, then the broker has issued an option with a maturity date T This option appears each day at the close and expires at

The Exchange

Offer .

When a contract

5,

=

14.2

52

(million) 10 20 30 40 50 60

CalI Values v, = v2 = 57o, 0, T 6 p,2

0.2, v, = v2 p,2 = 0, T 8

0.69 l.38

3.108 6.216 9.325 12.43 15.54 18.65

=

=

2.07 2.76 3.45 4.i4

.

.

allowing an exchange of shares of an unlevered firm, firm 1 against shares of another firm, firm 2, shareholders in the offered corporation get an option to exchange one share with a price Si against one share of price S,. If the offer expires at time t*, firm 1 has N shares and firm 2 has n shares, then the offer may increase the share price in firm 2 since its shareholders get the option to exchange their shares for something

14.2

OPTIONS

14.2.1

Analysis

Following Fisher TABLE

.

imtlated

is

and

WITH

UNCERTAIN

EXERCISE

PRICES

Valuation

(1978), suppose

the real stock price dynamics are given by dS as dt + as d Ws

(14)

=

=

where as and as are constant. Let K(t) be the real strike price at time i, though the call is exercisable at time E at a strike price K(T), which is unknown at time t. If Z(t) is the price of a specified index at time t, then K(t) might be equal to 1.17Z(t). Assume that the dynamics of K(t) are given by dK =

av dt +

Ük

d We

(15)

OPTIONS,

302

FUTURES

AND EXOTIC

DER1VATIVES

EKCi4ANGE,

TABLE 14.3 Fisher Option 10%, Uk Price: S 100, ru

where as and ou correspond respectively to the instantaneous expected rate of increase in the stated strike price and its volatility. Suppose that the two processes are correlated with a correlation coefficient pks 3S dWsdWe --pasdt

=

Volatility I0%

with

25%

are given by

rMo

50%

oMdWM

rMdt

=

pMk

dt

Uncertain

(19)

p=0.5

K

0./0

0.50

l.00

0.l0

0.50

l.00

90 100 I I0

9. I4 2.51 0.3I

7.94 3.30 0.99

5.74 I.89 0.00

8.89 I.95 0.12

6.52 I.73 0.00

0.00 0.00

90 100 I 10

9.73 3.5 l 0.84

10.45 5.93 3.I 3

10.18 6.35 3.79

9.08 2.38 0.26

2.94 0.72

5.l3 L26 0.00

90 i00 I lo

11.68

16.34 I2.02 8.74

19.38 I5.59 12.55

10.63 4.77 l.71

13.46 9.04 5.91

15.10 I 1.27 8.37

6.08 2.78

7.6l

r ty It is given by 30

The construction of a riskless portfolio à la B-S, comprising the stock, the hedge security that hedges against changes in the strike price and a call, gives the following call pricing formula:

C

=

SN(di) - Kexp

[-(rs - ak)T]N(d2)

-

20-

(22)

where

15

-

10

-

o di

ln

=

(rs -

+ v2

=

o

2)

as +

T

d2

,

=

di -

vn

.0

(23)

6 0.4

(24)

- 2psgasoy + o

5-

o.2

v2 is the instantaneous proportional variance of the change in the ratio S K. This 0 and 0, as formula reduces to that of B-S for a constant strike price and when as

where

=

=

=

3.05

and r constant, then re would be

Ms RMk

where b is the risk premium on the hedge se

Th

Exercise

Øk

p=0.I

rh

M d WM dW

>

303

Time to Maturity

If the hedge security exists (for example, a stock price index), then a portfolio can be constructed using the index. Otherwise, the Capital Asset Pricing Model can be used to If the dynamics of the market rate of return

OPTIONS

Call Values with

=

(16)

The real call price C is a function of the stock price, the current stated strike price and the expected real rate of return, rh, On a hedge security whose dynamics are given by dH Ûk d Wk rh dt (1ÿ) H

infer

START AND CHOOSER

FORWARD

V.

In the same context, the European put price with an uncertain strike price is N(di)] K exp [-(rh Øk)T][l N(d2)] S[l P =

¯

-

-

-

112 116

values with uncertain exercise prices are shown for parameters and Figure 14.1, and put values in Table 14.4. Since (22)gives the call price in real terms, we will consider its use for a fixed nominal strike price K. Assume Q(t)is the purchasing power of money with Q(0) l. Hence, the real strike price is KQ.If the dynamics of Q are given by

Call

=

=

Q

a

di + a

d W,

(26)

120-1.0

K: Striking price

(25)

in Table 14.3

various

p: Correlationwefficient

-0.6 --0.8

84 66 92 96 100 104108

FIGURE 14.1

.

Uncertain

'negative'

Exercise

Price Option

Call Values

of the expected rate of infiation, then op will be the volatility where ap is the of the rate of change in the nominal stock price, S/Q, and a will be the nommal interest rate on a default-free nominal bond. Hence, if v2 is replaced in the di and da in (25) the B-S formula is obtained with all variables expressed in nominal terms.

304

OPTIONS,

TABLE 14.4 Fisher Option Price: S 10%, Ûk 100, rk =

=

FUTURES

AND EXOTIC

Put Values with Uncertain 4% 0.25, as

DERIVATIVES

14.4

Exercise

IO%

25%

50%

START AND CHOOSER

OPTIONS

305

PAY LATER OPTIONS

Pay later options provide a certain insurance against large one-way price movements and are traded on stock mdices, foreign currencies and other commodities. The buyer of pay later options has the obligation to exercise his option when it is in the money and to pay of the importance of the difference the premium. The exercise takes place regardless and the strike underlying price price, the i.e. the amount by which the asset between option is in-the-money. Following Turnbull and Wakeman (1991),we use the following notation:

Time to Maturity =

0.I

.

p

=

0.5

K

0./0

0.50

/.00

0.10

0.50

/.00

90 100 I 10

0.50 4.02 I 1.98

4.95 I l.09 19.55

10.3\

0.25 3.46 I l.78

3.52

18.07 27.65

9.52 18.42

7.62 15.28 25.33

90 100 i l0

l.09 5.02 12.50

7.46 13.72 21.70

14.74 22.54 3\.59

0.44 3.89 I I.92

4.62 10.73 19.28

90 100 I 10

3.04 7.59 14.44

13.35 19.80 27.31

23.95

I.99 6.28 13.38

10.47 16.82 24.48

volotility

FORWARD

=

=

p

EXCHANGE,

31.77 40.35

.

Sr: the price of the underlying F: the current forward rate,

9.69 17.44

asset at the option's maturity date,

cy: the option premium paid at the option's maturity date.

27.12 19.67

date, the pay later European call option's pay-off is

At the option's maturity

27.45

cy(ST,

0, K)

36.17

max [S, - K - cy, 0]

=

(29)

Hence, the option pays out Sr - K - c, when S, > K, otherwise it has a zero pay-off. Applying standard arbitrage arguments, the value of the pay later European call option is given by

14.3

FORWARD

cs(S,, T, K)

START OPTIONS

Se 'arN(di)

=

"N(d2)

cr)e

- (K

(30)

with

Forward start options are options which give an answer to the following question: how much can one pay for the opportunity to decide after a known time t in the future, known date', to get an at-the-money call with time to maturity t with no additional as the cost Following Rubinstein (1991a)we use the following notation: 'grant

S,: the unknown underlying asset value d: 1 plus the known pay-out rate, C(S, K, r): the call value with a time to

di

In

=

r'

(r -

la')T

.

S,C(I,

1, r)

(27)

When we account for the pay-out ratio, the current value of the forward option is =

.

a

way

=

=

Se'

N(di)l

,_

N(d2

- K

=

The pay later call option formula can also be written

C(S,, S,, r)

(31)

F

N(di)

- K

(32)

r.

Since this option is homogeneous of degree l in the underlying asset price and the strike written value the of forward-starting at-the-money call be as price, a can =

di - o

=

When the option contract is initiated, the premium c, must be established in such value is zero, i.e. cs(S,, T, K) that 0. This implies cr

C(S,, S,, r)

d2

,

that the current contract's

after time t, maturity

+

K

a

Sd 'C(1, l, r)

(28)

Since all uncertainty

is resolved once the underlying asset price is observed, after time t, then C(1, 1, r) is known in advance and corresponds to the current value of an at-thein a simple buy-and-hold strategy by money call. This option can be easily replicated holding C(1, 1, r) shares from the current time to the grant date. It is convenient to note that the above results can be easi\y generalized to allow the granting of options which are proportionally in- or out-of-the-money, i.e. by introducing a constant ß in the call pay-off, C(S,, ßS,, r).

cç(S, T, K)

Se '"N(di)

=

as

"N(d2 - Ke

Te

"N(d

)

(33)

This price corresponds also to the value of a foreign exchange option less c, digital options. This decomposition offers a natural way for the hedging of the pay later option. In fact, it shows that the purchase of cr digital calls and the sale of standard calls with the same time to maturity and strike price give a perfect hedge. At the option's maturity date, the pay later European put option's pay-off is pp(ST,

0, K)

S, - pr, 0] (34) Hence, the put option pays out (K - S, - pr) when S, < K, otherwise it has a zero payoff. Applying standard arbitrage arguments, the value of the pay later European put option (seeTable 14.5 for selected values) is given by py(S,, T, K)

=

-Se

max [K -

=

'

'

N(-di)

+ (K - pr)e

"

N(-da)

(35)

306

OPTIONS,

TABLE

14.5

Pay Later

Premium

AND EXOTIC

FUTURES

DERIVATIVES

EXCHANGE,

START AND CHOOSER

FORWARD

2

Values V

Time to Maturity CT Volatiilly 10%

25%

50%

di

I

S2

0.I

I

0.5

90 100 I 10

13.53 7.\2 4.29

17.33 I l.17 7.77

28.25 24.99

90

I l.89

20.26

100 I 10

6.75 4.44

16.48 14.00

90 100 i10

17.34 13.70 I l.4I

36.l6 34.14 32.66

ln

+

05

0.69 2.29 9.42

22.66 53.51

52.62 52.06

r* + (a2)T

(r -

2.40 4.57 9.24

I

3.l6

8.71

12.54

12.25 16.98

16.41 2l.IO

'

=

cre

N(di)

e pre

8.0\ I1.67 16.70

19.09

23.80

26.28 3\.54

29.13

37.23

d2

,

=

di - o

K - F

"n(

d2)

'N(-di)

e

-

So s/i c,di

K +-

e

'regular

maturity. Hence, the chooser is neither a call nor a put. If the underlying asset price is low on the choice date, the holder will exchange his claim for a put; if it is high, he will trade it for a call. Hence, on the choice date, the value must be the maximum value between the call and the put where the values of these standard options correspond to their B-S values. It is important to note that the minimum value of the chooser must be at least equal to that of a standard call or put, whichever is greater, and its maximum value must be the sum of the call and the put or a straddle, 'choice'

(36)

Simple

14.5.1

(37)

(38)

Ve

=

(So

N)'

pS _

[C*(K,

T - t), P*(K,

T - t); t]

(44)

.

.

R: 1 plus the riskless interest rate, d: 1 plus the pay-out rate, S*: the unknown value of the underlying asset after the elapsed time t. parity theorem, the pay-off of the chooser can be rewritten as

Using the put-call =

'In(d2) =

(41)

r n(d2)

max

'Tn(d2)

S2a cre

=

where T - t is the time to maturity. Smee the buyer has the nght to choose between a call and a put before the chooser's matunty date, the value of the chooser must lie between the value of a standard option and a straddle. Following Rubinstein (1991a),we use the notation

c,

e

Options

c,

(39)

(40) -

Chooser

Chooser options allow the holder, immediately after a predetermined elapsed time, to choose whether the option is to be a call or a put. This is the principal idea on which a standard chooser is based. The pay-off of the standard chooser is

"n(d2)

K

(43)

So

'complex

These options are more expensive than standard options and their Greek-letter nsk measures are easily denved. They are given respectively by the following formulae: =

+ d2

'choice

9.88

.

e

(Sod)'

E

A chooser is a contingent claim that allows its holder at a certain date, known as the chooser' date', to trade this claim for either a call or a put. The claim is a when the call and the put have identical strike prices and times to maturity. The claim is a chooser' when the call and the put have different strike prices or times to

5.97

=

K - Se"

-

OPTIONS

CHOOSER

14.5

When the put option contract is initiated, the premium pr must be established in such a 0. This implies that that the current contract's value is nil, i.e. py(S,, T, K) =

-

§

3.67

5.9I I I.31

way

p,

=

n(d2)

PT

10.60 2.79 !.02

=

pre-'

307

OPTIONS

2 -+ Sa

Ö

(42)

[C*(K,

C*(K,

T), [C*(K,

T) + max

T)

[0, -S*d

S*d 6

+ KK

KK 0;

"];

t]

t]

(45)

It is convenient to note that this pay-off is similar to that of the following portfolio: •

d2

max

•

underlying asset S a long call with a strike price K, time to expiration Tand an with a strike price KR " ", time to expiration t and an underlying long put a

Sd

asset

308

FUTURES

OPTIONS,

Using this decomposition and arbitrage arguments, chooser is TN(x)-KR'N(x-o

Cs=Sd

AND EXOTIC

it follows that the

DERIVATI

of a standard

value

"N(-y)+KR'N(-y+aÑ)

)-Sd

ES

EXCHANGE,

START AND CHOOSEROPTIONS

FORWARD

with

(46)

ja2T,

y=x-aÑ

(47)

Table 14.6-shows a selection of simple option chooser values

Complex

Chooser

is

"

'choice

chooser implies the choice at a future date, known as the date', between a call and a put with a strike price Ki or K2 and a time to maturity Ti - t or T2 - t. In this spirit, the complex chooser can not be assimilated to a package of standard options and is identified with a compound option. Its pay-off is given by A complex

=

[C*(Ki,

max

Ti), P*(K2,

Using the put-call parity theorem, the formula presented complex chooser is

2);

t]

(48)

in Rubinstein

E

N2

¯X,

"

¯Ï29 Ñ2) + K2 r

l

Sed

ln

=

oT

by

'

=

+);2(T,

K,r

t

I0% 10% IO% 25% 25% <7 = 25% = a 50% a= 50% a=50%

- t)

for i

=

1, 2

(54)

where N2(a, b, p) is the bivariate normal distribution function. This formula is similar to the cornpound option formula. The first two terms of a call on a call and the other terms correspond to the correspond to the valuation valuation of a call on a put. The only difference lies in the determination of the critical underlying asset price X which is given by the equality between the value of a standard call, C*(Ki, Ti), and a standard put, P*(K2, T2), after the elapsed time t. This critical asset price level can be determmed by an iterative procedure.

,

-

y2 + a

,

p2)

(49)

proposed by Nelken (1993)for the pricing of compound options is quadrature. Usmg arbitrage arguments, the value of a European standard call option is given by its discounted expected pay-off, or The methodology

a numerical

based on

Values

for the following

which is written

=

(55)

"E[max(S, - K, 0)]

e

in an integral form as

=

c=e" =

'N(-z2)

Nelken's Approach

N2(-x + a

TABLE 14.6 Simple Chooser Option 4% S 10%, d 100, r parameters:

o a= a= = a = u

(52)

rl=L2

T -t)+(d¯

"N(zi -a

Kr

c

=

U2T

(1991a)for the

N2(x,yi,pi)-Kir-r'N2(x-ad,yi-oß,pi)

- Sd

"N(zi)-

zi and z2 are given

and where

Approach

ce=Sd

(51)

the solution to the following equation:

z,

cc

'a

Options

A complex chooser option is defined in the same way as the simple chooser except that the strike prices and/or the times to maturity for the call and the put are different. Rubinstein's

+

=

.

where Se

0=Sod 14.5.2

(50)

')

In

y,=-

/T2

p2 -

In(Sd '/Sr x

In

t/Ti,

=

pi

with

x=

309

K

T = 0. I

T = 0.25

T = 0.5

T = 0.75

T= I

90 i00 il0 90 100 I IO 90 100 I10

I4.824 7.2 6.073 18.046 13.293 13.2|2 26.836 24.84 25.671

i4.825 7.428 6.558 18.479 14.7\7 14.893 29.127 28.088 29.181

I4.845 7.944 7.236 19.468 16.487 16.851 32.223 31.819 33.147

I4.907 5.334 7.795 20.445 17.891 18.368 34.739 34.684 36.l75

i5.005 8.738 8.281 21.357 l9.088 19.65\ 36.898 37.089 38.710

max(S,-K)p(S,)dS,

(56) =

e

"

(S, - K)p(S,)dS,

where p(S,) refers to the probability that the underlying asset price will be S, at the maturity option's date. In a risk-neutral world, the underlying asset price at maturity is given by S, where µ

=

r - q -

jo

2

=

Soe"'

A change of variables

"

allows one to write

(57) the call price as

OPTIONS,

310

* ,, =

c

ln (K,/So)

e

FUTURES

- µt [Soe'"

" -

AND EXOTIC

DERIVATIVES

K]n(z) dz

•

(58)

.

In this context, the value of a European call is the same as that given m B-S (1973).Since at the choice date, the holder can decide for his option to be a call or a put, its pay-off is given by the maximum of both values, or .

cc

=

max [c(S,, Kc, Te - t),

p(S,, K,, T, - t)]

(59)

the indices c and p refer, as before, to the call and the put. The value of a complex chooser is given by its discounted expected pay-off:

where

"

cc

Since S,

=

=

e

max {c(S,, Ke, Te - t), c(S,, K,, T, - t)} p(S,)dS,

e

Soe"'*" "

cc

=

,

change of variables

max {c(See"'*"

'2,

(60)

gives the following integral:

Ke, Te - t), c(See"'"'

,

Ke, Te - t)}n(z)dz

(61)

Hence, the value of the complex chooser is based on the calculus of the above integral. This needs a quadrature algorithm. This technique does not require an initial guess as do method and the bisection method. the iterative Newton-Raphson

SUMMARY The option to exchange one risky asset for another is analyzed and valued. The identification of this option allows the pricing of several financial contracts. Examples include the performance incentive fee, the margin account, the exchange offer and the standby commitment. As we will see in other chapters, this concept is useful in the valuation of complex options such as options on the minimum or the maximum of several assets.

Options with an uncertain strike price are also analyzed and valued. Their prices are simulated and the effects of variations in parameter values are analyzed. The concept of an option with an uncertain strike price is useful for the pricing of bonds and bond options when inflation is accounted for. This chapter also presents a framework for the analysis and valuation of forward start options, pay later options, simple chooser and complex chooser options. Forward start, pay later options and simple chooser options are analyzed and valued. Then the approaches proposed earlier for the pricing of compound options are applied to the valuation of complex chooser options. Simulation results are also provided.

POINTS • • • •

FOR DISCUSSION

What is an exchange option? Give some examples of exchange options. Analyze the exchange option in the performance incentive fee. Analyze the exchange option in the margin account.

EXCHANGE,

• • •

FORWARD

START AND CHOOSER

What is an option with an uncertain strike price? What is a forward start option? What is a pay later option? What are the different forms of chooser options?

OPTIONS

311

Rainbow Options

CHAPTER

OUTLINE

This chapter is organized as follows: 1. In section 15.1, rainbow options are analyzed and valued using a continuous-time framework and a discrete time setting. In particular, analytic formulas are given for options on the minimum (themaximum) of two assets. Also, some of their properties are presented. An extension of the results in Stulz by Rubinstein (1991b)to the valuation of an option delivering the best of two assets and cash is proposed, and the generalization of the results to options on the minimum or the maximum of several assets is studied. 2. In section 15.2 simulation results are presented. 3. In section 15.3, some applications of rainbow options are presented. In particular, applications to currency bonds, multi-currency bonds, corporate option bonds, spread options, portfolio options, dual strike options, quality options with N deliverable options are analyzed and assets, the timing option in futures contracts, and wildcard valued.

NTRODUCTION Rainbow options, and in particular, options on the minimum or the maximum of two or more risky assets, have proved to be useful in the pricing of a wide variety of contingent claims, traded assets and financial instruments whose values depend on extreme values. For example, on the Eurobond market, the option bond gives the right to the bearer to choose among two or more currencies in which the payment is to be made. A discount option bond gives the right to the bearer to choose at maturity between two currencies at contracts, collateralized a predetermined exchange rate. Compensation plans, risk-sharing loans, and growth opportunities among other contracts, may present a pay-off function corresponding to that of an option on the minimum or the maximum of two or more risky assets, More generally, two-color rainbow options refer not only to options on the maximum (minimum)of two assets, but also to all options whose pay-off depends on two underlying assets: options delivering the best of two assets and cash, spread options, portfolio ootions, dual strike ontions. etc.

OPTIONS,

314

FUTURES

prices for options on the minimum o maximum assets. Using the Cox and Ross approach and a trick based on a device used by Margrabe (1978),Johnson wrote down the solution for the general case of an option on several assets. His result is useful in pricing, among other things, currency option bonds, portfolio insurance and the quality option in commodities contracts. Rubinstein (1991b,1994) also presented formulas for the valuation of rainbow options in a continuous and a discrete time approach. Some futures contracts provide the seller a certain fiexibility with regard to delivery. Examples of such contracts include the treasury bond futures contracts traded on the Chicago Board, on the MATIF, commodity futures contracts and agricultural futures contracts, among others. These delivery options concern the time, the place, the quantity and the asset to be delivered. They are known as the timing option, the location option, the quantity option and the quality option. For example, agricultural futures contracts include a quality option allowing the seller to deliver one of a number of grades of the underlying commodity. They also contain a timing option allowing the seller to decide when during the delivery month the delivery will take place. The wheat futures contract allows the short position to deliver any one of different types of wheat at any time during the delivery month. The treasury bond futures contract allows the short side to deliver any one of a predefined set of long-term government bonds. Another type of delivery option is the wildcard option. This option arises when the settlement price of a derivative security is fixed before the final exercise opportunity and when exercise closes the underlying asset position. For example, the T-Bond futures contract traded on the CBT has two wildcard options. In fact, since the settlement price of the contract is set at 2:00 pm and the short futures can wait until 8:00 pm to declare an intention to deliver, this gives rise to an end-of-date wildcard option. Also, since the T Bond futures contract is seven business days before the end of the contract's month, and the short position can wait until the end of the month to deliver, this gives rise to an endof-month wildcard option. The American options traded on the CAC40 index on the Pans Bourse also present end-of-day options since the settlement price of the option is set equal to the CAC40 index level at 5:00 pm while exercise is allowed until 5:45 pm. A similar example is given by options on the S&Pl00 index when the settlement price is equal to the S&l00 index level at 3:00 pm while the exercise can appear until 3:15. of these options includes Arak and Goodman (1986, The literature on the valuation 1987), Boyle (1989),Garbade and Silber (1983),Gay and Manaster (1986,1991), Kane and Marcus (1986),Livingston (1987), Valerio (1989)and Fleming and Whaley (1994), among others

Stulz

RAI9tBOW OPTIOblS

AND EXOTIC DERIVATIVES

315

(1982)and Johnson (1987)derived

=

cmin

of two risky

H)

[min( V,

max

Assume that the price dynamics of the two assets are given by d V/ V

µv dt + ov d Wv

=

15.1.1

Analytic

OF RAINBOW

OPTIONS

:

The Call on the Minimum

=

=

dP

=

Pv dV + Pa dH - P,dt + ([Pyv V2

If the portfolio comprises V, H, a riskless satisfy the following differential equation: -P,

=

(rP

Using equation

-

rPy V - rPaH)

(5) and

+ PoH2o

Following Stulz (1982),let Vand H stand respectively for the prices of two risky assets. At maturity, the pay-off of a European call on the minimum of these two assets is

+ 2Pys VHpvaasos]dt

asset and is self-fmancing,

([Pvy V2ag + PanH2o

-

then its

(4)

value

must

+ 2Pys VHpysavas]

(5)

the following boundary conditions: P( V, H, 0)

=

max

[min( V, H)

P(0, H, r)

K, 0]

-

A

(6)

0

=

P( V, 0, r)

(7) 2

0

=

(8)

Stulz gives the following formula for the pricing of a call on the minimum of two assets: cos(V,

H, K, r)

=

HN

y, + oun,

- Ke "N(yi,

VN

,

of

a

y2 + avs/r

d2 '

d4 '

a

a

}

y2, pon)

(9)

with

di

=

da

=

ln(V/H)

-jo2

ln (H/ V) -

d3

yr

=

a

2

=

(10)

jo2

QQ (12)

pysov - as RVH

H ¯

K)

(13)

V

on)r

(r

=

72 _of Two Assets

(3)

where µv, µH9 V and as are constants. The price of a European call on the minimum of Vand H, with a maturity date Tand a strike price K, denoted by cas,(V, H, K, T - t), is equal to the value of a self-financing portfolio which has the same value as the option at date T Denoting the value of that P(V, H, r), with r T - t, and using Itô's lemma gives the dynamics portfolio by P for P:

ln(V

Formulas

(2)

dH/H=µadt+oadWo

In(H/K)

VALUATION

(1)

.

d4

15.1

- K, 0]

(14)

(r -

jag)r

(15)

a( + op, - 2pynavan

and where N(a, ß, p) is the bivariate cumulative upper limits of integration and p is the correlation

normal

(16) distribution

coefficient.

where

a,

ß are

the

OPTIONS,

1816

FUTURES

AND EXOTIC

RAINBOW

DERIVATIVES

We now write in a compact form the price of a European call on the minimum of two assets as cass(V, H, K, r)

HN(ai,

=

a2, Íc) + VN

If the strike price is zero, formula

(9) reduces

(ßi, ß2,pe)

- Ke "N(yi,

y2, pys)

(17)

317

OPTIONS

H. Since these portfolios are worth the same at then portfolios A and B are worth K maturity, they must have the same initial value. ps,ax(V, H, K, r), and its value is The same proof applies for a put on the maximum, given by Stulz as -

to

pmax(V, H, K, r)

couss(V,H,0,r)=V-cs(V,H,l,r)=V-VN(d

)+HN(d22

with

=

e "K - cmax( V, H, 0,

T)

+ Ceaux(V, H, K, r)

(23)

Some Properties In(V/H)+(jo2g

dii

=

,

d22

=

du -

(19)

CE(V, H, 1, r) stands for the price of an option to exchange one unit of asset Hfor unit of asset V This formula is also given in Margrabe (1978) one where

The Call on the Maximum

of Two Assets

As for ordinary options, it is possible to obtain some parity relationships between options on the minimum, the maximum, the underlying asset and the interest rate. Let coax(V, H, K, r) be the price of a European call on the maximum of two assets V and H. Its pay-off at maturity is given by c

Stulz showed that the

value

ex

=

max

[max( V,

H) - K, 0]

(20)

of this option is

C(V, K, r) - cm,s(V, H, K, r) + C(H, K, r) (21) with strike price K C(V, K, r) stands for the price of a European call on asset V a and a maturity date r and the option This result can be easily verified. In fact, if V a K, V is the maximum, on the maximum pays at maturity V - K. In this context, a portfolio comprising a call on H, a call on V and a sale of a call on the minimum of H and V also pays V - K as C(H, K, 0). The same analysis applies when H is the maximum of V cas,(V, H, K, 0) and H. cmax(V, H, K, r)

Ef¶ecto¶o Change in the Correlation Coefficient When the correlation coefficient is equal to 1, the option value is highest since an increase in this coefficient increases the probability that the pay-off of V will be close to that of H. Hence, V is more likely to be high when H is high and the option's expected pay-off is high as well. It is convenient to note that the price of such an option may be nil even when the underlying assets have positive values. This may be the case when the correlation coefficient is equal to -1. Ef¶ect o¶o Change in Time to Maturity The prices of such options may be an increasing or a decreasing function of time to maturity. This result is different from that shown by Merton (1973a)for options on a single asset. Ef¶ect of a Change in H, V, r and the Strike Price The price of an option on the minimum is an increasing function of H, V and r. However, it is a decreasing function of the strike price.

=

where

Ef¶ect of a Change in Volatility It is convenient to note that the sign of the partial derivative of the option with respect to the variance is ambiguous. This results from the fact that an increase in the instantaneous variance of a risky asset can either increase or decrease the expected option's pay-off. For example the partial derivative with respect to V may be negative

when

Options

Delivering the Best of Two Assets and Cash

as > pysay.

=

Following Rubinstein The Put on the Minimum

(Ma×imum)

of Two Assets

Let pass(V, H, K, r) be the price of a European put on the minimum of two assets Vand H. Its price must satisfy the following relationship:

(22) e "K - csos(V, H, 0, r) + coin(V, H, K, r) This result can easily be proved by the construction of two portfolios A and B. Portfoho A comprises a put on the minimum of Vand H. Portfolio B comprises an option on the minimum of Vand H, a discount bond paying K at maturity and a sale of an option on the with a zero strike price. minimum Consider the pay-offs at maturity. If the minimum of Vand H is greater than or equal p in(V, H, K, r)

=

.

and portfolio B pays K-min(V,H) A is worthless K, then portfolio of Vand H is Vand is less than K, then If the minimum min(V, H)-K=0. portfolios A and B are worth K - V. If the minimum of Vand H is H and is less than K,

to

9,: S,,: R: d,: K: t: o p:

:

(1991b),we

will use the

following

notation

(i

=

1, 2):

price for asset i, the initial underlying the terminal price for asset i, 1 plus the riskless rate, 1 plus the pay-out rate for asset i, a fixed amount of cash potentially received at expiration, the time to maturity, the volatility of the underlying asset i, coefficient of the naturallogaHthnis of 1 plus the rat s of return the correlation of the two underlying assets.

Thepay-offofthisoptionisgivenby cb2ac =

m3X

[Sp, S,,, K]

(24)

318

FUTURES

OPTIONS,

AND EXOTIC

RAINBOW

DERIVATIVES

The value of this option is given by its expected pay-off discounted at the riskless interest rate under the appropriate probability: Ch2ac

R 'E[Max (S

=

,

S,,, K)]

OPTIONS

319

li

(25)

E

is

expectation

the mathematical Cb2ac

=

operator. This may be also written as

S2e', K]f(x,

max[Sie",

R

y)dxdy

12

(26)

f(x,

In(Sr,/Si),

=

f(x,

y)

(27)

=

2xto a2/l

p2

-

(x-µit)2

1 um

2p

1-p2

1

=

µi

I2

R

=

Sa •

/3

'

=

KR

ln (R/di)

-

jo2,

µ2

=

ln (R/d

)

> =

'

(29)

a

e"f(y)dy

p2B

f(y|x) dy N(-xi

and K, it is convenient

x

+U2

-x2

,

=

In(Sid

KR

')+(a

e"f(x) dx

,

p)

ln(S2d

KR

')

(37)

y

times the expected

fmal

value

of Sr, under the conditions

times the expected

final

value

of

=

to

Sr,

> K and

Sr,

> K and

ln(S d '/S2d

')

ln(S2d '/Sid

')

=

S R

'

R

'

ST,

under the conditions

Ï

Sg>S •

f(x y)dx

(28)

2

Since the option value depends on the positions of Sr,, S, break the pay-off into three components: Ii

yi, pi)]

S2d [N(y2) - N(-x2, y2, KR

e' f(x) dx

(34)

'

=

dy

with

with

•

f( y|x)

_

13

y) is the bivariate density function given by

-x

"

S2R

In(Sr /S2)

y=

ln(K/Si)

'""2/

=

with x

*

Sid '[N(yi) - N(-xi,

=

where

*

'

Si R

=

that K > Sr, and K >

times the probability

(x) (y)

1

'72"!

e

=

e

1/2"!

and the followmg conditional

x-µit

with

v

=

with

v

=

(3l )

(x y)

(y|x)

~1/2

2x(1

p2)a

-

e

e-w2/2

=

2x(1

-

¯ wa

With

p2)o2t

Using these densities li, I2 and 13 can be written

«2

p(y _ y ¿p2

1 - p2)t

w2

=

[(y -

#21)

a,

P(I

"1

agl as

+

jÏ

(39)

+

jÏv

(40)

-2paia2

(41) pa, - a

The value of the option delivering the best of two assets and cash Cb2ac

i

Options

on the Minimum

2

3

(42) is

(43)

of Several Assets

(Ma×imum)

Johnson (1987)extended the results of Stulz to the pricing of options on the maximum or the minimum of several risky assets. His model is based on some results that appeared in the B-S model, the Cox and Ross approach, Margrabe's model and Merton's model. Remember that in the B-S model, the stock price and the strike price are multiplied by N(di) and N(d2). In the Cox and Ross approach, N(d2) is considered as the probability that the stock price at expiration, ST, will be greater than the strike price K, given that the stock price is S today. If a trick based on a device introduced in Margrabe is used, then the N(di) term can be written down immediately without laborious calculation. In fact, a call option can be regarded as an option to exchange cash (thestrike price) for the common stock. Hence, it .

a_I

1

(a

(30)

densities:

=

a

po2 - a

ST,.

This decomposition allows us to write the option value as the sum of the three compo¯ nents li + 12 13 with the following densities: =

=o

+

- p2)t

-

#i t)]

(32)

,

(33)

.

.

OPTIONS,

320

FUTURES

AND EXOTIC

RAINBOW

DERIV

121

OPTIONS

The call price can be valued as in Margrabe by taking the stock price as a numeracy. measured in units of the stock price can be regarded as a European put on a risky asset with unit strike price and zero interest rate. with current price x (K/S)e price' is just the risk free In this context, the stock price measured in units of the asset. Using Itõ's lemma for dx and replacing the drift term with zero in the Cox and Ross approach, Johnson showed that N(d,) N(-dí) where '

=

,

'stock

=

d'= This procedure

Si, S2, S3, For n

=

is applied

to value

calls on the maximum

(minimum) of

n assets '

So, with a strike price K and a maturity date T 2, the formula presented by Johnson reduces to that of Stulz.

.

.

(d, D)

(d, C ) FIGURE 15.1

The First Move

.,

From the return (u, B), the second return which gives a total return of 15.1.2

The Discrete

Approach

(u, B){(u,

In this section, we present briefly the discrete time approach proposed by Rubinstein (1994)for the valuation of European and American rainbow options. These options can be valued by constructing a square binomial pyramid. We show first how to construct the binomial pyramid and then apply the approach to the valuation of an American call option on a spread. The following additional notations are used: R:

bi: ö2: K:

the annualized the annualized the annualized a fixed arnount

discrete interest return, discrete pay-out return for asset 1, discrete pay-out return for asset 2, of cash potentially received at expiration.

A),

(u, B), (d,

C),

(d, D)}

=

can again be

A),

(u, B), (d, C), (d, D)}

=

A),

(u, B), (d, C), (d,

D)}

=

(u, A), (u, B), (d, C)

(ud, AC), (ud, BC), (d2, C2), (d',

From the return (d, D), the second return can agam be which gives a total return of

(d, D){(u,

2

The First Move As in standard binomial trees, we assume the returns to the first asset to be either u or d with the same probability. When the first asset moves to u, the second asset has returns A or B with equal probability. When the first assset moves to d, the second asset has returns C or D with equal probability. Hence, starting at (1, 1), the four pairs (u, A), (u, B), attained with an equal probability of I /4. The first move can be represented as shown in Figure 15.1.

(d,

The Second Move From the first return (u, A), the second (u, A), (u, B), (d, C) or (d, D) which gives a total return of

return

C),

(d,

D)}

=

C) and

(d, D)

(u2, A2), (u2, AB), (ud, AC), (ud,

again

B

,

)

(u, B )

(ud, BD)

are

(44)

(d, D)

(ud, BC)

|

be

(d AD)

(u

AB )

(ud, AC)

(d, C) can

,

(ud AD )

(u, A )

assets.

2

2

2

(u

)

RGURE

15.2

,

(d

C )

The Second

Move

(d, D)

or

(d,

D)

(46) (d, D)

(ud, AD), (ud, BD), (d2, CD), (d2,D2) (47)

2

A

,

For the sake of simplicity, we standardize the returns of the two underlying assets to the pair (1, 1) and consider what happens in the first and second moves of the two underlying

(u, B), (d,

CD)

Note that a probability of l 16 is attributed to each of these 16 pairs. These different pairs can be represented as shown in Figure 15.2.

(u

A),

or

(u, A), (u, B), (d, C)

The Binomial Pyramid

(u, A){(u,

or

(u2,AB), (u2, B2), (ud, BC), (ud, BD) (45)

From the return (d, C), the second return can again be which gives a total return of

(d, C){(u,

(u, A), (u, B), (d, C)

,

CD

)

(d

2 ,

D

2

)

322

OPTIONS,

FUTURES

AND EXOTIC

DERIVATIVES

Constructionof the Pyramid

Let a horizontal slice of the square pyramid represent total returns after each move. When we set (1, 1) at the apex and the last move at the bottom of the square pyramid, then several paths through the pyramid lead to the same node. Since by assumption, AD BC, four paths reach the central node. When there are n moves, the total number of distinct nodes at the bottom is (1 + n)2. Now, to construct the appropriate move sizes in a square binomial pyramid, the following values of (u, d), A, B, C and D are used:

RAINBOW

OPTIONS

value

Finally, the option Csee

=

323

at the initial time is given by

max [(S2 - Si) - K,

C(u, B) + C(d, C) + C(d, D)]}/Rh

A)

({C(u,

(55)

=

"I

e" A

=

D

',

eu"""2

=

d

,

=

B

a

=

e""

"

C

=

""

e'""""

15.2

SIMULATIONS

(48)

15.2.1

Calls on the Minimum

(49)

Tables 15.1 and 15.3 show the effect of a change in the correlation coefficient for 10%, r 1700, k 1700, V 1700, r 0.25, as 0.3 and av 0.3 for Call H 122.75 and for Call H 122.75 and 35.62 respectively. For example, using Table V 29.64. Table 15.2 shows the effect of a change in the volatility 15.1, when p 0.7, c.;, 0.15. for the same parameter values except av =

where

=

=

=

=

µi

of,

In(rföi)

=

ö2) -

µ2 - M

h

=

(¤

t/n

(50) (51)

Using the above expressions, it is possible to proceed as in the standard bmomial model by starting at the end and discounting each four nodes into one at each move with the same probability

at each node

=

=

=

=

Coefficient: TABLE 15.1 Effect of a Change in the Correlation 10%, r 0.3, Call on H 1700, K 1700, r 0.25, oH 0.3, av Y= 122.75 =

=

(1/4).

=

=

=

=

=

=

H 1700, Y 122.75, Call on =

=

=

.0

p

of the Model

Application

Call H

of an American call option on a spread. At the expiration date, the pay-off of the option on a spread is

Consider the

valuation

Cs, When n

=

=

max

2, the move tree at expiration

C(u2, A2) C(u2, AB)

=

=

C(u2, B2)

=

C(ud, A C)

=

C(ud, BC)

=

C(ud, AD)

=

C(ud,

=

BD)

C(d2, C2)

=

C(d2, D2)

=

Sr

[0, (Sr,2 -

K]

)-

callV

0.4

0,5

0.6

0.7

0.8

0.9

l 22.75 I22.75

I22.75 I22.75 3.3 I

)22.75 I22.75 I5.64

| 22.75 I22.75 29.64

122.75 I22.75 46.64

l 22.75 I22.75 68.82

O

cas,

I

(52)

shows the following call option values:

2) - K, 0] max [(S2A2 _ S u2) max [(S2AB - S - K, 0] 2) [(S2B2 - K, 0] max - S ud) [(S2A S C max - K, 0]

-

max [(S2BC - S ud) - K, 0] max [(S2AD - S ud) - K, 0] max [(S2BD - Siud) - K, 0] max [(S2

2

Triax[(S2D2

-

-

Sid2) Sid2)

-

K, 0]

- K, 0]

When we go back one move, we get the following values:

TABLE I 5.2 Effect of a Change in the Volatility: 10%, r 0.25, «H 1700, r 1700, V 1700, K I22.75, Call on V 73.97 0.15, Call on H av ro

p Call H Call V com

H 0.3, =

=

=

=

=

.

=

=

=

=

0.6

0.7

0.8

0.9

122.75 73.97 0

122.75 73.97 5.67

122.75 73.97 |4.02

122.75 73.97 22.6 I

in the Correlation TABLE 15.3 Effect of a Change 1700, V 1700, K 1700, r 10%, Coefficient: H 35.62, Call on 0.3, av 0.3, Call on H 0.25, as r V 122.75 =

=

=

=

=

=

=

=

=

C(u, A)

=

C(u, B)

=

C(d, C)

=

C(d, D)

=

AD)]} R max [(S2A - Siu) - K,¼{ C(u2, A2) + C(u2, AB) + C(ud, AC) + C(ud, R^ max [(S2B - Siu) - K,¼{C(u2, ßA) + C(u2, ß2) + C(ud, BC) + C(ud, BD)]} max [(SaC - Sid) - K, max [(S2D -

({C(du,

CA)

C(du, Cß) + C(d2, C2) + C(d2, CD)]} R

Sid) - K,¼{ C(du, DA) + C(du, DB) + C(d2, DC) + C(d2, U

p Call H Call V c

0.5

0.6

0.7

0.8

0.9

0.99

35.62 122.75 0

35.62 122.75 5.06

35.62 122.75 13.f4

35.62 122.75 22.32

35.62 122.75 3I.91

35.62 122.75 35.6i

I22.75 I22.75 122.70

324

OPTIONS,

I 5.2.2

FUTURES

AND EXOTIC

DERIVATIVES

RAINBOW

Calls on the Maximum

TABLE I 5.7 Effect of a Change in the Correlation 0.3 0.25, ÜH 0.3, av I0%, r I700, r |700, K =

Tables 15.4-15.6 show similar data to Tables 15.1-15.3 for calls on the maximum. For example, using Table 15.4, when p 198.86. 0.8, ca,,x c c

TABLE 15.4 Effect of a Change in the Correlation Coefficient: I 0%, r 0.3, Call on H 0.25, «H 0.3, av I 700, K 1700, r V I 22.75 =

=

=

=

H 1700, V I 22.75, Call on =

=

=

=

p

=

=

=

325

OPTIONS

0.4

0.5

0.6

0

3.3 I I598.36 62.97

I 5.64 I609.07 64.58

I588.68 69.341

po

Coefficient:

H

=

I 700, Y

=

=

=

0.7 29.64 I62|.24 66.42

0.8

0.9

46.64 I635.68 68.983

68.82 i654.54 72.336

I.0 I22.70 1699.99 8.73

--

=

TABLE I 5.8 Effect of a Change in the Volatility: 10%, r 0.25, 1700, K 1700, V 1700, r H 0.I5 0.3, av as

-----

p Call H Call V casa,

0.4

0.5

0.6

0.7

0.8

0.9

I22.75 I22.75 245.50

I22.75 I22.75 242.I8

I22.75 I22.75 229.86

I22.75 I22.75 2I5.75

I22.75 |22.75 198.86

I22.75 I22.75 I76.68

.0

I

=

=

I22.75 122.75 I22.75

Poun

=

=

=

=

=

=

p Call H Call V c

0.7

0.8

0.9

122.75 73.97 196.73

122.75 73.97 191.05

122.75 73.97 182.7

122.75 73.97 174.12

15.3 15.3.1

=

=

=

=

r*:

S:

=

Call H Calli V cas,

0.5

35.62 122.75 158.38

0.6

0.7

0.8

0.9

0.99

APPLICATIONS Pricing

Currency

the the the the

Bonds

firm's value in domestic currency, bond's face value in foreign currency, riskless rate of interest in foreign currency, spot exchange rate.

35.62 |22.75 l53.3l

35.62 122.75 145.24

35.62 |22.75 |36.05

35.62 122.75 126.46

35.62 |22.75 122.75

B*(V

Tables 15.7 and 15.8 show similar data to Tables 15.4 and 15.5 for puts on the minimum K For example, using Table 15.7, when p 0.8, the discounting factor e 1658.026, cas, 46.64, c 1635.68 and p.,, 68.983. =

=

22.6 I 1639.83 40.8

=

=

S, K*, r)

(l/S)cssas(V,

=

H, 0, r)

(56)

In fact, Stulz showed that casos(V, H, 0, r) corresponds to a standard call with K 1, r 0, and underlying asset price V H. Since c,sise(V, H, 0, r) is a decreasing function of o2, the bond's value is also a decreasing function of the volatility of the firm's value and the exchange rate value. Using the interest rate parity theorem

Puts on the Minimum

=

I4.02 \631.78 40.26

At maturity, the bond's pay-off is given by min ( V, SK*) where SK* is uncertain. K* be the price of a zero-coupon Let H(t) bond at t paying SK* at T Let S(t)e B*(V/S, K*, r) be the value of the foreign currency bond (in foreign currency). It is possible to show that this bond's value is also equal to the product of a standard European call option and the inverse of the exchange rate, i.e.

=

15.2.3

5.67 1624.59 39.103

0.9

=

=

>

0 1618.032 39.99

0.8

A currency bond is a bond denominated in a different currency from that prevailing in the country in which the common stock of the firm is traded. To price this simple form of currency bonds, we use the following notation:

TABLE I 5.6 Effect of a Change in the Correlation Coefficient: H I 0%, I 700, V I 700, K I 700, r 0.25, as 0.3, Call on H 35.62, Call on 0.3, av Y V I 22.75 =

0.7

H 0.3,

V: K*: =

=

=

=

=

0.6

=

0.6

p cm c

TABLE I 5.5 Effect of a Change in the Volatility: 1700, K 10%, r 0.25, «H 1700, Y 1700, r 0.15, Call on H 122.75, Call on V 73.97 av

=

=

=

=

=

(t,

T)

=

S(t)e"

'

"

(57)

OPTIONS,

326

FUTURES

AND EXOTIC

DERIVATIVES

f(t, T) stands for the forward exchange rate at t for a delivery at T, it is possible to show that H depends on f(t, T). Using this relationship, H can also be written as

where

RAINBOW

OPTIONS

327

When V > SK* > K, the option on the minimum pays SK* - K, and when V > K > SK* it pays nothing. If SK* > V > K, the bond value is V and the portfolio K} value is K + {min(V, H) V. =

H

=

f(t,

T)e "K*

(58)

It follows that the higher the forward exchange rate and the higher the expected future spot exchange rate, the higher the bond's price. This result is immediate when changes in exchange rate are certain. In fact, in this context, the bond is assimilated to a bond in a domestic currency with a fixed strike price f(t, T)K*. Hence, an increase in the forward rate increases the bond's face value in domestic currency.

I 5.3.2

Bonds

Multi-Currency

g*) be the Let S4(Ss) be the current price of one unit of currency of counti) A (B) and rA riskLless inSt est rSateKinc' r) st nd fr the discount bond's price, giving the option to the KA* units of currency of maturity, either K units of domestic currency, or at get bearer, to * country A, or Ka units of currency of country B. A portfolio comprising a domestic * * discount bond, two options on SAK, and SaKa, with the same strike price K, and an * * and exactly the same pay-off as this option SaKa, gives the of minimum S,K, option on bond. If we define the values of two assets H and Vas ·

H

=

V then the bond's

value

exp

4

exp(-rat)SaKb

(59) 4

(60)

is

B(SAKi, where

=

(-r,*r)S,Ki

SaK,*, K, r)

=

cniax(V, H, K, r)+

Ke

"

(61)

cmax(V, H, K, r) stands for the price of a European option on the maximum of V date r.

and H, with a strike price K and a maturity

15.3.4

Spread

Options

A standard spread option entitles its holder the right to call or put the spread value against and corresponds in general to the difference between two a predetermined strike price prices. Examples of spread option include spreads in futures markets, bond markets and energy markets. Spread options in futures markets refer, for example, to the spread or the basis between Nebraska's corn and Chicago's corn. This spread may be due to location or grade, among other things. Spread options in bond markets may be due to the differences issued in different countries. Spread options in energy in yield spreads between bonds markets result from the differences between, for example, refmed and unrefined products, such as crack spreads. Spread options can be defined by using two underlying assets, contracts or commodities which are closely related. This correlation between the assets or commodities results from demand substitution or the potential for transformation. Spread options have the following pay-off: .

.

.

cspa

=

max

{0,¢(Se

-

S

)-

¢K}

(63)

-1

the binary variable ¢ takes the value l for a call and for a put. Even though the models proposed above allow the valuation of spread options, there are pros and cons to these approaches, since, as shown by Garman (1992),the valuation of these options is rather complex. Besides, they present the paradox of negative where

vegas.

15.3.5

Portfolio

Options

Portfolio options have the followmg pay-off 15.3.3

Corporate

Option

Bonds

=

c,

Let B(V, SK*, K, t) stand for the price of an option bond issued by a corporation. The issuer gives the bearer the option at maturity to choose between payments of K* units of the foreign currency and K units of the domestic currency. The bond's pay-off is equivalent to that of a portfolio comprising a discount bond with K*, with Se face value K, an option on the minimum of Vand H a strike price K and a sale of a standard European put on Kwith the same strike price. In this context, the bond's price is given by

where

15.3.6

max

{0,¢(niS,

+

n2ST,)

¢K}

-

(64)

to the number of units of the two assets

ni and n2 correspond

in the

portfolio.

Dual-Strike Options

=

Dual-strike options have the following pay-off: cm

B(V, K, SK*, r) At maturity, the currency

=

Ke

" -

option is worthless

P(V, K, r) + cos(V, when

H, K, r)

the firm's value is smaller than K-

(62)

=

max

{0,¢(Sc

- Ki),

¢(ST,

-

K2)}

where the binary variables ¢i and ¢2 take the values 1 or -1. For the valuation of these options, see the numerical techniques

(65) in Boyle et aL (1989).

FUTURi!BAlglþEKOTICOERIVATIVES

OPTIONS,

328

I 5.3.7

Options

Delivery

OPTIONS

329

computations involving integrations over n - 1 dimensional multivariate normal integrals. His simulations show that the impact of the quality option increases as the number of deliverable assets increases. For example, for 10 deliverable assets, the impact is nearly twice as large as in the case when there are just four assets.

Options

and Wildcard Assets

with N Deliverable

QualityOptions

RAINBOW

Consider a futures contract allowing the seller to choose at maturity between n assets for delivery. The seller will choose among these underlying assets and deliver the cheapest. When there is just one deliverable asset, a long forward contract to purchase that asset is equivalent to a portfolio containing a long European call and a short European put on one unit of the asset for which the strike price is the equilibrium forward price. When there are two deliverable assets, the forward contract value is equivalent to a long call on the minimum of the two assets and a short put on the minimum of these assets. These relations can be generalized for n assets. Following Boyle (1988),we use the following notations: '

-

in Futures Contracts

The Timing Option

Timing options are embedded in many futures contracts, offering the seller some flexibility concerning the timing of the actual delivery. In fact, delivery may take place on any business day within the delivery month. Following Boyle (1989),let the short position deliver the underlying asset at any time during the interval (t + Ti, t + T3) where t + T3 is the maturity date of the futures contract. Suppose delivery occurs at time t + T2 and Ti < T2 3Let F[t, (Ai), t+ Ti)] Fi be the futures price at time t of a futures contract on an asset with a current price Ai and having a maturity date of t + Ti. In a similar way, let F[t, (A2), t + T2] F2 be the futures price at time t of a futures contract on an asset with with price A2 current a a maturity date t + T2. Using arbitrage arguments, it can be shown that F2 must be equal to Fi. In this case, the timing option is nil and the short position should optimally deliver the asset at the first permitted opportunity. The value of Fi is obtained from the cost of carry model, so that =

t: t + T: A;(t): f[t, (Ai, A2,

the current time, the delivery date, the current asset price for i 1 to n, the forward price at t giving to the short side the t + B: choice among the n assets, A,), K, t TJ:a European call price on the minimum of n deliverable assets As), K, t + T]: a European put price on the minimum of n assets. =

n), ·

·

->

C

[t, (Ai,

A2,

P

[t, (A i, A2,

...,

=

'

...,

Since the interest rate r is constant, the futures price is equal to the corresponding forward price. In the case of two or more assets, the value of the forward contract can be expressed in terms of a long call and a short put as C

is,,[t,

(Ai, A2,

A,,), K, t+

•••,

T]-

Pass,,[t, (Ai, A2,

As), K, t+

...,

T]

(66)

At inception, the value of the forward contract is zero and the strike price K is equal to As), t+ T]. the forward price: f[t, (Ai, A2, By analogy with the simple case of two assets and using the extended put-call parity theorem m Stulz, expression (66)can be written as ...,

.

Coss,,[t, (Ai, A2,

A,), 0, t+

...,

T] - Ke

'

(67)

A,), 0, t + T) stands for the European call on the minimum where C (t, (Ai, A2, of n assets having a zero strike price. Since the forward's contract value at inception is zero, it is possible to write ...,

[t, (Ai,

A2,

...,

A,), t + T]

' =

e

C

,,[t,

(Ai, A2,

...,

An), 0, t+

T]

(68)

Hence, the forward price is given by the discounted value of the European call option on the minimum of the n assets with a time to maturity T + t and a zero strike price. For 2, it is possible to use the Stulz formula. For n a 2, the Johnson formula can be n =

used. To compute the value of the quality option, Boyle (1989) assumed that all assets have the and are equi-correlated, the same current price and volatility so he simplified

Fi

=

Ai(t)e'

Since the seller is committed to deliver during the delivery window, and there are no gains from waiting, F2 iS also equal to Fi. This result holds when there is no distribution to the underlying asset. However, if there is more than one deliverable asset, it is no longer true that the futures price when there is a timing option is equal to the futures price assuming earliest delivery. regarding In fact, when there is an uncertainty the asset to deliver, it could happen that the first asset is the cheapest to deliver under the futures contract without the timing option, whereas the second asset is the cheaper at time t + A2. Hence, there may be an interaction between the timing option and the quality option in the presence of at least two deliverable assets. Boyle (1989) developed a bmomial lattice framework to value the timing option where at each node, delivery is assumed to occur if it minimizes the contract value. He concluded that the impact of the timing option on the futures price is rather small. However, it is convenient to note that his model does not apply to Treasury bond futures contracts since it ignores coupon payments and it may be more appropriate to commodity .

.

.

.

.

futures. Analysis and Valuation

of Wildcard

Options

Anolysis of Wildcard Options Fleming and Whaley (1994)analyzed and priced wildcard options embedded in the S&Pl00 index option contract, OEX. The OEX wildcard options arise because the settlement price is given with reference to the S&Pl00 index level at 3:00 pm Central Standard Time (CST), at the close of the New York Stock Exchange, while the option holder may postpone the exercise decision until 3:20 pm. If the market rises (falls) during this wildcard period of 20 minutes, the holder of a

OPTIONS,

330

FUTURES

AND EXOTIC

DERIVATIVES

slightly in-the-money put option (or call) may find it optimal to exercise his option. Since the wildcard option is linked to the exercise procedure, it appears each day between 3:00 and 3:20 pm until the option's maturity date. Hence, there is a wildcard option sequence which must be taken into account when pricing S&Pl00 index option contracts. that builds upon the standard Fleming and Whaley (1994)provided a methodology CRR (1979)binomial model for valuing American-style options to price the embedded early exercise premium, which is options. They applied their model to the wildcard value with embedded wildcard American option the difference between defined as an options and an American option without this privilege. Their results show that the wildcard premium is an important component of an OEX option value. It accounts, for example, for about 12 cents of the value of slightly in-the-money call and put options. Also, wildcard premiums are approximately the same for calls and puts with the same and time to expiration. They show that the wildcard value increases with time moneyness and that it accounts for more than 2% of an at-the-money to expiration and moneyness option value. Their analysis suggests that the wildcard privilege is even more valuable for derivative securities for which the wildcard period is hours or sometimes days. proposed by Fleming and Whaley (1994) Valuation o¶Wildcard Options The methodology for the valuation of American S&Pl00 index options uses simultaneously the lattice approach proposed by CRR (1979)and a modified version of the B-S model. First, the number of time steps m in the lattice is selected in such a way that the end of each time step coincides with the end of each wildcard period. Hence, when valuing an OEX call which has an end-of-day wildcard option, the number m is set equal to the number of days to the option expiration. option is Second, at each node j in each time step m, the value of the wildcard calculated and added to the present value of the expected future's option value. When proceeding backward from the end of the tree to a node just before, an American call value is calculated using the following formula:

C

"'a

=

i

max [Su'd'"

'(pC,,,

K, e

-

(1 - p)C,,,

)]

=

max

RAINBOW

'"

-

K, A)]

A

At the node defined by

(m,j), W

'[pC,,,

e

(1 -

the value of the wildcard =

Se

"'N(di)

p)C,,, µ]

.

POINTS • • • • •

•

(70)

•

•

(71)

• •

with

•

di

=

In (S /(K + A)) -

t is the wildcard period. Now, the value of an American option

(r -

(Ü2y ,

•

d2

=

d,

-

pf

(72)

where

with embedded

wildcard

qasons ss calculated at

each node using the formula

C,,

=

max [S,, - K, A + W),,]

(73)

+

(1 - p)C,4, ]

to the pricing

of American options

(74) with

FOR DISCUSSION

What is a rainbow option? What is the pay-off for a call on the minimum of two assets? What is the pay-off for a put on the minimum of two assets? What is the pay-off for a call on the maximum of two assets? What is the pay-off for a put on the maximum of two assets? What is the pay-off for an option delivering the best of two assets and cash? options? Are there put-call parity relationships for extremum (maximum/minimum) of options on the extremum of two assets be extended to N How can the valuation assets? How can we identify extremum options in currency bonds? How can we identify extremum options in multi-currency bonds? How can we identify extremum options in corporate option bonds? What is the pay-off of a spread option? What is the pay-offof a portfolio option? What is the pay-off of a dual-strike option? What is the pay-off of a delivery option? What is the pay-off of a wildcard option? .

•

•

- (K + A)N(d2)

i

In this chapter, several forms of rainbow options and options on the minimum or the maximum of many assets are analyzed and valued in a continuous and a discrete time setting. First, using the approach in Stulz, options on the minimum or the maximum of two assets are studied and valued. Then options delivering the best of two assets and cash of several assets or the minimum are analyzed and valued. Also, options on the maximum are analyzed. Some simulations of option values are run. Finally, some applications to currency bonds, multi-currency bonds, corporate option bonds, spread options, portfolio options and dual-strike options, quality options with N options are deliverable assets, the timing option in futures contracts and wildcard provided.

(69)

option is

[pCoi

e

SUMMARY

• =

=

This simple methodology can be applied embedded wildcard options.

where

A

331

where

•

[SI

OPTIONS

.

I6 Extendible

CHAPTER

Options

OUTLINE

This chapter is organized

as follows:

1. In section 16.1, extendible options are identified, analyzed and valued. 2. In section 16.2, simple writer extendible options are analyzed and priced. 3. Finally, in section 16.3, extendible options are applied in the analysis of extendible bonds and extendible warrants. Simulations of option values are provided.

INTRODUCTION There are many

types of financial contracts

or contingent

claims allowing the issuer or

the contract's holder to extend the maturity date of the initial contract. For example, corporate warrants give the issuer the right to extend the maturity date of the contract unilaterally. Also, bonds with embedded options give the right to the issuer to extend the initial maturity date. More generally, in any financial contract with provisions of terms can imply the existence of payments, a renegotiation concerning a rescheduling of options with extendible maturities. These options can be extended by either the option holder or the option writer. When this option allows the holder to extend the initial maturity date T, to another date T2, he must pay an additional premium A to the option writer. Note that when pricing these options, the strike price is often adjusted from Ki to K2.

16.1

PRICING

16.I.I

The Valuation

Context

The valuation of these options is realized in the B-S context. In fact, following Longstaff (1990),the dynamics of the underlying asset are described by dS/S=adt+odW where a and a are constants.

(!)

334

OPTIONS,

The valuation

FUTURES

AND EXOTIC

DE

ATIVES

EXTENDELE

OPTIONS

equation that must be satisfied by the option price, V(S, t), is 'Û2X2Vss + rXVs - rV + V,

,

0

=

(2)

where subscripts refer to partial derivatives. Let CE(S, Ki, Ti, K2, T2, A) be the current value of an extendible call as a function of the two strike prices Ki and K2, times to maturity Ti and Ta, the underlying asset S and the premium A. At date Ti, the call's pay-off is

(3) {0,C(S, K2, T2 - Ti) - A, S - Ki} i.e. the option holder can choose between three pay-offs: the intrinsic value S Ki, zero, with a strike price or the difference between the premium A and a standard European call K2 and a maturity date T2 - Ti. This may also be written as CE(S, Ki, Ti, K2, T2, Á)

=

335

m3X

a2/2)T2

yi

=

(ln(S/I2)

+

Q+

y,

=

(ln(S/li)

+

(r +Ü2/2)T,)/

y

=

ya

=

(ln(S/K2) + (ln(S/Ki) + p

(10)

(r

+ o 2/2)T2)/

(1l)

(r

+
(12) (13)

=

with

-

K2, T2 Ti) - A], max [0, S - Ki]} (4) risky pay-offs: the pay-off of a This pay-off function corresponds to a maximum of two standard call and that of a call on a call. The pay-off looks like that of an option on the maximum of two assets. When A is positive, there is some critical value of S at Ti denoted by li below which the option is not extended, and another critical value I2 above which the option is again not extended. Hence, extension occurs when Sis in the interval [Ïi, I2 At Ti, the value of li is given by the solution to the equation CE

=

m3X

K2, T2 T,) -

A

=

(5)

70

and It must lie between A and A K2e 0; and when li > Ki, the extension privilege When A 0, li A sufficient condition for I, to be less than Ki is =

At Ti, the value of I2

iS

Extendible

worthless

is

'O A < K2 - K2e given by the solution to the equation

C(I2, K2, T2

16.1.2

-

(6)

=

The extendible call has some interesting properties. It is worth at least as much as an equivalent standard call without the extension privilege. It is greater than or equal to the maximum of a standard call, C(S, Ki, Ti) and a compound option on C(S, K2, T2 Tip when S Note that the call is worthless 0. When li is nil and I2 iS infinite, the extendible call is worth C(S, K2, T2). The extendible call price is an increasing function of S, r and o, and a decreasing function of Ki. The examples in Tables 16.1-16.4 simulate values of extendible call prices for different parameter values. Also, the effect of change in the parameter values on option prices is examined. =

TABLE 16.1 I 5%, Ti r =

Effect of a Change 0.25, T2 0.25, Ti =

=

,

T,)

CE

SN(yi, y4) - Ae

'T

N(y,

Ke

-

,

y2, -oo,

N(yi

,

y2

Premium 0.3

A: S

=

40, K,

=

43, K2

0.5

I.0

4l.I I 43.52 I.79 I-8I

43.44 42.84 I.79 I.79

TABLE 16.2 K, 43, K2 =

,

)

I

=

45.04 42.20 1.79 f.84

2.0

2.5

3.0

46.32 41.58 1.79 f.94

47.38 40.99 l.79 2.08

48.38 40.39 1.79 2.27

(2)subject to

y3, p)

-

-

=

.5

li 12 C(S, K

SN(y

in the 0.5, «

(7)

T,) -- 12 - Ki + A

C(S, Ki, Ti)

=

-----

Calls

Cs(S, K,, Ti, K2, T2, A)

•

N2(a, b, c, d, p) defined as the cumulative probability of the standard bivariate normal density with correlation coefficient p for the region [a, b] × [c, d] N(a, b) defined as the cumulative probability of the standard normal density in the region [a, b] C(S, Ki, Ti) a standard call option.

Premium A

The value of the extendible call given by Longstaff as a solution to equation condition (3) is

where

•

[0, C(S, {max

C(I,,

=

•

y4 ¯

(8)

Il I2 C(5, K CE

,

T;)

=

Effect of a Change in the Strike Price: S 45, r 0.25, T2 0.5, a= 0.3 15%, T, =

=

39.45 46.68 I.79 2.05

=

=

40.94 45.13 1.79 l.89

44.08 41.88 1.79 l.82

40,

50,

336

FUTURES

OPTIONS,

TABLE 16.3 Effect of a Change in the Volatility 0.25, T2 0.5, A I 15%, Ti K2 45, r =

=

=

=

AND EXOTIC

Parameter:

S

EXTENDIBLE

DERIVATIVES

40, Ki

=

OPTIONS

337

43

=

+ SN(y49

¯

Kie

"l

N(y4 ¯

2

N(yi O y2 ) Note that when li K the option is not extendible and its value reduces to P(S, Ki, Ti). When A 0 and li 0, the Pa option value is P(S, K2, T2L When A is positive and li 0, the Ps value is equal to that of a call on P(S, K2, T2 ¯ l The Pa option value is an increasing function of Ki, Ti, K2, T2 and o, and decreasing a function of S, r and A. =

0.ls

0.30

0.20

'

- Ae

Volatilitya 0.l0

2

=

0.35

0.40

-

,

,

=

0.50

=

=

I, I, C(5,

KI,

43.59 42.49 0.29 0.3 i

Ti)

CE

42.72 43.24 0.63 0.64

41.70 44.21 !.067 i I

39.45 46.68 2.04 1.79

.06

TABLE 16.4 Effect of a Change in Interest 0.3, A Ti 0.25, T2 0.5, « 1 =

=

=

Rates:

38.28 48.I4 2.57 2. I 8

S

=

37.10 49.76 3.10

2.58

40, K,

=

34.00 53.00 4.l8 3.37

43, K2

=

.

=

Interest Rote r

li I, C(5, K,, T;)

.

CE

10%

I2%

16%

IB%

39.90 45.64 \.60 (.77

30.72 46.0 I i.67 f.88

39.36 46.94 \.82 2.l0

39.l8

39.00

47.56 l.9I 2.33

48.33 1.99 2.36

16.2

SIMPLE WRITER

16.2.1

Simple

EXTENDIBtÆ OP1%ONS

45,

Writer

Extendible

Calls

Let Cw(S, Ki, Ti, K2, T2) be the value of a simple writer extendible call for which T, is the initial maturity date and T2 is the extended maturity date. If the call is extended by the writer, its strike price is adjusted from Ki to K2, and A is zero. The Cw payoff at Ti is

20%

Cw(S, K,, T,, K2, The use of equation

O ¯

(8) under

C(S, K2, T2 S - Ki

the restrictions

A

-

=

Ti)

if S Ki at Ti

0, I,

0 and 12

=

=

Ki gives the

followíng price: 16.1.3

Extendible

Cw(S, Ki, Ti, K2, T2)

Puts

Let Py(S, Ki, Ti, K2, T2, A) be the value of an extendible price of a B-S put. At maturity date, Ti, the pay-off of Ps is Po which

=

max

{0,P(S,

Ka, Ta

-

Ti) - A, Ki - S}

(14)

Ka, T, - T;) A], max (0, K, - S)] (15) This analysis is very similar to that of an extendible call and implies the existence of two values of the underlying asset in the interval (I,, I2) that must be determined from the following equations: =

max

[max[0, P(S,

P(Ii,

Ki - 11 A P(I2, K2, Ta - Ti) K2, T2 - Ti)

=

Ps(S, Ki, Ti, Ka, T

,

A)

=

for equation

(2)

subject

P(S, Ki, T;) SN(yi, y2, -oo, -

K2e

"

N(yi

,

y2 -

N(y3

-

-y4

,

,

(20)

p)

It is convenient to note that the price Cw is not always a monotone increasing function of the underlying asset price. The intuition is that if the call is near-the-money at Ti, the option buyer has a second chance to exercise the option at T2. This option may be both convex and concave with respect to the underlying asset. The examples in Tables 16.5-16.9 simulate values of simple writer extendible call prices for different parameter values. For example (Table 16.6), when S 100, Ki 100, K2 120, r 0.1, Ti 0.25, T2 0.6 and Cw 9.24, the bivariate normal l is nbl 0.1357, and the bivariate normal 2 is nb2 0.11156. Reducing T2 from 0.6 to 0.5, Cw falls from 9.24 to 9.09. =

=

=

=

=

=

=

=

=

(16)

A

(17)

=

The solution given by Longstaff extendible put pnce s

- K2e

put and P(S, K, T) be the

can also be written as Py

C(S, Ki, Ti) + SN(ys, -y49

=

to conditior(14)

for the

16.2.2

Simple

Writer

Extendible

Let P,(S, Ki, Ti, K2, T2) be the date, its pay-off is given by

value

y3, p) ,

-oo,

73 -

'

A

Pw(S, Ki, Ti, K2, T2)

=

Puts of a simple write extendible P(S, K2, T2 Ti) Ki - S

put. At the maturity

if S > Ki at T if S< Ki at Ti

(21)

TABLE

100

100

I10

0.25

100

100

95

0.25

o

nb'

nb2

10%

0.4

0.1964\

0.1813

9.16 9.69

10%

0.4

0.3225

0.3212

12.64

T2

r

0.60 0.50 0.60 0.50

T

K

K

S

OPTIONS

339

TABLE I 6.9 Effect of a Change in interest tol0%:S=IIO,K,=90,K2=50,r=IO%,T,=0.5,T2=1

Price and Maturity

in the Strike

Effect of a Change

16.5

EXTENDIBLE

AND EXOTIC DERIVATIVES

FUTURES

OPTIONS,

338

C .

Volatility Cw

.

0.10 25.33

0.20 32.54

0.15 28.79

Rates from

0.25 35.72

0.35

20%

0.40 42.8\

38.39

12.17

This pay-off corresponds to two components: a standard put and the value of the extension feature. Substituting A 0, li K, and 12 oo in equation (18)gives the Pw

----

=

=

=

value: TABLE

of a Change

Effect

16.6

Strike

in the

Price,

Date

=

K

K2

12

Ti

nb2

r

a

ad 0.2016

0.1685

K2e The examples

4.33 6.\3

100

\00

I20

0.25

0.60 0.50

10%

100

100

I20

0.25

0 00

10%

0.3

0.t357

0.II156

\00

90

100

0.25

10%

0.3

0.1109

0.I 195

100

95

100

0.25

0.60 0.50 0.60

IO%

0.3

0.\930

0.\990

100

110

130

0.25

0.90 0.50

IO%

0.3

0.2023

0.1591

13.54 I3.7 \ 10.75 10.58 4.54

100

90

80

0.50 0.25

1.50 0.50

10%

0,3

0.3426

0.4050

3.29 23.38

S

K

K

95

60

90

50

110

90

50

Price,

Maturity

16.3

APPLICATIONS

16.3.1

Extendible

16.10

5

Effect of a Change

in the Strike Price and Maturity

Ki

K2

Ti

T2 0.75 0.50 0.75 0.50 0.75 0.50 0.75 0.50 0.75 0.50

nb

nb2

Cw

100

\00

100

0.25

0.50

l.00 0.50 l.00 0.50 l.00 0.50

IO%

0.3

0.3570

0.4667

100

90

100

0.25

10%

0.3

0.2720

0.3780

35.62 26.56 38.39 28.70

100

80

100

0.25

20%

0.5

0.4442

0.6107

56.07

100

I l0

120

0.25

100

120

140

0.25

0.\0 33.59

=

r

o

obi

ob2

IO%

0.4

0.2997

0.2732

4.l8 4.|6

10%

0.4

0.445

0.446

5.58

IO%

0.4

0.565

0.613

9.07 8.49

IO%

0.4

0.265

0.204

10%

0.4

0.\88

0.l25

16.63 I7.86 33.20 36.23

Pw

-

40.48

=

=

=

T2=I Volatilit Cw

=

Bonds

a

=

puts

=

As shown in the analysis and valuation of compound options, the stockholder's claim on the assets of a levered firm may be assimilated to a call on the value of the firm with a

Date, Volatility

TABLE 16.8 Effect of a Change in Volatility: 0.5, 20%, Ti S I 10, K, 90, K2 50, r =

=

=

r

0.25

y4 ¯

,

16.12 simulate values of simple writer extendible

=

=

=

T

0.50 0.25 0.50

+

=

T

0.15 110

Asset

(22)

N(-y3

=0.2732.

16.32

in the

in Tables 16.10

SN(-y3, y49

-

100, Ki 100, for different parameter values. For example (Table 16,10), when S 0.10, Ti 0.25, T2 0.75, o 0.4 and Pw 4.18, nbi K2 100, r 0.2997 and nb2 Reducing the time to maturity from 0.75 to 0.50 year (9 to 6 months) reduces Pw from 4.18 to 4.16.

TABLE TABLE I 6.7 Effect of a Change Rates and interest

'

Cw

0.3

o.so

1)

P(S, K2, T2 ¯

----

S

100

2)

Pw(S, Ki, Ti, K2,

Volatility and Maturity

0.15 39.61

0.20 43.83

0.25 46.82

0.35 51.037

TABLE 16.1 I Effect of a Change r= IO°/o, T, 0.25, T2 0.5,«= 0.3 =

S w

95 25.29

96 25.03

5.46

in the

Asset

Price:

K2

=

120,

K,

=

90,

=

97 24.46

98 23.89

99 22.72

100 22.I I

101

102

21.51

20.90

103 20.27

104 19.64

106 19.07

I10 16.40

OPTIONS,

340

TABLE 16.12 K2

=

100, K,

=

0.I 0.0

o Pw

AND EXOTIC

FUTURES

Effect of a Change in Volatility: S 0.25, T2 0.50 10%, Ti 80, r 0.2 4.09

0.15 \.85

=

100,

=

=

=

DERIYATIVES

0.25 5.78

0.3 6.97

0.35 7.83

strike price equal to the debt's face or nominal value and a maturity date equal to that of the outstandmg debt. Many corporate bonds have embedded options allowing the issuing firm to extend the maturity date of the original debt. By analogy, extending the life of the debt is equivalent extendible call to extending the time to maturity of the call. Hence, stockholders have an additional time gives maturity them time extending the to debt's since the value, firm's on valuable. applications particularly Other this around. privilege is Hence, the firm turn to of extendible option analysis are possible in pricing the firm's capital structure.

I 6.3.2

Extendible

Foreign Currency Options and Hyb rid Securities ,

,

y

a

,

CHAPTER

y

OUTLINE

This chapter is organized as follows:

Warrants

Warrants are often issued by corporations with simple or complex provisions. Some longconsequently be modeled as term warrants stipulate a change in the strike price and can when the strike price changes from extendible calls. In fact, if you denote by Ti the date then warrant, of the at Ti the warrant holder must Ki to K2, and by T2 the maturity date strike price Kr. If the warrant is not the whether exercise his warrant at decide or not to 0). exercised at Ti, it is extended to T2 (A This type of warrant can be valued using the formula for extendible calls. For other applications, see Longstaff (1990).

1. In section 2. In section 3. In section 4. In section results are

17.1 equity-linked Forex options and quantos are analyzed and valued. 17.2, the valuation of hybrid foreign currency options is described. 17.3, several forms of capped options are studied. 17.4 some hybrid instruments are analyzed and valued. Also, simulation provided.

=

UMMARY valuedIn this chapter, options with extendible maturities are identified, analyzed and values their and are First, the general context is provided. Second, options are priced and simulated. Third, some applications are presented. In particular, extendible bonds extendible warrants are analyzed and identified to options with extendible maturities.

POINTS • • • • • • •

What What What What What What What

FOR DISCUSSION

is the pay-off of an extendible call?

is the pay-off of an extendible put? is the pay-off is the pay-off is the pay-off is the pay-off are the main

call? a simple writer extendible writer extendible put? simple a an extendible bond? an extendible warrant? characteristics of options with extendible

of of of of

maturities?

INTRODUCTION For more than 15 years, foreign currency markets have been characterized by wide price changes. The high volatility of exchange rates has exposed treasurers and international investors to a high level of currency risk. Currency options markets have developed to provide new means of dealing with this growing risk. Currency options are traded on several security exchanges throughout the world. A sizeable over-the-counter market has also developed, offering a variety of specialized currency options. Among such specialized options are the so-called quantos, hybrids, ratios, capped options, and so on. It is often interesting for investors to link a strategy in a foreign equity and a currency to create pay-offs corresponding to a foreign equity option struck in foreign currency, a foreign equity option struck in domestic currency, an equity-linked foreign exchange option, a fixed exchange rate foreign equity option, and so on. This last option is known as a quanto option. An example of a quanto option is the dollar-denominated option on the Nikkei 225 allowing its holder to participate in the foreign equity market without exposure to currency risk. Ratio options, capped options and hybrid securities are also traded in different forms in OTC markets. Ratio options are options on the ratio of two asset prices, index levels, commodities, etc. An example of a ratio option is given by the dollar-denominated European option on the ratio of the German DAX stock index to the French CAC index. Also, as a special

OPTIONS,

342

FUTURES

AND EXOTIC

DERIVATIVES

option on the ratio of two US or Japanese index levels case, there is a dollar-denominated options may involve more than two assets and several sources of or stock prices. Ratio the different underlying risk from the assets, the exchange rates and the interest rates from couSntrieesfinancial possible pay-offs at maturity. contracts impose a cap or a floor on the reduced. A capped option can be is risk the issuer's By limiting the possible pay-offs, the range forward Examples include option. and call put of combination a analyzed as a a option note. and the index currency bonds collar, indexed the contract, resemble commodity Also, several firms have issued some financial contracts that securities, known as linked exchange principle rate the futures and debt, for example commodity futures and options with those of combining traditional which erls, are notes receive at maturity a pay-off, which is debt instruments. These notes allow the holder to value nommal of the notes. either larger or smaller than the face or based options some use of the put-call parity on are These hybrid foreign currency and simultaneously sells a call buys corporation put exporting a example, relation. For an of a straight put, but then the from the bank so that the overall cost is less than the cost of the foreign currency above the appreciation exporter loses any potential gain from an strike price of the call. compromise between fkxibility, Currency risk management has thus become a delicate tailoring an protection and cost. The achievement of such a trade-off amounts to anticipaconditional his the of on investor, needs instrument that perfectly matches the .

.

17.1.1

OPTIONS

CURRENCY

FOREIGN

Equity Option Struck

The Foreign

C' ,*

EQUITY-LINKED AND QUANTOS

FOREIGN

EXCHANGE

OPTIONS

the B-S formula for stock options also As shown in Garman and Kohlhagen (1983), where the foreign interest rate replaces applies to the valuation of options on currencies in a foreign stock and a the dividend yield. When an investor wants to link a strategy equity option struck of foreign options: different types a currency, he can use at least four struck in domestic currency, fixed exchange in foreign currency, a foreign equity .option options, or an equity-linked foreign rate foreign equity options, known also as quanto valued in this section. options are analyzed and option. These different types of exchange

in Foreign

Currency

X* max [S"

=

-

K', 0]

-

the equity price in the currency of the investor's country and K' is a foreign currency amount. The spot exchange rate expressed in domestic currency of a unit of stands m front of the pay-off to show that the latter must be foreign currency, X converted into domestic currency. The domestic currency value of this call option is given by where S

is

,

Ci

K'Xr*

S'Xd 'N(di)

=

'N(di

my/t)

-

(2)

with In (S'd '/K'r*

di

'

=

(3)

s

.

where os. is the volatility of S'. This option can be easily hedged by an stocks and B' units of foreign cash, with

As B

17.1.2

The Foreign

,

-K'r

=

=

d 'N(di) 'N(di

smaam A

in

(4) (5)

)

- as

Equity Option Struck in Domestic

Currency

When an investor wants to be sure that the future pay-off from the foreign market is meaningful when converted into his own currency, then the foreign equity option struck m foreign currency is appropriate. This option has the following pay-off for a call:

C

=

max [S'*X*

- K, 0]

(6)

where K the domestic currency amount. For the foreign option writer, the pay-off is given by is

Cl*

17.1

343

W h en an mvestor in a foreign equity is not interested m the risk borne from the drop in t¡exnchhangerate,ohe snapyainvest in a foreign equity call struck in foreign currency. This

tions.

marketing these hybrids, although Banks worldwide have been quite successful in traded on organized theoretically they can be replicated by combinations of instruments Citibank the range forward option from cylinder or markets. A typical hybrid like the customize the hedge by selecting the holder to allows its Brothers contract from Salomon available on an organized exchange. appropriate strike and a maturity that might not be mostly European options, which make them Besides, hybrid foreign currency options are potential for a wide variety of The counterparts. American less expensive than their the French hybrid currency specifications has attracted many customers. In that respect, actively competing among thembeen have French banks since market is interesting, in such a way that selves to market several hybrid contracts. These contracts are designed referred and they upfront to as zero-premium cost are the buyer does not have to pay any options. hybrid foreign currency

AND HYBRID SECURITIES

=

max [S''

-

KX'*

'

0]

23

1/X. X' corresponds to the exchange rate quoted as the price of a unit of domestic currency in terms of the foreign currency. It is convenient to note that this payoff corresponds to that of an option to exchange one asset (K units of our currency) for another asset (a share of stock). The value of this option is given by

where X'

=

,70)

C'*

=

S'd 'N(d2) - KX'r

'N(d2 - o

g)

with

da

In(S'd '/K'X'r =

')

9)

OPTIONS,

344

FUTURES

AND EXOTIC

DERIVATIVES

FORE1GN CURRENCY

MS

AND HYBRID SECURITIES

OPTIONS

I

oi.+oi,-20s'x-Us

ass-=

(10)

x'

B'

-

C,

=

'N(d2 - Kr -

'N(d2)

S'Xd

)

¤s

vit) - as

exp(-ps'xus-Jxt)N(d3

-

r*

X

ps,,, is the correlation coefficient between the rates of return on S' and X'. If we multiply this formula by the exchange rate and substitute l/X for X', we get the domestic value of this option, i.e. where

'

rd

KS' =

As

(20)

C3/S'X

=

_

(21) (22)

B'X

(11)

with '

ln (S'Xd

d2

Kr

')

+ as

(12)

2pssora

a

=

os

(asx

(13)

This option can again be easily hedged by an amount A in stocks and B' units of foreign

Rate

The Fixed Exchange Options

(7.1.4

are proposed

Two approaches

EquityDptions

Foreign

orquanto

for the pricing of this option:

Reiner's

approach and

Jamshidian's approach.

cash, with

d 'N(d2)

A= B'

'N(d2

-Kr

=

-

os

)

(14)

Reiner's Approach

(15)

When an investor wants to hedge away the exchange risk on his investment in a foreign equity and fixes in advance the exchange rate at which the future pay-off will be converted in the domestic currency, he can invest in a quanto option or a fixed exchange rate foreign equity call with the following pay-off:

This formula is equivalent to that of B-S with S'X replacing S and as x replacing a. It is asset S'K This as if the B-S risk-neutral pricing approach were applied to the underlying allows the derivation of simple rules that can be applied for the valuation and the hedging ofthe two following options

C4*

=

Ïmax {S'* -

K', 0}

=

Xi is the rate at which the conversion will are equivalent even though the strike price for the first is in foreign terms and in the latter in domestic terms. We rewrite the pay-off in reciprocal units:

where

17.1.3

Equity-linked

Foreign

Call

Exchange

When an investor wants to place a floor on the exchange component of his investment in foreign equity, he can use a combination of a currency call and an equity forward to foreign exchange call with the following pay-off: create an equity-linked Cl*

S'*

=

max

[1

KX'*, 0]

=

S'd 'N(d3) - KS'X'

/

C'*

ln(Xr*

'/Kr

)

(-ps

)

xos ax

(17)

pssos.ext

=

ÏX'

=

S'Xd

'

N(d3) - KS'

=

ÏX'e"max

(24)

{S'e" - K', 0}

exp(-psxagoxt)N(d4)

S'

(25)

(18)

(a

K'r

'N(d4

- os

v't)

(26)

with

d4 exp( -ps

-

.

+

The domestic value of this call option is C3

max {S'* - K', 0}

u and v stand for the natural logarithm of 1 plus the returns of S' and X'. Using the joint distribution for u and v, the value of this option in foreign currency given by Rubinstein and Reiner (1991a)is

C' =

ÏX'*

and express it in the following form:

with

da

=

where

rd

exp

C'*

(16)

The foreign value of this call option is given by

C'

(23) {S'*Ï- K, 0} be made. Note that the two pay-offs max

ayaxt)N(d

- a

)

(19)

This option can again be hedged in an unusual form by an amount A in stocks, B' foreign cash and ß in domestic currency, with

'/K'r*

')-Ps'xasoxt S

The domestic value ofthis option is rd

in

ln(S'd

C4

=

XS

-r*

'

exp(-psxoraxt)N(d4)-

K'r 'N(d4 -

s./0

(28)

OPTIONS,

346

FUTURES

AND EXOTIC

DERIVATIVES

FOREIOMØdRRENCY

Ï

rd

exp(-pyxoraxt)N(d4)

=

X

\r* B'

-AyS',

=

B

=

(29)

C

(30)

Jamshidian'sApproach

AblD NYERID SECURITIES

347

"

This option can be hedged in an unusual form by an amount A in stocks, B' in foreign cash and Bm domestic currency, where A

OPTIONS

.

dc(t)

c(to) +

hm c(t)

=

=

max (f(T)

0) - K,

'

(36)

Since f(T) Sr, this is exactly the option pay-off and the option price must be equal to C(to). This general result can be applied to options on stocks and zero-coupon bonds as =

wcIl as quanto and ratio options. of quanto and ratio options is based on the following The hedging and valuation theorem: a dynamic trading strategy in (only) international T-maturity forward contracts for which the net dollar result at time T is [f( T) f(to)] replicates the dollardenominated T-maturity forward contract with terminal price f( T) dollars, and the later price is f(to) dollars (seeNelken (1996)). Consider a quanto option having the following pay-off at the maturity date: -

The approach proposed by Jamshidian (1994) needs the stiategy and the application of Itô's lemma to show that lent to that of the call. Accordingly, the option price investment for the strategy. We illustrate this technique with respect to a off in dollars:

construction of a self-financing the strategy's pay-off is equiva is equal to that of the initial

'generic'

Co

=

Coo

option with the following pay-

max [Sr - K, 0] at the maturity date T, referred

(31)

where S, is the underlying variable to as a T-maturity underlymg forward contract The option value can be expressed in terms of the dollar price of this forward contract. forward price, f(t), where by definition, This latter is referred to as the underlying f(T) SrThe question is how to value and hedge this generic option in terms of the underlying forward contract? An answer can be given in a B-S and Merton economy when f(t) is continuous and has dynamically the underlying a deterministic volatility. Therefore, we need to replicate under of other forward the assumption that the contracts in terms forward contract instantaneous covariances between foi,«ard prices are deterministic. Let

where Sr refers to the price S, dollars. To apply the generic of the yen-dominated denote by g(t) the yen

underlying

=

max [Sr - K, 0]

(37)

forward contract denominated in dollars

with terminal

option results, we have to replicate this forward contract in terms with terminal price S, T-maturity forward contract yen. If we forward price of this contract, then

=

c(t)

=

f(t)N(di)

- KN(d2)

(32)

+ v(t)

(33)

with

ln(f(t)/K)

di

=

d2

=

v(t) In(f(t)/K)

v2(t)

-

v(t) =

var,

pg)

-

f(s)

(34)

=

=

g(t) exp

df cov,

(s) dg(s)l

(38)

,

where the covariance between the forward exchange rate and the actual forward price is to be deterministic. We denote by S(T) the exchange rate, and consider an investor who is long n(t) f(t)/(f,(t)g(t)) actual T-maturity forward contracts at any time t, to< t< T. The forward exchange rate contracts at time t + dt at the then investor shorts n(t)dg(t) prevalent forward exchange The net result of this rate, f,(t+dt)=f(t)+df,(t). strategy at time Tis

assumed

=

i

r

S(T)

n(t)dg(t)

-

r

n(t)dg(t)[S(T)

-

f,(t+dt)]=

(35)

consists of holding C(to) Consider at time to a dynamic trading strategy until c(to) and T, T-maturity bonds N(d,) units of the underlying Pr(to)c(to) dollars cash, forward contract. In this expression, P,(t) stands for the dollar price at time t of a US Tmaturity bond. of this position to Applying Itõ's lemma to equation (32) shows that the contribution i result interval time T at the strategy over an (in dollars) [t, + dt] is given by N(d,) d f(t) dc(t). Accordingly, the net pay-off at time Tis which

f'

f(t)

n(t)dg(t)[[s(t)+dfs(t)]

f(T)

-

f(to)

(39)

=

n(t)dg(t)[f,(t) since df(t) + df,(t)]. ST, Using the theorem, and the fact that f(T) it follows that this strategy g(T) replicates underlying entered the forward contract at time la and the later price is f(to) =

=

dollars. The same approach can be applied

to ratio options.

=

OPTIONS,

348

17.2

ANALYSIS AND VALUATION OPTIONS

17.2.1

The Range Forward

FUTURES

AND EXOTI

TIVES

FOREIGN

CURRENCY

OPTIONS

AND HYBRID SECURITIES

sale of a call with a strike price Ki (< f). Hence the contract's value is

OF CAPPED

X(S, T, Ki, K2) In a range forward contract, the buyer and the bank agree on two prices, Si and S2, at the mception of the contract. At the maturity of the forward contract, the buyer will purchase the foreign currency either at Si if the spot price is less than Si, or at S2 if the spot price is greater than S2, or at the spot price if it is between Si and S2. The two prices Si and S2 are set such that no money changes hands at the inception of the contract. Like a range forward contract, a participating forward contract guarantees a minimum exchange rate for a forward sale and a maximum exchange rate for a forward purchase. Besides, the seller (buyer)gets a participation in the foreign currency appreciation (depreciation).Obviously, there is a cost to this participation. Since the contract is structured with no upfront payment, the cost of the seller's upside participation is that the minimum exchange rate guaranteed through a participating forward sale will necessarily be greater than the outright forward price. Analogously, the maximum exchange rate guaranteed through a participating purchase will necessarily be greater than the outright

forward price.

and a purchase of a put with a strike price K2

(> f)

V(S, T,

=

Contract

349

f) - c(S,

T, Ki) + p(S, T, K2)

(42)

By construction, the initial premium for the range forward contract is zero. When the buyer takes a strike price, the bank picks the other, so that the contract's value is zero: X(S, T, Ki, K2) Hence, the value of a

range currency option pnces, where

=

p(S, T, K2) - c(S, T, Ki)

forward contract

c(S, T, Ki)

=

(43)

is given by the difference

Se '"N(di)

'N(d2)

- Kie

between two

(44)

and

p(S, T, K2)

=

"N(-d2)

Ke

-

Se

'

N(-ds)

(45)

.

with d where i

=

r*+¾a2)T

ln(S/K,)+(r =

,

d

=

d - a

5

(46)

1, 2.

A conditional forward purchase contract is similar to an outright forward purchase, except that the long side of the contract has the right to pull out of the forward purchase by paying a fee to the short side of the contract on the maturity date. The contract can also be designed in such a way that there is no cancellation fee simply by guaranteeing a buying price lower than the forward price. Similarly, a conditional forward sale contract is equivalent to an outright forward sale, except that the short side of the forward contract has the right to pull out of the agreement by paying a fee to the long side of the contract. To give an example for the use and valuation of a range forward contract, consider a when the spot rate is 1.4 manager of a US firm intending to buy pounds in three months, dollars per pound and the three-month forward rate is 1.38 dollars per pound. He can buy value a range forward contract, taking two rates 1.32 and 1.45, so that the initial contract is nil. At maturity, if the spot exchange rate is below 1.32, the buyer pays 1.32. If it is above 1.45, he pays 1.45. If the spot exchange rate is in the interval (1,32, 1.45), he pays

Take, for example, a firm issuing a bond for one year where interest rates are adjusted semi-annually with reference to the LIBOR. If in six months, the LlBOR is above (below) 9%, the firm pays (receives)the difference to the bank. Hence, the cost of financing in variable rate ranges between 6% and 9%. The collar can be regarded as a difference between two European option prices. Its value is given by

the spot rate. At the maturity date, T, the contract's value is assimilated and a short put, p. Hence, the contract's value is

Ki K2

V(S, T, K)

=

17.2.2

X(r,

T, Ki, K2)

=

p(r, T, Kr) - c(r, T, Kz)

(47)

where r

to a position in a long call, c,

The Collar

is the six-month LIBOR, is the price of a discount bond yielding 9%, is the price of a discount bond yielding 6%.

(40)

c(S, T, K) - p(S, T, K)

where

17.2.3

fstands for the forward exchange rate. A range forward contract can also be regarded as a combination of two portfolios. The first portfolio corresponds to a long position in a call and a short position in a put. The second portfolio comprises a cap which is placed on the range of possible pay-offs, by the

Standard Oil issued notes in June 1986 with a nominal value of 1000 and a maturity date of 1990. These notes have an implicit capped option. The additional amount of the mdexed note is given by 170 times the amount (if any) by which the price of a barrel of petrol exceeds 25 dollars at the maturity date. Hence, if the barrel's price at that date is 20 dollars, the option is worthless. However, if the barrel's price lies between 25 and 40 dollars, the option pay-off is (40 25) × 170. This indexed note is regarded as a sale of a call and a cap on the possible pay-oíTs at

S stands for the spot rate. At the contract's initiation, the strike price is given in a way such that the contract's value is nil, or V(S, T, K) where

,

=

c(S, T,

f) -

p(S, T,

f)

=

0

(41)

Indexed

Notes

-

OPTIONS,

350

FUTURES

AND EXOTIC

FOREIGN

DERIVATIVES

maturity. The embedded option is analyzed as a purchase of a European call with a strike value of the mdexed price of 25 and a sale of another call with a strike price of 40. The note is given by

X(S, T, Ki, K2)

c(S, T, Ki) - c(S, T, K2)

=

(48)

Table 17.1 lists the values of the indexed note X(S, T, K,, K2) for

various

[l69/S - l]l000

-

B(T) - [c(S, T, K) - c(Y, S, 2K)]I000/K

(49)

where

B( T)

the option,

value without

is the note's market

is the spot exchange rate (dollar/yen), K is the strike price (dollar/yen) 1/169, T is the maturity date of the ICON, c(S, T, K) is the European currency call (in dollars)

S

=

TABLE I 7.1 Values 3 years, 45, T K2 =

=

Petrol Price

16 18 20 22 24 30 35 40 45

of r

=

Notes: Indexed 40% 9%, «

PRIClNG HYBRID FOREIGN OPTIONS

K,

=

17.3.1

Hybrid

Tailoring Approach

Foreign

Implicit Option Value X(S, T, Ki, K2)

\.68 2.47 3.40 4.46 5.64 6.9 1 il 15.23

0.67

l.OI l.4l l.84 2.28 2.75 3.20 4.52 5.51 6.39 7.\2

19.51

2.89 3.7 \ 6.70 9.72 13.\2

23.96

16.84

CURRENCY

Currency

firm) aims at the

Options:

the Briys-Crouhy

On the one hand, corporate treasurers claim that plain vanilla put on call options are expensive. On the other hand, standard zero-premium hybrid foreign currency options cost nothing to the buyer to initiate, but he or she has to give up some capture'. There are at least three ways of dealing with this issue. The first way to lower the cost of the put is to slightly reduce the level of protection, as in the put with proportional coverage in Figure 17.1. If the contract expires in-the-money, the pay-off is limited to a percentage beta of what it would be for an ordinary put: P

=

ß(K

- S),

0

<

ß Gl

(50)

The second way to reduce the cost of the protection of the put is to bound the pay-off of the put from above as shown in Figure 17,2. with disappearing deductible'shown The third way corresponds to the so-called in Figure 17.3. These basic protection schemes are not mutually exclusive and can be combined without difficulties. A tailored hedge can then be set up to provide the exporter with the desired trade-off between protection, flexibility and cost. A general pricing formula for put contracts can be given along the lines of Briys and Crouhy (1988)and adapted to the exact specifications of the hybrid instruments to be priced. As long as the pay-off function of the put is given as a stepwise function, it is straightforward to separate it into decreasing continuous segments as in Figure 17.4. 'put

30,

Option 2 Value c(S, T, K2)

l.06 1.56 2. I8

351

The buyer of a hybrid security (a corporate treasurer of an exporting best trade-off between protection, flexibility and cost.

P S K

Payoff Spot rate at expiration Strike price B Proportional coverage rate

=

Option / Value c(S, T, Ki)

.22

AND HYBRID SECURITIES

'upside

where S stands for the yen/dollar spot exchange rate. If, at the maturity date, the exchange rate is 169 yen dollar, the above amount is received. If the exchange rate is 159 yen dollar, the bearer receives $937.11 (1000 62.89). However, if the exchange rate is below 84.5 yen dollar, i.e. 169/2, the bearer receives nothing. The bearer receives a minimum of zero and a maximum of $1000. The ICON's value is given by =

OPTIONS

petrol

prices. Another example of indexed notes is the index currency note, hereafter ICON, issued value of $1000, payable in by the long-term credit Bank of Japan. Each ICON has a face pay-off: following the rise 10 years, and gives to

V

17.3

CURRENCY

p

=

=

=

=

.

,

ß<1

0

RGURE

17.1

Put with Proportional

K

Coverage

OPTIONS,

352

P S K B

=

=

=

=

FUTURES

AND EXOTI

EillIVATIVES

FOltig

N CURRENCY

Payoff Spot rate at expiration Strike price Lower bound of the spot rate at maturity under which no additional coverage is provided

P

OPTIONS

AND HYBRID SECURITIES

P S K

=

L

=

=

Payoff Spot rate at expiration Ky Spot rate range for the payoff function Payoí\ at maturity date for S K Slope of the payoff function =

' =

a

353

=

L

B

0

FM

17.2

Put with

Bound

K

0

S

FIGURE

Payoff

17.4

Generalized

a P Payoff SK caetex¢ration r ke pr D Deductible

Ki

_

=

-1,

S

Payoff Segment

L

=

0, K

=

0 and j

=

1 with Ki

=

K.

=

Using the risk-neutral approach, the current premium Briys and Crouhy (1988)is

=

for this pay-off segment gíven in

Kj

P,

=

e"

(Ly - ayKj+

ayS,)L'(S,)dS,

(52)

where t is the time to maturity, S, is the foreign currency spot price at maturity date, Rd iS the domestic riskless interest rate, L'(S,) is the risk-neutral log-normal density function of the currency

p K

o FIGURE 17.3

Put

with

Disappearing

s

Deductible

Integrating this expression gives P,(So, t)

I 7.3.2

General

=

(L; - a;Ky)e

')

''[N(d

-

N(di)] + aySoe

')

'[N(d

-

N(d()]

(53)

where

Pricing

The pay-off at maturity is give

by

d,y

0 P,(So, t)

price at the

maturity.

=

L

O

-

a;K, + a,S,

if S,< Ky i if K, a S, a K if K, < S'

1 =

-

o

(51)

where j is the jth segment of the overall pay-off profile and S, is the foreign currency price at maturity date. to the case with lt is convenient to note that the ordinary put corresponds

In

So

/

-K

Ü2

ra -

r, - -

2

t

,

di

=

d + avit

(54)

with So the current spot exchange rate and re the foreign riskless rate. The total premium for a hybrid put contract is then P

=

Py(So, t)

(55)

0, and j

0, K i l, L, When a that in Garman and Kohlhagen =

=

=

-

1 with Ki

=

(1983) for the

Py(So, t)

=

AND EXOTIC

FUTURES

OPTIONS,

354

DERIVATIVES

CURRENCY

FOREIGN

K, the general formula reduces to

=

- See

'U

=

So + K

\ y/i a

ra - re - 2

N(-di)

di

,

1 with K, 0 and j -1, L, K - D, K,_; When a the premium for the put with a disappearing deductible: =

=

=

/(So, t)

=

=

355

Payment

on Re-

verse

(56)

ExchangeRate

where

d2

AND HYBRID SECURITIES

TABLE 17.2 The Principal Perls in Year 5

currency put:

Ke '2'N(-d2)

OPTIONS

=

=

d2 + a

(57)

D, the general forrada

Ke N'N(-d2) - Soe I'N(-di)

gives

(58)

Yen/$

$/Yen

40 60 69.475 92.6338 138.95 277.9 555.8

o.025000 0.0 I6 666 0.014393 0.010796 0.007 198 0.003 598 0.00\ 799 0.000 000

oc

Amount in $ 0 0 0

500 \000 I 500 1750

2000

where

d2

ln

=

Df

\

ra

-

ry -

2,

t

,

di

=

d2

(59)

Till@LE 173

Yen

17.4 17.4.1

FINANCING Specificities

of the

The principal characteristics

HYBRID SECURITIES:

WITH

PERLS

Product of these notes are as follows:

Issue: $100 000 000 Interest: 10|% per annum, payable in US dollars

• •

• •

amount of each $1000 note will

Principal amount: the principal an amount equal to $2000 minus 138.950 yen. Investment grade rating: type AAA Time to maturity: 5 years

17.4.2

Cash-flow

be paid

0.02500 0.01666 0.01439 0.0 1079 0.00719 0.00359 0.00179 0.00000

1000(I38 950)(l/60)

Components Pay-of¶on Short Contract

Principal Amount 1000 1000 1000 1000 1000 1000 1000 1000

Poy-of¶on Long Call

(2473.76) (\315) (1000) (500)

|473.76

500 750 1000

315* 0 0 0 0 0 0

Cumulative 0 O 0 500 500 1000 1750 2000

-2000

at maturity in

Identification

the principal

=

Yen

$

oc '3IS

Table 17.2 gives the year 5 prmcipal payment on the security expressed as a function o 0.01439 the yen dollar and dollar/yen exchange rates. When the exchange rate is above

(dollaryen),

$

40 60 69.475 92.6338 138.95 277.9 555.8

Issuer: Sallie Mae

• •

The Product

17.4.3

Valuation

of Perls

Following Barnhill and Seale (1990),the pricing of perls consists in the valuation of the product components. When the yield on the 5-year The coupon amount is. $54.375, or [(1000)(10|%)/2]. Sallie Mae straight bond is 8.2% (p.a.),the bond's market value is given by 54.375

is zero. For exchange rates below that level, the increasing to $1000 at 0.0143996, and in the extreme to

principal amount is received, $2000 should the yen become worthless. semi-annually. Per $1000 face value, the 10(% Sallie Mae reverse perls return $54.375 value of 138.950 the minus dollar equal $2000 the principle maturity, amount is to At of minimum with zero. a yen, d Table 17.3 identifies the pay-offs on the combination in year 5. The pay-offs correspon and call option. long forward contract a to a straight debt, short a

1000 =

payment

(1.041)'

(1.041)!0

$l l08

(60)

The value of the forward contract is given by Vi

where

=

1000 × 138.950 (Fe

=

1000 × 138.950(0.007

-

Fe) 196

-

0.009074)

=

-$261

(61)

OPTIONS,

356

FUTURES AND EXOTIC DERIVATIVES

FOREIGN

POINTS

Fe is the forward exchange rate, Fe is the equilibrium forward exchange rate at date of issue, given by

• •

Fe

=

S(1

-

rf

/(l

+ r*)'

=

(0.007 1968)(l.082)'

(1.033)'

=

0.009074dollar/yen

•

(62)

• •

where

• •

S is the spot exchange ratc3 r is the interest rate for 5 years on the US government securities, r* is the interest rate for 5 years on the Japanese government securities.

•

.

•

The option is priced using a modified version of the B-S model: c=Se''N(di)-Ke"N(d2)

(63)

where

d

ln(S

K)+(r

r* +)o2)T ,

d2

=

de-

dÑ

(64)

where r is the US interest rate (0.082) r* is the Japanese interest rate

(0.033),

t is time to maturity (5 years) S is the spot price, 138.950 yen or $2000, a is the volatility (12%). Using the above formula, the call price is $1. Knowing the bond value ($1108),the forward contract value (-$261) and the option value ($1), the value of perls must be

$848. i UMMARY

I

Q

CURRENCY

This chapter stresses the wide applicability of the option pricing theory when it comes to protecting or covering financial assets or a foreign currency position. The current trend in profiles that are well suited to financial institutions is to propose tailored conditional of customers. Several contracts have been described match the needs and expectations above to fulfill this goal. This chapter presents and analyzes equity-linked Forex options, quantos, range forward contracts, indexed notes, perls and several hybrid securities. These contracts have some characteristics shared by caps, floors and collars and their valuation sometimes reduces to the valuation of standard options. Some examples are given and simulation results are provided.

What What What What What What What What What

OPTIONS

AND HYBRIO SECM

FOR DISCUSSION

do we mean by an equity-linked foreign exchange option? is a foreign equity option struck in foreign currency? is a foreign equity option struck in a domestic currency? is an equity-linked foreign exchange call? are fixed exchange rate foreign equity options or quanto options? is a range forward contract? is a collar? is an indexed note? are the specificitics of perls?

357

I8 Binaries and Barriers

CHAPTER

OUTLINE

1. Section 18.1 presents examples of simple and complex binaries and barrier options. It also analyzes the main characteristics of these options. 2. Section 18.2 develops the valuation framework for binary barrier options. 3. Section 18.3 presents the formulas for the valuation of inside and outside barrier options. of standard and 4. Section 18.4 develops the framework for the analysis and valuation exotic options using simple binary options. 5. Section 18.5 is devoted to the analysis and valuation of options with continuous strikes and barriers. 6. Section 18.6 uses static hedging for the valuation of barrier options.

INTRODUCTION Recent innovations in OTC options markets involve not only certain relations between the underlying asset price and the strike price but also the number of time units for which a certain condition is satisfied. This corresponds, for example, to financial assets which are traded within a specified range. Standard calls and puts which have been traded for a long time are not the simplest and easiest financial products to value. For example, the value of a standard option can be obtained as a limit of a sum of binary or digital options. In fact, binary options which are considered as exotic options are much simpler to value than standard options. The value of a simple binary or digital call is given by discounting its pay-off under the usual B-S arguments of risk neutrality. The pay-off of this option is given by an amount A if the underlying asset price is greater than the strike price, and nothing otherwise. This pay-off corresponds to that of vanilla binaries or digitals. Structured products with embedded digitals are much more interesting than vanilla digitals. There are many types of range structures: range binaries, at-maturity range binaries, rebate range binaries, mandarin collars, mega-premium options, limit binary wall options, mini-premium options, boundary options, corridors, options, hokey-cokey options, dual-factor options, volatility options, and so on. In their complex forms, complex digitals or binaries may be presented as compound digitals, boolean digitals and

OPTIONS,

360

FUTURES

AND EXOTIC

corridors. Compound digitals obey the same principle as compound options and take different forms: quanto digitals, barriered digitals and options on digitals. Exotic options refer to options with some special features and are often traded over the counter, i.e. they are not exchange traded options. Barrier options are path-dependent options which are in life when they knock in and disappear when they knock out. A path-dependent option is an option whose pay-off depends on the history of the underlying asset price. In general, an upward movement of price underlying by followed asset the a downward movement is different from a downward movement followed by an upward movement. This is a main property of pathdependent options. Barrier options are sometimes referred to as knock-ms or knock-outs when the underlying asset hits or does not hit the barrier. The standard form of barrier options refers to European options which appear or disappear (ms or outs) when the underlying asset reaches a certain level known as the barrier. This barrier or knock-out level is set below the strike price for the call and above whenever option comes into existence the it for the put. For example, an in-barrier underlying asset value hits a specified level. The right to exercise an out-barrier option is forfeited when the barrier is hit. The holder of a down-and-out option can no longer exercise his option as soon as the underlying asset price hits some specified down-and-out .

.

.

.

.

.

.

.

barrier. When a barrier option is alive at the maturity date, its pay-off is identical to that of a standard option. If the underlying asset does not hit the barrier during the option's life, the pay-off of a knock-in option is nothing. When the option fails to knock in, a rebate is paid at expiry. If the underlying asset hits the barrier during the option's life, the knock-out option disappears. When the option knocks out, a rebate is paid. In the absence of rebates, the pay-off of a knock-in plus a knock-out option with the same underlying asset, strike price and barrier is equivalent to that of a standard option. Hence, it is sufficient to value knock-ins in order to get the values of knock-outs. Barrier options may also be defined in other forms where the barrier is above (below) both the strike price and the current underlying asset price. Barrier options can be ranked into four categories: •

Regular barrier options which come into existence when the barrier is out-of-theunderlying spot price. money with respect to the current Reverse barrier options which come into existence when the barrier is in-the-money with respect to the current underlying spot price. Double barrier options which come into existence when the barrier option has knock-in underlying asset price. or knock-out levels in either side with respect to the Continuous or soft barrier options are barrier options which do not appear or disappear suddenly. .

•

-

•

•

AND BARRIERS

BINARIES

DERIVATIVES

It is convenient to note that the main results obtained for standard options can easily be transferred to exotic options. The approach proposed by Bowie and Carr (1994)allows the use of static hedging as opposed to dynamic hedging for the valuation and managein the management of ment of some exotic options. This is a signficant improvement exotic options, since static hedging allows savings on transaction costs and the continuous

361

monitoring of an option position. This chapter analyzes these financial products and their applications and gives their valuation formulas.

18.1

ANALYSIS

18.I.I

Standard

OF BINARIES AND paamilinas

Binary Options

An option which pays out a fixed amount if the underlying asset price hits a certain predetermined level is a digital option. It is sometimes referred to as a bet option or a binary option. Hence, digital options have fixed pay-outs and are either or These options are something of a straight bet on the position of the underlying asset price 'on'

'off'

-

with respect to the strike price. In their simplest forms, the pay-off of these options is cash or nothing, or asset or nothing. In this spirit, a binary call is a bet on whether the asset price would be above the strike price. A binary put is a bet on whether the asset price would be below the strike price. Roughly speaking, digital options can be ranked into two categories: all-or-nothing digitals and one-touch digitals. .

.

.

.

Option

AII-or-Nothing

An all-or-nothing digital option is an option which pays out a set amount if the underlying asset is above or below a certain specified level at the option's maturity date, An all-ornothing call (put) is an option giving the right to its holder to receive a predetermined all', if the underlying asset goes above (below)the strike price. How far the amount, underlying asset price is above or below the strike price is not important since the pay-off will be at the maturity date. or 'the

'nothing'

'all'

One-Touch

All-or-Nothing

Option

A one-touch digital option is an option which pays out a set amount if the underlying asset price hits a certain specified level at any time during the option's life. A one-touch all-or-nothmg call (put) is an option giving the right to its holder to receive a all if the underlying asset goes above (below) the strike predetermined amount, price at any time during the option's life. How far the underlying asset price is above or below the strike price is irrelevant since the pay-off will be or .

'the

.

,

,

.

'all'

18.1.2

Complex

Binaries

and

'nothing'

Range Options

A Range Binary In this contract, the investor specifies a currency range for a certain predetermined interval of time. If the spot remains within the specified range during that period, the investor receives a fixed amount of any premium invested.

OPTIONS,

362

FUTURES

AND EXOTIC

DERIVATIVES

BINARIES AND BARRIER$

363

A Limit Binary Option

Example

When the $/DEM spot rate is 1.5, the investor receives, for example, six times any premium invested if the spot remains in the range 1.2-1.8 for six months. This range binary option can be constructed, for example, by a long position in a six-month $1.2 call, DEM put, with a double knock-out boundary at 1.2 and 1.8 and a long position in a six-month $1.8 put, DEM call, with a double knock-out boundary at 1.2 and 1.8.

A limit binary option is similar to a range binary option which pays out if both sides of the specified range trade. A Boundary

Binary Option

The pay-off of a boundary binary option is an amount 4 are hit and an amount B if neither side of a piedetermmed

it

both sides of g epeeriod range iange

trades

A Wall Option A Rebate

which pays off a fixed amount each day whenever the spot predetermined rate. certain above (below)a

A wall option is an option

Range Binary

By analogy to the range binary, in a rebate range binary the investor specifies a certain interval of time. If the spot remains within the currency range for a predetermined specified range during that period, then the investor receives a fixed amount of any premium invested. If one of the two boundaries is hit, the investor recuperates the initial premium invested. A Mandarin

Wall Option

A knock-out

wall option pays off a fixed amount each predetermined rate until a prespecified

(below)a certain A Mini-Premium

Collar

ln a mandarm collar, a collar is contracted m order to protect an underlying the net premium earned is mvested in a range bmary. .

position and

.

.

day whenever the spot fixes above level is traded in the market.

Option

option is an option for which the buyer pays nothing at initiation and A mini-premium with respect to pays fixed set amounts whenever the spot rate moves out-of-the-money the option's strike price. .

Example

Example An investor who wants to hedge the sale of a $ position against DEM can buy a $1.5 call and sell a $1.4 put. This operation might yield a net premium of 0.0035 DEM $ which will be mvested m a l.40-l.50 range binary with a pay-out ratio of 14 Hence, the investor is hedged through the collar and the position shows a gain of if neither 1.4 nor 1.5 trades. 0.05 DEM/$ (0.0035(14)) .

.

A Mega-Premium

A Knock-out

ßKes

.

.

An investor who intends to buy a $1.5 call DEM put for six months when the $ DEM spot rate is 1.45 would pay $2.95%. option, the investor pays nothing Using a mini-premium initially and pays each of the trades: 1.43, 1.40, when 1.37. Hence, the followmg levels $1.40% minimum potential premium is zero and the maximum potential premium is $4.2% (1.43 1.4 + 1.37) This pay-off can be constructed by a portfolio comprismg a long $1.5 call, DEM put, and short positions in a one-touch digital with a strike price of 1.43, a onetouch digital with a strike price of 1.50 and a one-touch digital with a strike price of 1.37. .

Option

An investor can sell two knock-out options with strike prices at the range extremities and invest the resulting premium in a range binary. If the range binary knocks out, the investor option. This is the characteristic will be short of an at-the-money of a mega-premium option.

A Hokey-Cokey

Option

A hokey-cokey option is an option which knocks in one side of the underlying and knocks out on the other side.

gpat rate

OPTIONS,

364

FUTURES

AND EXOTIC

DERIVATIVES

BINARIES AND BARRIERS

exercise respectively Example

Boolean

at the maturity date, at any time before maturity or at some specified

maturity.

times before

The hokey-cokey option refers to a situation in which an investor is short an at-themoney spot $ put DEM call for three months with a knock-out level at 101 and a knock-in level at 89. When the $/DEM spot rate is at 99, the premium earned can be $0.75%. In this structure, the investor keeps the premium when the spot trades at 101 at any time durmg the three months and also when the spot does not trade at 99 at any time durmg the three months.

365

The Up-ond-out

Put 'up-and-out'

'down-and-out'

In the late 1980s call options put options emerged, while had been traded even before 1970. An up-and-out option, known also as a knock-out option, an over-and-out option, an over-the-top option, a barrier option, a vamshing option, or an extinguishing option, is simply a put option with a special feature: if the underlying asset rises above a certain level during the option's life, known as the knock-out boundary, the option becomes worthless. Hence, an up-and-out put is an option which becomes out when the underlying level or knock-out boundary. asset price reaches a certain level, known as the

Digitals

'out'

Boolean digitals are in the form of

'and/or'digitals.

Corridors

Example

A corridor can be defined as a contract which stipulates a range for a currency pair and for which the buyer accrues a fixed amount each day if the spot fixes within the specified

Suppose you buy an at-the-money European up-and-out index put when the CAC40 index equals 1900, the maturity date is in a year and the knock-out boundary is 1950. If, until the maturity date, the index does not rise above 1950, then the option P3Y¯out is identical to that of a simple European put option. If the index level exceeds 1950 at some point in time during the option's life, then the option holder receives nothmg.

range.

Corridors are presented in several forms: non-accruing, with a fixed strike price or with floating strike price. The holder of a corridor receives coupon accruing whenever the a underlying asset is within a specified range or band. Coupons accrue often on a daily basis. Complex corridors may be presented in the forms of quanto corridors and ratchet corridors. Ratchet corridors can be assimilated to a strip of calls on digitals.

The Up-and-in

Put

The up-and-in put is the inverse of an up-and-out put. If at any time during the option's life, the underlying asset rises above a certain prespecified level, called the level, the option is worth something, otherwise its value is zero. If the underlying asset reaches or exceeds the level, the option becomes a standard put, otherwise it is worthless.

Example

'in'

An investor who is long 1.51-1.61 corridor for three months, when the $/DEM spot rate is at 1.54, accrues a fixed amount every day whenever the spot fixes within the specified range. The corridor yields a minimum return of zero and a maximum return equivalent to twice the original investment

'in'

The Down-and-out

call

The down-and-out call is defined in a similar way as a down-and-out put, except for the level. If the underlying asset price rises above that level, the down-and-out call becomes a standard call; if it falls below that level the call is worthless. Now, we can define a down-and-out call for which the knock-out level is out-of-the-money. The following data regarding the OTC currency option market are used to give examples of barrier options: 'out'

18.1.3

Barriers

and

Structured

Barrier

Options

Barrier options allow the option holder to go some specified levels of the or underlying asset only on specified days during the option's life. That day may be the last day of each month, of each quarter, the Friday before the last Monday of each month, and 'out'

'in'

so on. than and down-and-out options more expensive These specifities make up-and-out standard options. Also, they make up-and-in and down-and-in options cheaper than standard options. These options may be European, American or quasi-American, allowing

• • •

å

•

The The The The

spot sterling dollar exchange rate is 1.6 dollars per pound sterling. three-month dollar interest rate is 5%. three-month sterling interest rate is 10%. three-month implied volatility is 10%.

OPTIONS,

366

FUTURES

AND EXOTIC

BINARIES

DERIVATIVES

The Down-and-in

Recall that a European currency call gives the right to its holder to buy sterling and sell dollars at 1.6 S/pound sterling. The holder exercises his option only if the spot rate is above 1.6 at the maturity date.

Call

.

'in'

Using barrier options, it is possible to construct several forms of synthetic forward Synthetic forward structures can be presented in structures in currency OTC markets. currency markets m the forms of trigger forward contracts, at-maturity trigger forward contracts and forward extra contracts.

A down-and-out call gives the right to the holder to buy sterlmg and sell dollars at 1.6 with a knock-out level of 1.54. This level is below the strike price and the option is out-of-the-money. What happens at the option's maturity date depends on whether the spot rate trades at 1.54 during the option's life. lf at the maturity date the spot rate is above 1.6 and it never trades at 1.54 during the option's life, then the option is exercised. Otherwise, if the spot rate is below l.6 and it never trades at 1.54, then the option is not exercised. However, if at the maturity date the spot rate is above 1.6 and it has traded at 1.54 during the option's life, the option is not exercised smce the contract is terminated. If at that date the spot rate is below 1.6 and it has traded at 1.54, then the option is not exercised.

-

A Trigger Forward -

Contract

-

This is a synthetic forward contract which disappears if a certain predetermmed rate trades at any time durmg the contract's life. This synthetic forward contract can be created by long and short positions in barrier put and barrier call options. .

Example When the $/JPY spot rate is 100.50 and a three-month rate is 98.50, an investor can sell US dollars at 102.5 and the trigger forward contract will be alive as long as 93 does not trade over the three-month period. If one buys a three-month US$ put, JPY call, with a strike price at 102.5 and a knock-out level at 93, and sells a three-month US$ call, JPY put, with a strike price at 102,5 and a knock-out level at 93, he constructs a trigger forward contract.

Call

We use the same data to define the up-and-out call when the knock-out level is in-thecall is a call that ceases to exist whenever the underlying asset money. An up-and-out price reaches the knock-out level, which is above the current underlying asset price and the strike price.

At-maturity Example An up-and-out call gives, for example, the right to the holder to buy sterling and sell dollars at 1.6 with a knock-out level of 1.68. This level is above the strike price and the option is in-the-money. What happens at the option's maturity date depends on whether the spot rate trades at 1.68 during the option's life. If at the maturity date the spot rate is above 1.6 and it never trades at 1.68 during the option's life, then the option is exercised. If the spot rate is below l.6 and it never trades at 1.68, then the option is not exercised. However, if at that date the spot is above 1.6 and it has traded at 1.68 during the option's life, then the option is not exercised since the contract is terminated. If at that date the spot rate is below l.6 and it has traded at 1.68, the option is not exercised. is very low when compared to standard options The price of an up-and-out-call because its pay-out is no longer infinite and is limited by a barrier above the strike price. Since, the boundary or the knock-out level is known in advance, it is possible to hedge and value these options.

367

The down-and-in call is the mverse of a down-and-out call. It is a contingent claim which becomes a standard call when the underlying asset price goes below the level and it is worthless whenever the underlying asset does not go below that level.

Example

The Up-and-out

AND BARRIERS

Trigger Forward Contract

This is a synthetic forward contract which disappears if the spot rate trades below (or above) a certain predetermined rate at the maturity date. This synthetic forward contract can be created by long and short positions in barrier put and barrier calls. Forward Extra Contract The investor has protection from a long option position with a predetermined strike price for zero cost unless a specified level is hit. If this specified level is traded, then the right to exercise the option becomes an obligation through a synthetic forward contract.

Example

)

When the $/DEM spot rate is 1.5, an investor has the right to buy US dollars at l.52. If the spot trades at any time at 1.38, then the investor is locked into a forward contract at 1.52. This synthetic extra forward can be constructed by forming a portfolio which comprises a $1.52 call, DEM put, and the sale of a $1.52 put, DEM call, with a knock-in boundary at 1.38.

OPTIONS,

368

Characteristics

of Barrier

FUTURES AND EXOTIC DERIVATIVES

BINARIES AND BARRIERS

18.2.1

Options

The possibility that the barrier option disappears makes its price less than standard or vanilla options. The price of a European up-and-out put is less than that of a standard European put since up-and-out put options have some probability of going out before the maturity date when compared to standard puts without a knock-out boundary. When we refer to the definition of an up-and-out put option, then some rational restrictions on its price are straightforward. First, if the knock-out boundary is high, then its price is more expensive than an up-and-out put for which the knock-out boundary is lower. Second, long-term up-and-out puts are cheaper than equivalent short-term European up-and-out puts. This result is also intuitive since the longer the time to maturity, the greater is the probability of hitting the knock-out boundary. When the underlying asset price gets in- and out-of-the-money, the up-and-out call gets cheaper because at the knock-out level the option is worthless. When the volatility increases, the underlying asset is more likely to hit the knockfinish out-of-the-money. So, a higher volatility induces a lower option out level or premium. The delta of the up-and-out call is positive as the underlying asset moves in-theas the asset price approaches the barrier. This money and then becomes negative change in sign implies that the delta is zero at a certain time, which implies no hedge at all. When the maturity gets longer, the option price decreases since there is an increasing probability that the underlying asset price reaches or exceeds the barrier. When the time to maturity is shortened and it remains less than three or four days to the maturity, the option's delta decreases, approaching minus infinity. This gives rise to particular hedging

difficulties. markets, it is For some barrier options, such as those traded in foreign exchange sometimes stipulated that only hourly closmg prices might cause the option to disappear This feature induces that intra-hour knock-outs are not taken into account. Also, there is another feature characterizing these options that may imply some hedging difficulties: the rebate. In fact, when the underlying asset price hits the barrier, a fixed payment is made at the same moment. Hence, another decomposition of the option price may be provided: a first term which is due to exercising the option and a second portion of the option price arising from the rebate.

Path-independent

Cash-or-Nothing

369

Binary Barrier

Options

In their simplest forms, the pay-off of a binary call is zero when the underlying asset terminal price S, is below the strike price K, and is predetermined a amount A if the underlying asset terminal price is above the strike price. The pay-off of the binary call is ccon

pean

OF BINARY BARRIER OPTIONS

Following Rubinstein and Reiner

(1991a,b),

we use the followmg

R is l plus the riskless interest rate, d is l plus the pay-out rate, H is the barrier, S(r) is the price of the underlying asset after elapsed time r, asset at expiration S, is the price of the underlying t, y, ¢ are binary variables taking the value 1 or -1.

notation:

if S,

<

K

else

if

0 A

=

S, a K

(2)

else

These options can be valued with respect to classic formulas à la B-S for standard options. Recall that the values of standard options are given by

C

=

¢Sd

'N(¢x)

'N(¢x

- ¢KR

¢¤

-

)

(3)

with

ln o

\KR

f

+

j¤ Á

(4)

-l

and ¢ 1 for a call and for a put. The B-S formula comprises two parts. The first term, Sd 'N(¢x), corresponds to the present value of the underlying asset price conditional upon exercising the option. The second term, KR 'N(¢x - ¢o f), refers to the present value of the strike price times the probability of exercising the option. The value of a cash-or-nothing call is =

=

cco,

The value of a cash-or-nothing

Asset-or-Nothing

VALUATION

0 A

=

The pay-off of a binary put is zero when the underlying asset terminal price S, is above the strike price K, and is a predetermined amount A if the underlying asset terminal price is below the strike price. The pay-off of the binary put is

AR 'N(x

U

-

)

(5)

put is peos

18.2

Options

AR 'N(-x

=

+ o

)

(6)

Options

Th ese binary options are similar to cash-or-nothing options except that in their pay-off the predetermined amount is replaced by the terminal asset value. The pay-off of the asset-or-nothing call is cs,

The

value

conditional

of this option upon exercising

() S,

=

if S, GK else

is given by the present the call, or cao,

=

Sd 'N(x)

value

of the underlying

asset price

(8)

AND EXOTIC

FUTURES

OPTIONS,

370

DERIYATIVES

SlNARIES

AND BARRIERS

e2µa/<>2 -v /2 og/25te

The pay-off of the asset-or-nothing put is zero when the underlying asset terminal price S, is above the strike price, and is the terminal asset price S, if the underlying asset terminal price is below the strike pric

h(r)

sS, a K S, value

The value of this option is given by the present conditional upon exercising the put, or

of the underlying asset price

p.=Sd'N(-A)

=

where a where

e

-

=

In (H S),

=

a

v

2t¡a - r¡µt

u

,

ln(H/S),

=

v

(16)

=

This last density is known as the first passage time density. Now, we denote by l

(10)

x=-

S

in

y

K

S

In

x,=-

Aa't

y

=

AÛ2

-H

a

hi

=

1

+Aa2t

-

a

Gap Options

AÜ2

n

give the folowing pay-off for a call:

Gap options are gingssegg

z

if S, SK

0 S, - A

=

c """

In

=

+ b

t

(20)

else with

K). Note that the pay-off of a gap call corresponds refers to the difference (A to the difference between the pay-offs of an asset-or-nothing call and a cash-or-nothing. Therefore, its value is given by

The

37l

'gap'

-

.

cga,

=

Sd 'N(x) - AR 'N(x - a

(12)

)

.

This formula is like that of a standard call except for the cash amount replacing the strike price The pay-off of a gap option put is if S, > K Pean S, - A else =

Note that the pay-off of a gap put an asset-or-nothing

(0

to the difference between the Hence, its value is given by

corresponds

put and a cash-or-nothing.

pg,=-Sd

'N(-x)+KR

sometimes

[2A]=

Following underlying underlying expiration

pay-offsof

[4A]

'N(-x+av'i)

(14)

Rubinstein and Reiner (1991a,b), let f(u) be the density of the risk-neutral asset return u. Let g(u) stand for the density of the natural logarithm of the asset return when it hits the barrier but ends up below the knock-out level at that the underlying asset starts at S above the barrier H). Let h(r) be the

(given

(u)

=

e

72

where v

=

[3AJ Sd =

Binary Barrier Options

density for the first time r when the underlying functions f(u), g(u) and h(r) are given by

=

=

In(R d)

a2

(15)

[4C]

KR 'N(¢xi

=

=

=

binary

KR 'N(¢x -

=

[2CJ

[3C]

Sd '(H S)2*N(ty yi),

-

'N(qyi

are

¢o v'i) ¢a i) (22)

KR '(H/S)2(1 I)N(yy KR '(H/S)26

options

-

-

ga

t)

77a

t)

Using the above notations, Tables 18.1-18.4 give the binary barrier call and put value formulae for in-cash or nothing (Tables 18.1 and 18.2) and in-asset or nothing binary barrier call and put value (18.3, 18.4), and Tables 18.5-18.8 give the formulae for out-cash or nothing (18.5, 18.6) and out-asset or nothing (18.7,18.8). 'in'

'out'

TABLE Nothing

18,1 Call

Down

(or

condition Down

Up

µ

[I C]

Sd 'N(¢xi),

'(H/S)"N(gy),

asset crosses the barrier. These density

,

Sd 'N(¢x),

(21)

Ü2

As in Rubinstein and Reiner (1991b),28 types of path-dependent valued with 44 different formulas. We define the following notations =

2

µ2 + 2 ln (R)o

b=

,

'

«2

(13)

replacing the strike price.

Path-dependent

a=

'

«2

[ l A]

This formula is like that of a standard put except for the cash amount sometimes

18.2.2

A=l+-

.

Up)-and-in-Cash

callValue

or

¢

y

HH

[3CJ

K S

H H

[IC]

K

H

[2C]

-

-

l I

[3C] + [4C]

[4CJ

-I

I

i

371

OPTIONS,

TABLE Nothing

18.2

Down

Up

TABLE Nothing

Up

TABLE Nothing

Up

DERIVATIVES

BINARIES AND BARRIERS

Put Value

S K KH 5 K K

18.3 Call

H H

[lC]-[2C]+[4C] [3C)

Down

S K K 5 K K

I - I -I -l

-1

Up

- I

Up)-and-in-Asset

(or

Down

Up

S K K S K K

¢

y

Nothing

,

[lA]-

[3A] [2A]+[4A]

I I

[lA] [3A]+

I

Down

I

,

H H H H H H

Down

H H H H H H

Put Value

H H

[lA]-

(or

[4A]

[2A]

-I

K S K K

H H H H

[4A]

[lA] -[2A] [3A]

[3A]

(or

[2C] - [4C] [lC) - [3C]

TABLE Nothing

1 - I - I

0 [lC] - [2C] [4C]

Down

Down

Up)-and-out-Asset

(or

or

Colf Volue

H H

[lA] - [3A] [2A] - [4A]

H H H H

=

[ \A]

-

¢

y

0 [2A] + [3A] [4A]

I I

I I

- I

I

18.8 Put

Down

S K

H H

K

H H H H

5

(or

Up)-and-out-Asset Put Value

[lA] - [2A] [4A] 0

[3A]

y

or

¢ 1 -1

-

[2A] - [4A] [lA]-[3A]

-l -I

-l -I

¢

y

I I

-I

S K K 5 K K

K K

or

I I

18.3 [3C] -

- I - I -I -I

Call

Cond tion

-I

-i -l

[I C] [3C] [2C] - [4C]

- I

¢

Up)-and-out-Cash

CollVolue

\

or

y

- [3A] [\A]

Up

I

Up)-and-in-Asset

Put Value

or

¢

y

[I C] - [2C] + [3C] [4C] 0

Condition

Down

Condition

(or Up)-and-out-Cash

or

Up

TABLE 18.5 NothingCall

S K

TABLE 18.7 Call Value

Down

Put Condition

Down [2C] - [3C] + [4C] [IC]

H H H

S> H K> H K H K< H

18.4 Put

¢

9

373

TABLE 18.6

or

Nothing

Condition Down

Up)-and-in-Cash

(or

Condition Down

AND EXOTIC

Put Condition

Down

FUTURES

I

VALUATION

OF BARRIER OPTIONS

In the following analysis, a distinction is made options. We define the following notations:

between

'in'

barrier and

'out'

barrier

'N(¢x - ¢KR

'N(¢x)

[1] ¢Sd =

=

[3]

=

¢Sd '(H/S)"N(77y)

-

'N(¢xi

¢KR

[2] ¢Sd 'N(¢xi) -

DEglyATIVES

)

¢a

glgetARIES AND BARRIERS

TABLE 18.9

"N(yy

'(H/S)

[5] [6] wherex, xi,

=

=

KR '[N(yxi R[(H

-

S)""N(yz)

-

+ (H/S)"

"N(r¡yi

y, yi, z, 1, a and b are defined as in equations

callVolue pay-off

max (0, S* - K) R(at expiry)

=

K
pay-off

|

=

e×Piry)

Down

18.10

The option buyer pays for the down-and-in call up-front and does not receive the call until the underlying asset price hits a prespecified barrier or knock-out boundary, H. After with strike price K and time to an elapsed time rt, the holder receives a European call a rebate Re at expiration t - I when the barrier is reached, otherwise he receives expiration Following Rubinstein and Reiner (199]a),the pay-off of a down-and-in call can be

=

(0, S* - K (at expiry)

(max R

if Br if Vr

« «

t, S(r) < H t, S(r) > H

I i

-i

Put Value

S> H

pay-off

S*) max (0, K R (at expiry)

=

H Up

if Br if Vr

[2]pay-off

S< H

.

4

t, S(r) GH t, S(r) > H i

[5]

5*) max (0, K R (at expiry)

=

i

if Br a t, S(r) > H Vr a t, S(r) < H

If

-1 -l

[t]-[2]+[4]+[5] [3]-[5]

K>H K
¢

y

-l -l

(24)

for some rs on) S(r) GH for some rs (conditional

(conditionalon) S(r)

GH

t, S, t, S,

> 4

K, pay-off K, pay-off

=

=

S, - K. 0.

asset price can end up above the call, when K < H, the underlying For an up-and-in barrier having first hit the barrier. This gives the holder a positive pay-off. Also, it can The end up below the barrier but above the strike price. This gives the holder nothing. asset price ends up above the barrier holder receives the rebate when the underlying without touching it before expiration. The difference between the down-and-in call and the up-and-in call is that in the latter case the underlying asset starts out below the barrier. and Using notations [1]-[6] Tables 18.9 and 18.10 give the down (or up)-and-in call put value formulae.

18.3.2

'Out'

Barrier

Options

> H and K < H.

Given this pay-off, we must distinguish between two cases: K asset price can end up above the For a down-and-in call, when K > H, the underlying strike price having first reached the barrier. This gives the holder a positive pay-off. Also, it can end up above the barrier without ever having hit the barrier before maturity. This gives the holder a rebate. This may be written as If If

I

a t, S(r) > H if Vr a t, S(r) < H

as pay-off

•

i i

Put

(or Up)-and-in

Options Down

written

i i

(18) (21). Condition

'In' Barrier

¢

if Br < t, S(r) GH if Vr a t, s(r)> H

[l]+[5] [2]- [3]+ [4]+ (5]

K>H K< H

TABLE

18.3.1

y

[3]+ [5] [i]-[2]+[4]+[5] S* max (0, - K) if Br

( R(at

tjov/i)]

-

2r¡bov/Ï)]

N(yz -

Call

(or Up)-and-in

K> H

(23) Up

(H/S)2'

Down

S> H

Down

r¡ovít)

-

"N(yy;-yopli)

[4]=¢Sd'(H/S)"N(tyy)-¢KR'(H/S)2' r¡ap/t)

375

Condition

¢«vít)

-

¢KR

-

AfdD EKOTIC

FUTURES

OPTIONS,

$7ig

each down-and-in and up-and-in option a down-and-out and Tables 18.11 and 18.12). up-and-out option an (see When the rebate 47 is nil, the pay-off from a standard option equals the pay-off from a down-and-out option plus the pay-off from a down-and-in option. This parity relationship holds for both put and call options.

It is possible to associate to

TABLE

18.1 I

Down

(or Up)-and-out

CallValue

Condition Down

S> H

pay-off

=

S< H K> H K
f

i

K>H K< H Up

Call

max (0, S - K) R(at hit)

if if

Bi s t, S(r) t, S(r)

vr a

> H «

H

[l]-[3]+[6] [2] [4]+ [6] pay-off

=

ma×(0, S - K) R (at hit)

¢

y

I |

i |

i

i I

if Br a t, S(r)< H E t, S(r) > H

if VI

[6] [l]-[2]+[3]-[4]+[6]

_

-l

376

OPTIONS,

TABLE

18.12

Down

(or Up)-and-out

Condition

S> H

Down

pay-off

=

S< H

pay-off

K> H K< H

=

BINARIES AND BARRIERS

AND EXOTICM1VATIVES

18.3.3

Put Put Value

K> H K
FUTURES

(0, K - S*) R(at hit)

r¡ if

max

if

l

¢

Br a t, S(r)

vr a t, [i] - [2]+ [3]- [4]+ [6] [6] max (0, K S*) if Br < t, if Vr < t, ( R (at hit) + [2]- [4] [6] [i] - [3]+ [6]

> H S(r) GH

Outside

377

Barrier

Options: 'mside'

.

an Alternative

.

.

Approach

.

.

In classic barrier options, an barrier is defined with respect to the underlying asset price. Some other barrier options are defined with respect to a second variable, an barrier which determines whether the option is knocked in or out. For example, Bankers Trust International proposed to investors in October 1993 a call on a basket of Belgian stocks where the knock-out level corresponds to an appreciation by more than 3.5% of the Belgian Franc. and its location with respect to the Using the nature of the barrier, or with options, it is possible to structure underlying asset, inside barrier as or four different outside barrier call options and four different outside barrier put options. option plus an option with the same strike price, barrier and Recall that an maturity date have the same pay-off as an otherwise identical standard option. So there is no use in valuing the eight options separately. In fact, it is sufficient to value the or the options to get the prices of the others by the stated relationship. Hence, the value of a down-and-in call is given by the difference between the price of a standard call and that of the corresponding down-and-out call. of options with an outside barrier in a B-S economy is analyzed in The valuation Heynen and Kat (1994a).They consider these options as options on two underlying assets variable', and V S and V where S defmes the actual option pay-off, called the variable', stands for the defining whether the option is knocked or Following Heynen and Kat (1994a),we assume the following dynamics for the two 'outside'

I -i I -I S(r) < H S(r) > H

'in'

-

-I

I - i -i

'out',

'down',

'up'

'out'

'in'

'in'

'out'

The proof of this relationship is as follows. Consider an investor who holds a portfolio comprises an otherwise down-and-out option and a down-and-in option. If the underlying asset price never reaches the barrier, then the portfolio's pay-off is identical to that of a standard option. If the underlying asset reaches the barrier, the down-and-in option gives a standard option and the down-and-out option disappears. The main difference between down-and-in and down-and-out options is that for the former the rebate is received at the maturity date and for the latter it is received when the barrier is hit. Since the time the barrier is hit is unknown, the valuation of down-and-out options also needs the knowledge of the density of the first passage time r for the barrier to be hit by the underlying asset Since we need to value the rebate associated with the down-and-out option, its present value is given by its expected value discounted by the interest rate raised to the power of the first passage of time: which

'pay-off

'barrier

underlying

'in'

assets:

S

dln

=

dln

[6]

=

Rr

h(r)dr

R[(H/S)""N(yz)

=

+ (H

S)""N(yz

- 2pbo

)]

'out'.

=

-

µi dt + oi dWi

(28)

µ2 dt + «2dW2

(29)

(25) We denote by

y equals l if the barrier is hit from above and -1 if it is approached from below It is convenient to note that the value of an up-and-out put is given also in Cox and Rubinstein (1985) and Hudson (1991) and indirectly in Conze and Viswanathan (1991) who present the following formula: where

di

d' P

=

SN(-x) + KR 'N(o

-

x) - [-S(S/H)

"N(-y)

KR '(S/H)

2'"'N(a

-

l =

aid

=

di +

y)]

(26)

1 ei

=

ln . 2p

aiÖ ln

U2

So -K

In

+

(µi +ai)T

/ H \

LVo

H - 0

-

In

H -

,

(µ2

d2

,

d'

=

2p

d2 + ai

Û162

N 62

9

ciÑ

di -

=

In

¯

/ H\

LVo

El

(30)

RÛl

with

2 ei

=

ei

We also denote by ri This formula applies only for a pay-out protected not account for dividends.

European upend-out

put, i.e. it does

pl(u,

ti>

(belowpl(u,

ti

2 =

e2 +

H

In -

of time for V through the barrier H and by of In (Sr/So) and V not hitting the barrier H from above > T)) during the option's life. It can be shown that

T), the

the

e'

nrstpassage

joint density

pl(u,

r,, > T)

µ, T

N

=

2 -

¢

,

ln - µ¡ T - 2p (H

Nx

exp

In (H/ Vo) - µ2 T


AllllI¥ATIVES

ANÐ EKOTIC

FUTURES

OPTIONS,

378

BINARIES

18.4 ,

In (H

Vo) + µ2
cri

The pay-off of a binary option is given by an amount A if the binary condition lis,,«, satisfied. Using the standard B-S approach, the valu of a binary call is given by

Ñ

The option pricing formulas can be obtained by taking the discounted expected pay-off under the appropriate probability measure Hence, the values of a down-and-out outside call and put are given respectively by =

coco

e

"

=

The values integrals:

of

=

cooo

puoo

=

e

(K - Soe")p

l (u, r|>

(32)

T)du

(33)

outside call and put are given respectively

by the foMowing

So)

'

e

an up-and-out

1 (u, ti

> T) du

ln(K

poco

(See" - K)p

"

7 (u, ri

(See" - K)p

ln(K

So)

in (K

so)

'

(K - Soe")p

e

l (u, r'|>

>

¢ei,

->¡¢p)

-

in

2'

exp

(34)

T)du

(35)

N(17d',

In

C si -with µ

i

-

'AE

[lgs,, x,]

=

r

,

AN

\Kf

µ

v'

¢e',

is

(37)

ln (R) + ¾a2. ln this section, we show how to represent and value standard options, down-and-in and knock-out range options in terms of elementary options, switch options, corridors digital options.

18.4.1

=

Example

I: Standard

Options

of strike prices K, increasing by K an amount Ax, i.e. Ko K + Ax, Ka K + 2Ax, K, K + iAx. If we take this sequence with the _ 3mOunt the following pay-off is Ax and add the corresponding binary conditions, Obtained for an infinite number of strike prices:

Consider a sequence

T) du

Using some transformations and tedious calculations, it can be shown that the values down-and-out and up-and-out calls and puts are given by

r¡So N(7¡ds

379

ANALYSIS AND VALUATION OF STANDARD AND EXOTIC OPTIONS USING BINARY OPTIONS

¢p

Vo)

AND BARRIERS

y

=

=

,

.

=

.,

of

When Ax tends to zero,

i

em

the limit, the above conditiori can be rewntten as Its,

-r¡¢p)

dx

(39)

K)1ts,> x,

(40)

Note that the value ofthis integral is(S, ---7¡¢µ)

-

ye "K

N(17d2,¢e2, -17¢µ)

-

exp

2

in

N(77d',

¢e',

(36)

where v¡ takes the value 1 for the call and -l for the put and ¢ takes the value l for an up-and-out option and -1 for a down-and-out option. The correlation coefficient between the two processes is very important in this formula. When p 0, the formula reduces to a standard B-S call which multiplies a probability factor. For in-options, the probability corresponds to that of knocking the option in. For out-options, the probability corresponds to that of not knocking the option out· When p 1, the formula reduces to that of an inside barrier option. there is a higher When the pay-off and the barrier variables are negatively correlated, pay-off vanable the knocked-out call is that probability moves into the as a down-and-out coefficient yields a higher option price. For a in-the-money area. A higher correlation coefficient, the lower the option price. down-and-in call the higher the correlation

-

where the indicator variable takes the value l if S, > K and 0 if S, < K. This condition is simply the pay-off of a standard call we take several multiplicative and additive combinations of binary options, then it is possible to represent the pay-offs of exotic options in terms of digital options.

=

18.4.2

Example

2: Down-and-out

Call Options

=

The down-and-out by the condition

call with a strike price K and a knock-out

(Sr - K)1,s,,

x,1
barria Hean be represented

(41)

FUTURES

OPTIONS,

380

AND EXOTIC

DERIVATIVES

BINARIES AND BARRIERS

The condition

Using Fubini's theorem for the calculation of the term under expectation, linf(r,0stúT,SJH)

'

(42)

Q is

e

dr

N

=

r=ñ,

value of the switch option is given in Pechil When So, EK the option value is

(43)

Gro +

the set of all rational numbers. allows one to get the prices of down-and-out call options.

µ

r - To

a

<=r

have

r - To di

o

(49)

(1995).

A(T - To)N(d

r

we

)

representation

-ro)AK

r

18.4.3

In(Sro/K)

The

1(sem

¡Q0,T]nQ

This

1,s -

U

"

E

is equivalent to

where

381

Example

3: Switch

Options

r

By contrast to the preceding examples where time is fixed and summation is done over K, now the strike price is fixed and summation is done over time. When summation is done options'. When summation is over K, we refer to the derivative assets as options'. This terminology done over time, we refer to the derivative assets as is introduced in Pechil (1995). number of time units r, as summation If we fix the strike price and use an increasmg index, then the choice of a pay-out amount, AAr, and an increasing sequence to 0, iAT, Ar, r2 3Ar, nr, with nAr < T< (n + 1)Ar, 2Ar, 73 r, r, ri gives the following discrete time condition:

+ r

N(di)

a2

'

A

T

"I"

e

2µ2

"

- r

N(d

A

2u2

N(di)

(50)

) "i/2

"'A

e

'quantitative

'qualitative

.

moglwhen Se

K the option value is

C

.

=

r

"

Are + r

A(T - To)N(di)

=

=

=

=

=

...,

Its

A

,

«,Ar

(44)

i.e. in the limit, the above condition

When At tends to zero, continuous time as

can be rewritten

in

rT

lys,x,dr

A

U2

- r

=

...,

(45)

*

+ r

o

=

"

r

L'

Are + A

)Ar

E(1,s

(46)

r-a-r'A

N(di)

,

N(-4)

2µr >A

"

e

"

- To)

'A a

(51)

N(-d2)

e

2

with di

In order to value the switch option in a B-S economy, suppose that a certain time To elapsed, and the option has paid at least an amount Aro with to < To. The value of the discrete switch option is given by

Cdso

r

'AK

=

+

T - To -

K* T - To K

d2

,

__

In

a

Note that this formula is

useful

for the

-

a l

4

=

valuation

T - To

K -S, of corridor

-

T - To

(52) (53)

options.

This gives the following option value: Cdso

=

r

"

'°'

N

Aro + A istar

The value of the continuous

ri

In (S, K) "V'iAr - To

18.4.4 iAr - To Ar .

a

switch option is given by -r

Coo bi

=

r

"

"'

lis,

Aro + AE "

A

dr

Example

4: Corridor

Options

(47)

48)

Recall that a corridor option gives the right to its holder to receive a set amount for each day the underlying asset trades within a given range. The pay-off of a corridor is determined by the product of an amount A by the time units Ar such that S,,, lies in the range [K,, Ka]. It can be shown that the pay-off of a corridor is given by the difTerence between two switch options,

382

OPTIONS,

FUTURES

AND EXOTIC DERIYATIVES

The following condition presents the pay-off for the corridor: A

lix,

se,,Ar

<s

-

A

Dis,

,

x,, -

les

Example

5: Knock-out

383

Examples ,

y,,]Ar

(54)

Note that this pay-off is exactly the difference between the pay-offs of two switch options Hence, the value of the corridor is given simply by the difference between the formulas for switch options applied for two strike prices Ki and K2.

18.4.5

BINARIES AND BARRIERS

Range Options

Recall that a knock-out range option is an option which pays out a set amount for each day the underlying asset trades withm a specified range. This option is no longer m life when the underlying asset moves out of the range. The discrete time pay-off corresponding to the knock-out range option can be written .

as

A call with a strike range of $40-$50 has at the maturity date a similar pay-off as a portfolio of calls for which the strike prices range from $40-$50. A continuous barrier option is a barrier option for which the barrier extends fiom $80-$90 and which has similar characteristics as a portfoho of barrier options for which the barriers range evenly between $80 and $90. A continuous strike option is an option in which the contract stipulations allow a continuous strike price.

weighted portfolio of options that are otherwise identical but differ only with respect to the strike price. Let KL Stand for the lowest strike price and AK be the space between . Table 18.13 shows the portfolio's pay-off for a range of endme strike prices. The matrix m stock prices. The rows correspond to the pay-off profile of each option in the portfolio. The portfolio's pay-off is given in the bottom row, It is convenient to note, for example, that when the final underlying asset price is 25, triangle'. The pay-off the sum of the pay-offs of each option in the portfolio forms a ¾(Sr-KL2 AS AK gets very fOr S,>KL. tends to zero, the small and is area approximating portfolio's pay-off tends to that of the continuous strike option. For more details, see Ross and Hart (1994)and Carr and Bowie (1994). 'near

lix,

Fa(S) --

s

,

(s,,,iss,,Ar

(55)

or in a continuous time form: 1(Ki<

Fe(S)=

TABLE I8.13 AK I, Start =

Inf

{S,[sSup(S,}«K:)dr

(56)

(K, 57)

Option The above functionals give the following conditions ,ar

FalAFe(S),

i.-ir-i

!

2

s]3AFe(S)

Hence, the use of elementary options or binary options allows solutions for most of the complex options-

(57) one to get closed-form

i 3 4

5 6 7

8

18.5

CONTINUOUS STRIKE AND CONTINUOUS BARRIER OPTIONS

18.5.1

Definitions

.

and

Analysis

barrier options or soft barrier options are barrier options which do not appear or disappear suddenly, since they stay alive to reflect the minimum or maximum level of the underlying asset through time. Options with continuous strike prices and barriers help of portfolios with in to overcome some hedging problems that arise in the management and out options. This is made possible since these options offer a gradual rather than an instantaneous knock-out. of this type of option, we consider an equally In order to illustrate the mechanism

Portfolio at 20

9 10 Il Portfolio

20 2l 22 23 24 25 26 27 28 29 30 3\ X

17

l8

Pay-off l9

20

0 0 0 0 0 0 0 0 0 0 0 0 00000001 0 0 0 0 0 0 0 0 0000000000I2345 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000000000000000 0 0 0 0

Constituent

per 2l

22

23

24

25

26

I

2

3

4

0

I 0

2 I

3 2

0 0

0

0 0

0 0

6 5 5 4 4 3 2345678 I 2 0 I

0 0 0 0

0 0 0 °

0 0 0 0

0 0 0 0

0 0 0 0

|

3

6

10

15

0

0

Option

Call

0 0 0

0 2l

Pay-off:

27

28

29

30

3/

7 6 5

8 7 6

9 8 7

10 9 8

II 10 9

3 2

4 3

5 4

6 5

7 6

0 0 0 0

I 0 0 0

2 I 0 °

3 2 I ©

4 3

28

36

45

55

2 I 66

Continuous

18.5.2

Valuation

of the Continuous

Strike

Option

The continuous strike option, Cso, can be hedged with a portfolio of European options in a static hedging strategy. It can also be hedged in a dynamic strategy with a position of A units of the underlying asset and borrowing a cash amount. Since the

OPTIONS,

384

option can be hedged, it can be discounted value. Using the following notation:

FUTURES

valued

easily

AND EXOTIC

in a B-S context

BINARIES AND BARRIERS

DERIVATIVES

using the expected

Example When K > H and the initial underlying asset price is 100, we can denne a barrier call. If the lowest underlying down-and-out from asset price to $75 $90 to a range attained has been 90 or above, the option is still alive. If the lowest underlying asset If the lowest underlying price attained has been 75 or less, the option is worthless. asset price attained has been 87, then 20%, i.e. (90 87)/(90 - 75), of the option

S,: the value of the underlying asset at maturity asset, a: the volatility of the natural logarithm of the return of the underlying thheerik-free rate n numeraireccurrency fore netnamoeusm *: ed yield for the sasse H: the barrier for a discrete barrier option, U: the upper limit of the barrier range of a soft barrier option, L: the lower limit of the barrier range of a soft barrier option,

Cso

=

([S'e"

2"

given in Rubinstein and Reiner (199la)between the upper and lower limits of the range option. Recall that the value of a down-and-in call with K > H, in the absence of rebate, is

strike call option is given by ""'N(x

)-

+ o

2KSe "'N(x)

+ K2e "N(x - a

-

disappears.

maturity,

the value of a continuous

given by

)]

(58) "'

with

co,(K > H) l x

=

-

In

385

S

+

-

(r -

r*

jo2

+

(59)

"

N(y) - Ke

Se

=

N(y - a

y __

In

SK

a

la

1

,

r -

=

r*

+

The value of the soft barrier range down-and-in call option for K Continuous

Strike

Range Options

cssao,

The continuous strike range option, Csao, is an option for which there is a lower and an upper limit on the strike price. The pay-off of a continuous strike range option is similar to that of a portfolio with a long position in a Cso with strike price K and a short position in a Cso with strike price Kc. The pay-off of a continuous strike range option is

Csao

=

(max{S, -

KE,

0}2

-¡rnax{S,

0}2 - Ka,

(60)

cssaa;

•

10

=

-

N

Soft Barrier

Options

.

.

A barrier option can be transformed to a continuous barrier option by keeping a single strike price and allowing for proportional knock-in and knock-out of the barrier. This allows the option to get gradually in or out. The value of a barrier range option can be easily obtained by integrating the formula

U - L

Se

(U)] - µ,N[y2(U)]

(63)

'S

2(A +

()

Ai

(64)

A

N[yi(L)]

-

-

+ µsN42(L)]

(65)

Of

csekoic

18.5.4

is givën by

(SK)"i

l -

2

(U2 KL

L

(62)

with Ai

Buy a Cso with a strike price KL and sell a Cso with a strike price Ko. An equally weighted portfolio of European calls with strike prices varying from Ko.

> H

coi(K > H) dH

=

U

¼

or

The pay-off to this option can be easily replicated by one of the following strategies: •

(61)

with

Since the option pay-off involves S2, the above formula can be used to valueanireplicate power options. The Cso does not have an upper limit on the strike price.

18.5.3

)

=

'

Ke "S

i

(SK) ,

A2

(66)

with U2 A2 where

=

N[y3(U)] - µxN[y4(U)]

L2 -

SK

N[y3(L)]+

µgN[y4(Q]

(67)

OPTIONS,

386 l

yi(H)

=

In SK o v/t µs

+(A-

exp {-((1-()(A

=

DERIVATIVES

(H2 In H2

l

ya(H)=

AND EXOTIC

FUTURES

la

,

1)o

,

y2(H) -- yi(H)

y4(H)=

+()o2t},

µ

-

y3(H)

-

G+

(1- )a

exp {-((A -()(A

=

o

(68)

l +()c;2t}

BINARIES

AND BARRIERS

387

struck at H 2 K where the barrier H corresponds to the geometric mean of the strike roes This relationship allows one to hedge a short position in a down-and-in call (strike K< H) by a long position in K/H puts with a strike price equal to H2/K.

Example

the difference between this formula and the European B-S Using call gives the European down-and-out barrier range call formula. the parity relationship,

Consider a call struck at 4 and a put struck at 1. When the underlying asset price is (at the barrier), then how many out-of-the-money puts struck at l should you have in order to get the equivalent value to an out-of-the-money call and to implement the hedge? 2. Note that this mean of 2 The geometric mean of the strike prices is corresponds to the barrier. So the number of purchased puts is K/H 2 4 2 and the put strike price is H'/K, or 22/4 1. Hence, the sale of the down-and-in call with the strike price K can be hedged by the purchase of two puts with strike

2

18.6

OF BARRIER OPTIONS

STATIC HEDGING

=

=

The following analysis concerns the static hedging and valuation

18.6.1

Hedging

At-the-Money

Down-and-in

of down-and-in calls.

Calls

When the strike price K equals the barrier H, the down-and-in call is at-the-money. This is a simple case The sale of a down-and-in call can be hedged by the sale of an otherwise standard put with the same underlying asset, strike price and time to maturity as the down-and-in call. If the underlying asset price remains above the barrier, then the put and the down-andin call are worthless. However, if the underlying asset price is at the barrier or hits it for the first time, then standard puts and calls have the same value. The hedger can sell the standard put and buy a standard call simultaneously since these options have the same value. Hence, the sale of a down-and-in call can be exactly hedged by the purchase of a standard put. If we denote by co(K, H) the value of a down-and-in call with a barrier H and a strike price K, then its value prior to hitting the barrier is simply equal to that of a standard put, P(K) for H K. This can be written as =

cen(K, H)

=

P(K)

for H

=

K

(69)

This relationship applies only when the strike price equals the barrier

18.6.2

Hedging

Out-of-the-Money

Down-and-in

=

=

Cans

When K < H, the strike price is less than the barrier and the down-and-in call is out-ofi.e. selhng . the hedge . the-money. The same hedging strategy as in Section 18.6.1 apphes, as the barrier is hit and doing nothing when the asset price is above the barrier. However, when the barrier is hit, the emergent call is out-of-the-money and the put-call parity does not apply. This is because the call and the put have different strike prices. It is possible m this case to use the results in Carr (1994)concerning the put-call symmetry. The put-call symmetry shows that the pay-off from a call struck at K when the underlying asset price is equal to the barrier H is equivalent to the payoff from K/H puts

price 1. If the underlying asset does not hit the barrier (2), then the call and the two puts worthless. When the barrier is hit, the two puts have the same value as the call are and the hedger can sell the two puts and buy a call at zero cost. Hence, a short position in two standard puts is an exact hedge for the sale of a down-and-in call. The following relationship is verified: coi(K, H)

18.6.3

Hedging

In-the-Money

=

\H)

P

HEK

\ K /

Down-and-in

(70)

Calls

When K > H, the down-and-in call is in-the-money and has a positive intrinsic value at the barrier. We define a down-and-in bond as a security that pays $1 at maturity if the barrier is hit before the maturity date. If we construct a portfolio comprising H K down-and-in bonds, then it presents the same value at the barrier as the emergent call. It is shown in Rubinstein and Reiner (1991a)and in Bowie and Carr (1994)that a down-and-in bond can be synthesized from standard and binary puts. Also, a binary put, Ps(K), struck at K can be expressed as a function of standard puts in the following form: -

.

Ps(K)=

hmn

n

2

P K+-

l n

-P

K--

1 n

(71)

It turns out that the down-and-in bond with barrier H can be synthesized by a long position in two binary puts struck at K and a short position in (1/H) standard puts struck at H. Hence, an investor who is short (1 H) standard puts at the initial time and covers his position when the barrier is hit, ensures a self financing transaction at the barrier. The down-and-in bond value is

388

Boi(H) If P(H) is replaced can be

written

OPTIONS,

FUTURES

2Ps(H)

P[H]

=

AND EXOTIC

DERIVATIVES

BINARIES AND BARRIERS

SUMMARY

(72)

by an average of puts struck just above and below H, then Boi(H)

as

Î

i

(PH PH

+ Boi(H) If one substitutes puts:

=

2Ps(H)

(71) in (73), it

-

n

_ _

n

lirn

(73)

2

-

I

4

is clear that the down-and-in bond is only a function of

standard

Ba(H)

=

lim

(7

l

1 + -

PH

n -

/

'i

-

m

l

PH

n

l

(74)

The rebate, Rey, associated with a down-and-in call can be synthesized from a long position in Rey standard bonds paying $1 at expiry and a short position in Rey down-and-

in bonds: Ru(H)

=

Reye

"' -

RevBo,(H)

(75)

As a consequence, the rebate can be expressed solely in terms of standard puts. The above analysis now allows the valuation of a down-and-in call when the strike price is less than the barrierHence, a short position in a down-and-in call (strikeK, barrier H > K) can be hedged by a long position in a standard put (strike price K) and in H K down-and-in bonds.

-

2

This chapter analyzes simple and complex digital options, digital range options and barrier options. In particular, it illustrates by simple examples, elementary digital call and put options, all-or-nothing options and one-touch all-or-nothing options. Then, some examples of complex binary and range options and structured currency products are presented: range binaries, rebate range binaries, mandarin collars, mega-premium options, limit binary options, boundary binary options, wall options, knock-out wall options, mini-premium options, hokey-cokey options, boolean digitals and corridors. Also, differup-and-in, down-andent kinds of barrier options are studied. Examples of up-and-out, out and down-and-in calls and puts are given. Then, examples of synthetic forward structures in currency OTC markets are analyzed. These structures are based on combinations of barrier options. Some characteristics of barrier options are studied. It turns out that binary options are useful in the representation of the pay-offs of simple in Pechil, the chapter analyzes and and complex options. Using the representation provides simple valuation formulas for down-and-out options, switch options, corridors and knock-out range options. Second, analytic formulas for the pricing of binaries and barriers are presented. Also, a framework for the analysis and valuation of continuous strike options, continuous strike range options and soft barrier options is presented. of Finally, we present some static hedging techniques for the analysis and valuation barrier options.

POINTS

Example

• •

An investor selling a down-and-in call with a strike price of 4 and barrier of 7 can hedge his short position by the purchase of a standard put struck at 4, and 7 4 3 down-and-in bonds with barrier 7. If the spot remains above the barrier, the down-and-in call, the down-and-in bonds and the put are worthless at maturity. If the barrier is hit, the two down-and-in bonds and the standard put are used to 4). Since the exact hedge for finance the purchase of a standard call (strikeprice the sale of a down-and-in call is a portfolio with a long position in a standard put (strikeprice 4) and 3 down-and-in bonds with barrier 7, then the value of the downand-in call before the barrier is hit is equal to that of the portfolio. =

=

• • • • • • • • • • • • •

If the hedge involves

just standard

cix(K, H)

=

puts, the following relationship

(H - K)ßoi(H)

+ P(H)

H

applies: >

K

•

(76)

where

Boi(H) is given in (74). Bowie and Carr (1994) show that for n the bid-ask spread of these options

• • •

=

10, the relative

error

in the hedge liestithin

389

• •

FOR DISCUSSION

What is a binary option? What is a barrier option? What is a path-dependent option? What is a path-independent option? What is an all-or-nothing option? What is a range binary option? What is a rebate range binary? What is a mandarin collar option? What is a mega-premium option? What is a limit binary option? What is a boundary binary option? What is a wall option? What is a knock-out wall option? option? What is a mini-premium What What What What What What What

is a hokey-cokey is a corridor? is the pay-offof is the pay-off of is the pay-off of is the pay-offof is the pay-off of

option? cash-or-nothing options? asset-or-nothing options? gap options? the down-and-in cash (at hit) or nothing when S> H? the up-and-in cash (at hit) or nothing when S< H?

390

• • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • • •

FUTURES

AND EXOTIC

DERIVATIVES

the down-and-in asset (athit) or nothing when S> H? the up-and-in asset (at hit) or nothing when S < H? the down-and-in cash (atexpiry) or nothing when S> H? the up-and-in cash (at expiry) or nothing when S< H? is the pay-off of the down-and-in asset (at expiry) or nothing when S> H? is the pay-offofthe up-and-in asset (at expiry) or nothing when S< H? is the pay-off of the down-and-out cash or nothmg when S > H? up-and-out cash or nothing when S< H? is the pay-offofthe is the pay-off of the down-and-out asset or nothing when S > H? is the pay-offof the up-and-out asset or nothing when S< H? What is the pay-off of the down-and-in cash-or-nothing call when S > H? What is the pay-off of the up-and-in cash-or-nothing call when S < H? What is the pay-off of the down-and-in asset-or-nothing call when S > H? What is the pay-off of the up-and-in asset-or-nothing call when S< H? What is the pay-off of the down-and-in cash-or-nothing put when S > H? What is the pay-off of the up-and-in cash-or-nothing put when S< H? What is the pay-off of the down-and-in asset-or-nothing put when S > H? What is the pay-off of the up-and-in asset-or-nothing put when S< H? What is the pay-off of the down-and-out cash-or-nothing call when S > H? What is the pay-off of the up-and-out cash-or-nothing call when S < H? What is the pay-off of the down-and-out asset-or-nothing call when S > H? What is the pay-off of the up-and-out asset-or-nothing call when S< H? What is the pay-off of the down-and-out cash-or-nothing put when S > H? What is the pay-off of the up-and-out cash-or-nothing put when S < H? What is the pay-off of the down-and-out asset-or-nothing put when S > H? What is the pay-off of the up-and-out asset-or-nothing put when S < H? What is the pay-off of a down-and-m call? What is the pay-off of an up-and-in call? What is the pay-off of a down-and-m put? What is the pay-off of an up-and-in put? What is the pay-off of a down-and-out call? What is the pay-off of an up-and-out call? What is the pay-offof a down-and-out put? What is the pay-off of an up-and-out put? What is the link between standard and exotic options and binary options? What is a continuous strike barrier option? What is a continuous strike range option? What is a soft barrier option? How can we implement static hedgmg of barrier options?

What What What What What What What What What What

• •

OPTIONS,

is the pay-off of is the pay-off of is the pay-off of is the pay-offof

?

Look back Optio ns

CHAPTER

OUTLINE

This chapter is organized as follows: 1. Section 19.1 analyzes and gives some examples of lookback options. 2. Section 19.2 presents valuation formulas for European lookback options. In particular, standard lookback options, extreme options, limited risk options and partial lookback options are valued. Also, simulation results are provided. 3. Section 19.3 presents valuation formulas for an exotic timing option, with simulations. 4. Section 19.4 studies the strike bonus option.

NTRODUCTION Some financial contracts allow the holder to receive, at the maturity date, a pay-off dependmg on the maximum or the minimum of the reahzed values of the underlying asset durmg the contract's life. The underlymg asset may be a spot asset, a forward or a futures contract, commodities, indices and so on. These options are negotiated either m organized markets or m the OTC markets and may be embedded in some contracts issued by financial institutions and firms. Hence, a lookback call is defmed as an option whose strike price corresponds to the minimum price recorded by the underlymg asset durmg the option's life. A lookback put is defined as an option whose strike price corresponds to the maximum price recorded by the underlying asset during the option's life. These options are interesting but are more expensive than standard options. Therefore partial lookback options are designed to reduce the costs attached to lookbacks while preserving their main characteristics. This can be easily achieved by reducing the period during which the underlying asset is monitored. Hence, a partial lookback option allows its holder to purchase a fraction of the underlying asset above a certain minimum. .

.

.

.

.

.

.

'full'

.

There are many

reasons

.

.

which

incite investors

.

.

.

to buy and sell lookback

options.

Perhaps the main aim is to gain a lot of money and to avoid transaction costs resulting from the replication of these options by some strategies. Generally, investors who resort knowledge about an asymmetry in the to lookback options think that they have of the dynamics of the underlying asset price, the price paths and the 'superior'

'volatility

volatility',

among

other thines.

OPTIONS,

392

FUTURES

AND EXOTIC

DERIVATIVES

When an investor has some prior expectations about future prices and believes that upward and otherwise they will downward, he might have some incentives to purchase and hedge at-the-money lookback calls and to sell and hedge lookback puts. Also, when his prior expectations about future prices let him believe that they will upward and downward, he might be incited to purchase and hedge lookback puts and to sell and hedge lookback calls. This strategy is based on the fact that the market tends to go up slowly and fall suddenly. Goldman, Sosin and Gatto (1979)and Goldman, Sosin and Shepp (1979) analyzed and valued standard lookback options which entitle the holder to buy the underlying asset at its realized minimum value during a certain period Harrison and Kreps (1979)and Harrison and Pliska (1981)showed that in complete markets, it is possible to implement a self-financing strategy to duplicate the pay-off of an option using the underlying asset and riskless discount bonds. The current option value is given by its expected pay-off discounted to the present under the appropriate probability measure. Hence, these options can be priced in a B-S economy using martingale techniques. They can also be valued using lattice approaches and numerical techniques· Whatever approach is used, the specificities of these options are reflected in their terminal and intermediate pay-offs. 'drift'

'gap'

LOOKBACK

393

OPTIONS

When the option was purchased gold was $405, and on the maturity date $395. If the highest price achieved during the option's life is $430 and the option is cash-settled at maturity, then the settlement value is

'drift'

(430-

'gap'

19.1

ANALYSIS

OF LOOKBACK

19.1.1

Examples

of Standard

Lookback

A lookback option gives its holder the right to buy (sell) a fixed amount of an asset at the best price which occurs over the option's life. Hence, a lookback call (put)involves the right to buy (sell)at the lowest (highest)price. The Lookback

Lookback

The holder of a nine-month European lookback call has the right to buy at maturity period. 10 000 ounces of gold at the lowest price which is attained over the nine-month The contract stipulates as well that the holder is entitled to the favorable strike price of $377 whenever the asset price fails to reach this level. Suppose that when the option is purchased gold was $405 and on the maturity date $395. Suppose that the lowest price achieved during the option's life is $380. If the option is cash-settled at maturity then the settlement value is

in-the-Money

Lookback

min

(380,377)]l0 000

Put

European lookback put has the right to sell at maturity The holder of a nine-month period. ounces of gold at the highest price which is attained over the nine-month

14¾

=

$180 000

Puts

The holder of a nine-month European lookback put has the right to sell at maturity 10 000 period. The ounces of gold at the highest price which is attained over the nine-month stipulates as well that the holder is entitled to the favorable strike price of $432 contract whenever the asset price fails to reach this levet Suppose that when the option is purchased gold was $405 and on the maturity date $395. If the highest price achieved during the option's life is $430 and the option is cashsettled at maturity, then the settlement value is

[max(430, 432)

19.1.2

The Floating

Strike

- 395]l0 000 with

=

$370 000

floating strike prices or with fixed strike

Lookback

For a floating strike lookback, the strike price is unknown before the maturity date. For a corresponds price realized by the to the minimum underlying asset durmg the option's life. For a lookback put, the strike price corresponds to the maximum price realized by the underlying asset during the same period. Hence, a A floating strike lookback call gives the right to the holder to buy at the minimum. floating strike lookback put gives the right to the holder to sell at the maximum. When the underlying asset price drops and then rises, the call will pay-off from the lowest price realized. Since the strike price drops with the underlying asset value, these

lookback call, the strike price

(395-380)l0000=$150000

$350000

Calls

There are two types of lookback options: pnces.

Call

The holder of a nine-month European lookback call has the right to buy at maturity 10 000 ounces of gold at the lowest price attained over the nine-month period. Suppose that when the option is purchased gold was $405/ounce and on the maturity date it is $395. Suppose that the lowest price achieved during the option's life is $380.If the option is cash-settled at maturity, then the settlement value is

The Lookback

In-the-Money

OPTIONS

Options

=

If we denote by S,s,, and Sas, the lowest and highest underlying asset price achieved at any instant t between the current time to and the maturity date T, then the lookback call pay-off can be written as Sr The standard lookback put pay-off is Smax Sr. - Sass.

[395-

In this section, we describe some lookback options with respect to their terminal and intermediate pay-offs. These options are often structured in order to offer, for a higher initial premium, some or all of the potential value of the option during its life.

395)l0 000

394

OPTIONS,

FUTURES

AND EXOTIC

options offer a suitable solution to the market entry problem and are more expensive than standard options. When the initial underlying asset price corresponds to the minimum value recorded during the option's life, then the floating strike lookback call pay-off is equal to that of a standard call.

19.1.3

Fixed Strike

LOOKBACK

DERIVATIVES

.

|

Lookback

.

ANALYTIC OPTIONS

FORMULAS

FOR LOOKBACK

When the dynamics of the underlying asset are given by the following familiar equation:

=

max

[0, Sr -

K,

f(.) -

K]

dS,

P

=

max

[0, K

- S,, K -

f(.)]

S,

(1) (2)

rS, dt + o S, d W,

=

(3)

then

and in the following form for the put:

value

Long a lookback call.

19.2

Strategies

C

f(.) is the

•

.

.

The value of path-dependent options is a function of the underlying asset, time, the strike price, and a function f(.) specifying the option in question. Most lookback options are defined as calls or puts for which the pay-off can be written in the following form for the call:

where

Short a synthetic forward contract at spot, i.e. selling a European call and buying a European put.

It is convenient to mention the Arcsine law for lookback options. The Arcsine law refers to the distribution which predicts the timing of the minima and the maxima of the underlying asset values during a certain period of time. This means that the underlying asset price attains a minimum or a maximum near the beginning or the endpoints of the lookback penod rather than at some time in the middle of the period. Hence, when an investor sells a lookback call and a lookback put, it is not impossible that one extreme value be attained at that time and the other extreme value at the maturity date. For more details, see Garman (1993).

.

19.l.4

as follows:

•

Â

For fixed strike lookbacks, the strike price is known m advance. The call option pay-off is given by the difference between the highest value of the underlymg asset during the option's life and the fixed strike pnce. The put option pay-off is given by the difference between the fixed strike price and the lowest value of the underlying asset recorded durmg the option's life. When the final underlying asset price is the maximum value recorded during the option's life, then the fixed strike lookback call's pay-off is equal to that of a standard call. .

395

A lookback put strategy is constructed

.

Lookbacks .

OPTIONS

-

1

=

Soexp

[(r

2

a2)t + aW,]

(4)

(seeChapter 2 for more details). Using the following notations for the over the interval [ti, t2þ

maximòm

and the

minimum

M

of the underlying asset, determined in a way defined in the issue of

max {Ss

=

é

[ti, t2D

=

m

min {Ss

é

Di, t2]}

(5)

the financial asset.

Hence, a lookback option is an option for which the function f(.) corresponds to the maximum or the minimum of all values attained by the underlying asset during the option's life. The maximum or the minimum can be calculated in a continuous time .framework as it can be sampled at different times. An average strike lookback call (put) of the underlying asset is defined by the function f(.) giving the minimum (maximum) value over the option's life A lookback call strategy is constructed as follows: •

•

Long a synthetic forward contract at spot, i.e. buying a European call and selling a European put. Long a lookback put.

value of S which can be achieved by The pay-off of this strategy is the maximum exercising the lookback put and by paying the strike price K, which corresponds to the

then X,

Y,

19.2.1

=

In

=

So

Standard

ln

=

max {X|s

Lookback

=

So E

[0, t]}

a2)t+

(r y,

=

ln

a W,

=

So

(6) min {X)s E

[0, t]}

(7)

Options

Since the pay-off of a standard European lookhack call at the matunty date S, - m its current value is

is

given by

FUTU

OPTIONS,

1996

c

SoN(d")

=

- e

"m

N(d"

ES AND EXOT1C

DERIVATIVES

LOOKSACK

OPTIONS

on)

-

e "(M

=

c

397

"M - e

SoN(d') - K) +

N(d'

-

an)

2r/
+e

-d

N

So

1

d"=-

ln

a

+rT+)a2t

m°

(9)

Since the pay-off of a standard lookback put at the maturity date current value is -SoN(-d')+e

e

.

is

given by M

S, its p

=

-

S

2r

+e'tiV(d')

-

114)

N

M

+e"N(d')

d' a

(10)

When K

>

p

with

_,,7,2

a2

So

So

-

2r

-K

N

SoN(

d") + e

2r

-d+-Ra

-e"N(-d)

(15)

the put's value is

m =

"

on)

+ e "KN(-d

SoN(-d)

e +e

d'

Let (K - m )* stand for the pay-off of a put on the minimum at time T. When K
UN)

N(-d'

M

(8)

So -

On the Minimum

So

.

p=

'

-errN(-d")

+

e "(K

-

m°

)-

"m"

an)

N(-d"

2ra2N

d'=

rT+(o2T

ln

(N

-e"N(-d")

-d"+

So

+e

6)

Note that these options correspond to ordinary options as in B-S formulas plus another term corresponding to the specificities of their pay-offs. 19.2.3 19.2.2

Options

on Extrema

Limited

Risk Options

When m and K are constant, then the pay-off of a limited risk call at time T is

at the maturity

The pay-off of a call on the maximum K M the current call price is c

=

=

c

On the Maximum

SoN(d) - e "KN(d

date i

is

(M, r

-

K)*. When

The call's current c

- o

=

value

So[N(d) - N(d

N

So -

K)*1M tri

is nil when M To > m. When M°To

)] -

e "K[N(d

)-

- a

N(d,

the call's current value is

Gm

-

on)]

) N

m

e

(Sr -

+e"N(d)

d-

2d, -d-

o

m

(12)

"K - e

N

\m)

2d,,, - d -

-N

d,,-

N

d,, -

-a

(18)

with with

d,, d

When K < M

=

the call's value is

ln

+

¡Û2

(13)

1 =

-

The pay-off at the maturity date of p

ln imited

=

So -

+ rT

a2 T

(19)

sk put is

(K - Sr)*14,

(20)

OPTIONS,

398

when

p

=

m° < m, the put -

is worthless. When

-2d,,

N

- n

m

DERIVATIVES

LOOKBACK

m, the put's current value

is

19.2.5

N(-d,,

)]

Tables 19.1 and 19.2 show the effect of changing maturity date and interest rate on standard lookback options. Tables 19.3 and 19.4 show the effect of changing maturity

e "K[N(-d

So[N(-d) - N(- d,,,)]

AND EXOTIC

FUTURES

)-

a

d

d,

N

-

+ a

Simulations

TABLE 19.1 Effect Maturity Date: m, =

-

N

K

-2d,,

d

-d,,,

N

-

399

+a

(2r/ol)+l

e

OPTIONS

(21)

These options are issued in foreign exchange markets and also in stock index markets.

19.2.4

Partial

Options

Lookback

date their pay-off

These options differ from lookback options smee at matuilty (S, - Am

=

c

with 1

)

is

d"-

-a

-2r/a2

a2 -2r

e

Se

AS

N

-mo,

-d

ln(A)

,,

-

o

2r -vT a

,

I Y 2r

N

d,,

=

0.2, r

in the 0.1

=

(5, T)

0.25

0.5

I

80 85 90 95 100 105 I 10 115 I20

31.77

31.63 22.19 16.37 13.56 13.19 \4.69 17.54 21.29 25.59

32.95 25.41 21.17 19.45 19.64 21.27 23.95 27.38 31.34

TABLE Interest "N

-Am°e

d"-

«

21.02 \3.78 9.85 8.95 10.50 13.74 17.94 22.63

22)

l

The current value of a partial lookback call is c=SoN

of a Change

100,

ln (X)

19.2 Effect Rates: m, =

of

a

100,

«

=

Change in 0.2, T 2 =

(5,r)

/0%

is%

IB%

80 85 90 95 l00 105 110 115 120

37.03

48.59 38.91 34.49

57.07 43.64 37.79 36.25

3\.71 29.16 28.57 29.45 3\.4l 34.17 37.53 41.33

33.34 34.29 36.58 39.77 43.54 47.7\

37.24 39.73 43.14 47.13 51.47

At the rnaturity date, the pay-off of a partial lookback put is p

=

(AM

with 0

- Sr)

A

(24)

l

TABLE 19.3 Effect of a Change 10% 25%, r So 100, « =

The current value of a partial lookback put is p

=

SoN

AM e "N

-d'

2r

S

e-cr 2r/

M

d'

<

N

d'

ln(A)

2r

o

a

-

e"12'7"

d'

N \

«

(25) When A

=

1, these options become standard lookback options.

in the Maturity

Date:

M,

=

=

=

(K, T)

0.25

0.5

0.75

I

3

80 85

28.95 24.07 19.19 14.32 9.44 5.33 2.69 l.22 0.49

33.19 28.43 23.68 18.92 14.17 9.89 6.58 4.18 2.45

36.64 3\.99 27.35 22.71 18.08 13.78 I0.22 7.37 5.18

39.64

57.28 53.58 49.87 46.l7 42.46 38.84 35.39 32.\4 29.08

90 95 100 105 I I0 I 15 120

35.\2 30.59 26.07 21.54 17.29 I3.6 I 10.52 7.99

100,

in Volatility: M,

19.4 Effect of a Change 10%, T 0.5

TABLE 100, r

AND EXOTIC

FUTURES

OPTIONS,

aoo

DERIVATIVES

100, So

=

LOOKBACK

OPTIONS

(gg

TA

=

E

.8

Efl

8.51

28.43 23.68 18.92 14.17 9.89 6.58

7.06

4.39 1.80

4.03

3I.34 26.59 22,15 18.3\

onkbbaecSkl9 icdalls 5 aind 19.6ad on the map ereast eect odf r ia cha i g in ate volatility on imite and 19.8 show the effect of changing time to maturity and optionvalues.

is

=

g,

=

r)

I.0

Effect of a Change in Interest 0.25 0.2, T 100, « 0.10

0,l2

0./4

0./6

8 9750

9.2I 4.89

9.47 5.15

9.74 5.38

0.5

TABLE 19.6 Effect of a Change 100, r 0.I, T 0.25 100, m,

I.00 I.05

in Volatility:

0.is

0./6

0.25

0.30

7

7,48

10.77 6.57

12.56 8.44

3.15

TABLE 19.7 Effect of a Change in Time 10%, 10%, « 100, r to Maturity: S =

m,

=

100, K

(5, T)

90 95 io0 105 i 10 Ils

S

=

=

=

o)

Rates:

=

=

=

=

=

30 0.5

0.75

9782

667

4.27

7.39 4.86 2.80 \.42 0,64 0.26

6.18 5.1I 3.89 1.75 f f.l5

0.25

.83

4.75 4.73 4.34

3.72 3.02 2.33 l.73

=

30

=

s

=

0.25

16.56 12.18

7.47 3.44 l.08 0.18

s

=

0.5

s

=

0.75

13.96 H.44 7.72 4.31 1.96

6.06 6.09 5.4I 4.38 3.28

0.72

2.30

0.01

0.21

I.53

0.00

0.05

0.96

c

19.3 ANALYTIC OPTION:

TABLE 19.5 S 100, m,

100, K

80 85 90 95 100 105 I10 H5

36.10

15.85 I2.2l

=

(5,o)

45.61 40.85

39.25 34.49 29.74 24.98 20.22

33.19

30.29 25.54 20.78 I6.03 11,27

27.53 22.78 18.02 13.26

80 85 90 95 100 105 I 10

m,

40%

30%

20%

IS%

10%

(K, a)

TM= Oa.2 hrangel0°o

0

=

=

FORMULA FOR AN EXOTIC TIMING THE BELLALAH-PRIGENT MODEL

This section compares the valuation of a lookback put option with that of an option which, at pay-off, gives its holder the difference between the maximum value recorded during the option's life and an initial value based on the underlying asset price at the time of initiation. We call this latter instrument an exotic timing option. Although the pay-off on the exotic option corresponds to a lookback option, a major advantage is that, in contrast to the lookback option, the holder does not run the risk of zero value when the maximum value is achieved at maturity. The exotic timing option has some appeal to investors, especially those who believe they have special skills at market timing, because even a small rise in the asset price allows the opportunity for the investor to lock-in a profit. This exotic option gives its holder the difference between the maximum value of the underlying asset during the option's life and the initial asset value. Since the highest price of the asset during the option's life cannot be less than the initial price, this difference cannot be less than zero. The value of this exotic timing option can be decomposed into two parts: a lookback put option and a forward contract on the underlying asset. The lookback option, which has a known solution, gives its holder the difference between the maximum value of the underlying asset during the option's life and the asset value at option expiration. Similarly, this option can not have a value less than zero. The lookback put is then like a conventional put with an exercise price equal to the maximum value of the underlying asset during the option's life. The key difference between the exotic timing and lookback put options is that the va\ue of the maximum difference is based on an initial asset price for the former and a terminal price for the latter. The exotic timing option has some appeal to investors, especially those who believe that they have special skills at market timing, since the option is at-the-money when it is purchased and will finish in-the-money if the asset price moves higher any time during the life of the option. The exotic timing option can be created as an independent instrument or can be

OPTIONS,

402

FUTURES

AND EXOTIC

DERIVATIVES

embedded in bond issues where the underlying assets may be as diverse as an index price, a share of stock, a currency, an interest rate, a commodity or any other tradable asset.

LOOKBACK

19.3.2

OPTIONS

Simulations

40;

and Option

Characteristics

It is useful to compare the exotic timing option sensitivities with those of lookbacks and standard puts. For the sake of clarity, the exotic timing option is rewritten as Analysis

19.3.1

and Valuation

Coax(0)

The main difference between lookbacks and the exotic timing option can be simply illustrated using a three-month option on the CAC40 index. Assume the CAC40 is at 1800 and over a subsequent 90-day period it reaches 2200. This latter value corresponds to the maximum. The net change is +400 points multiplied by the size of the contract, 200 FF, which gives a total gain of 80 000 FF. An investor who buys an exotic option contract will have 80 000 FF. Now, assume that the investor who believes he has superior market timing skills buys the lookback option on day 30 and also buys an exotic timing option on the same day .

is at 1400. At maturity, the pay-off from the lookback is zero. The pay-off from the exotic tíming size of the contract, o tion is given by the net change (+800 points) multiplied by the 200 FF, which gives a total gain of 160 000 FF. Assume 0 denotes the initial time, T denotes the contract's maturity date, and r denotes T r. S represents the underlymg the present time. The remaining time to maturity is t asset price, r is the riskless interest rate, and a is the instantaneous standard deviation. The maximum value of the underlying asset in a given time interval is

when the mdex

.

=

M(r)

-

max {S(ö)}

=

(26)

When pricing this exotic timing option, the value of an asset paying the realized maximum over a given period must be determined, then the current value of the underlying asset is subtracted from it. In a risk-neutral world, the option value is given by (27) Ceux(r) e "E [M(T) - S(T) + [S(T) - S(0)]\F,] probability risk-neutral where the conditional expectation Eç is taken with respect to the and the available information F,. The pay-off corresponds to that of a lookback put =

Q

initiated at time 0 and a forward contract. The exotic timing option value is S(0)e-"] P(r) + [S(r) where P(r) is the lookback put given by equation (14)in Conze and Viswanathan At r 0, equation (28) is rewritten as

Cs,x(r)

=

(28) (1991).

"]

P(0) + S(0)[I - e

=

(31)

This expression demonstrates that the exotic option is more expensive than lookback puts and standard puts with a strike price S(0). When the time to maturity tends to infinity,

Ceux(0) - P(0)

S(0)

=

(32)

.

This result demonstrates that a long position in a perpetual exotic timing option and a short position in a perpetual lookback put is equivalent . to a position in the initial underlying asset. The option s delta is greater than that of a lookback put and a standard put:

Acao

=

Amo, +

[l -

e

"]

(33) .

.

The gammas of the option and the lookback put are nil m this context. The option's derivative with respect to the volatility parameter is positive and equals that of alookback put. It is. given by Aa,so,

=

a

-S(0)e

e"N(di)

_

-

N di

2r

(34)

r

The option's derivative with respect lookback put:

to the interest rate parameter "

Pc.a=pe
8

(35)

to the time to maturity

=

is greater thanthat of a

Geo, + rS(0)e

is positive and greater than

r

(36)

.

illustrative

For purposes, formula (29) is used to simulate the exotic timing option values. When the underlying asset price S is 100, the results for different levels of r, o and T are reported in Table 19.9. Simulations show that exotic option values are increasing functions of the underlying asset price, the interest rate, the volatility parameter and time to maturity.

=

Coax(0)

=

S(0)[1

- e

"] -

e ,rS(0)N(-d3

S(0)N(-d,)+

'

+ S(0)e

c"N(di)

"

2r

- N di -

where

di

r

og

=

/2ry

+ o

il

)

19.4

ANALYSIS AND VALUATION BONUS OPTION

OF THE STRIKE

09) In the context of lookback strategies, the roll-over strategy gives rise to the strike bonus option. As soon as a short position in a lookback call is initiated, the investor must buy a European OTC call with the same strike price and maturity date as the lookback option. Therefore, he must fund the price of this standard cal. When the underlying asset price attains a new low, the position must be adjusted by selling the OTC call and buying a new call with the corresponding new lower price and a shorter time to maturity This 'old'

and N(.)

is the cumulative normal density function.

404

OPTIONS,

TABLE 19.9 Comparison tion Values, Cmax(0), with Values, P(0)

(a)Change

in interest

(T

rate

FUTURES

of Exotic Lookback

=

0.2, S

(r

=

0.I, o

0.2, 5 =

=

C,c,(0)

P(0)

8.49 I0.98 I 2.48 I 5.88 18.98

6.02 7.058 7.607 8.654 9.463

0.25 0.40 0.50 0.75 I.00

following notation:

use the

(1993),we

peo: the

cso,

CLos

Lo:

=

'

=

Furthermore

we

define

100) ,

T

Option

value of a European lookback call (put), the value of an American lookback call (put), csso, Psso: the value of a strike bonus call (put), S(r): the current underlying asset price at time r, T - r: the option tenor t d: the underlying asset yield rate, with ö r - d.

100)

6.02 5.\3 4.4 1 3.83 3.36

9.29 9.89 10.59

(b)Change in time to maturity

Following Garman

P(0)

8.49 8.82

0.i0 0.\5 0.20 0.25 0.30

41115

OPTIONS

The Strike

19.4.1

Timing OpPut Option

C,,,(0)

r

LOOKBACK

DERIVATIVES

*

0.25, a

=

AND EXOTIC

r}}

H, max {Ho, max {So, 0 sus guarantee price Ho, min {Lo, min {So, 0 Gus : L, tee price Lo. =

r}}

=

When t

=

(T

0.25, r

=

=

0.1, S

C,,,(0)

a

8.49 10.54 I 2.68 |4.88 17.06

0.20 0.25 0.30 0.35 0.40

=

100)

ÑLo

6.02 8.071 I0.2 I i I2.4I I l4.591

At any time before maturity, must be

CLo

strategy must be repeated until the lookback call's maturity date, each time a new low is attained. The investor must fund the costs of these successive roll-over strategies. The hedgeability of lookback options gives rise to what Garman (1993)calls the cash bonus' option, which the present value of all the roll-over replication fiows. Hence, a strike bonus option is simply a claim to the cash flows implied by the implementation of a roll-over when replicating a lookback option. The possibility of rise to the costs of the rolling over gives minimum attammg a new maximum or a new strategy and to the bonus' option. In order to finance and implement a roll-over strategy which replicates the lookback call's pay-off, the investor forms a portfolio which comprises a long standard call and a strike bonus option. This latter option is used to pay for subsequent roll-overs. Since the strike bonus option capitalizes the costs of the rolling over strategy, the value of the lookback option corresponds simply to the value of a strike bonus option and that of a standard option struck at the realized minimum or maximum: c + csro

(38)

H, - S,

=

lookback call option pay-off

a S - L,,

(39)

0, the terminal boundary condition is the same as the European lookback call, i.e.

=

At any time before maturity,

S, - Lr (40) T, the American lookback put option pay-off is

0 sus

PLo

'strike

=

(37)

S, - L,

=

T, the American

0 mus

CLo

CLo

price or the

and for the European lookback put

I

'strike

of the maximum

0, the terminal boundary condition for the European lookback call is

P(0)

At t

'capitalizes'

the higher

the lower of the minimum price or the guaran-

=

cea

(c)Change in volatility

=

At t 0, the terminal put, i.e. =

=

a H,, - S,,

boundary condition pLo

A slight modification

is the same as for the European

T

T

(41) lookback

(42)

of the following lookback pay-off: max [Sr - min (Lo, L,), 0]

(43)

gives the fixed lookback call pay-off: cea

=

max

[ max

(Ho, H,) - K, 0]

(44)

When He a K, the value of a fixed lookback option is equivalent to that of a portfolio which comprises a lookback put and a forward contract. The fixed lookback option can be expressed as

cas

=

=

(Ho, Hr)

max

[ max

max

[Ho - S,, H,

e

1

407

In (S L) + =

(¢ + ö +

(«2

(54)

(45)

K]

-

OPTIONS

Zi

19.4.3 '"N(Ze) - e

N(Z, + lay't)

" =

LOOKBACK

max [He - K, H, - K]

=

S,] + [Sr

-

1

DERIVATIVES

Garman gives the following formula for the strike

Using standard arbitrage arguments, bonus call when i is not zero: c suo

K, 0]

-

AND EXOTIC

FUTURES

OPTIONS,

406

\L

(46)

The Hedging of Strike Bonus Options

The decomposition of a European lookback option into a standard option and a strike bonus option gives the following formula for European lookback calls and puts: CLo

=

C

CSBo

with

ln(S L)+(ö+)a2)t

(55)

(47) UMMARY

The hedging ratio of the strike bonus option is

Asuo

-1)

"(1 =

X

e

When A is equal to zero, there is a singularity is used instead for the strike bonus option: csso

"a

Se

=

Ao

N(Z

\L/

)-

(49) (50)

POINTS

(48)

N(Z,)

hole' and the following formula

'black

or a

t[n(Z,)

e

This chapter defines and analyzes by simple examples several forms of path-dependent options, i.e. full and partial lookback options with Roating and fixed strike prices. First, standard lookback options, extreme options, limited risk options and partial lookback options are analyzed and valued. Simulation results are presented. Second, valuation formulas and simulations are presented for the exotic timing option. Lastly, the strike bonus option resulting from roll-over replication strategies of lookback options is analyzed and valued with respect to lookback options.

+ Z, N(Z,)]

The hedging ratio for the black hole is given by Asco

=

"[a

n(Z,)

e

(Zio

-

1)N(Z,)]

Garman gives the following formula for the strike

Using standard arbitrage arguments, bonus put when Ais not zero:

• • •

"

psso

[N(Z,,

e

=

+ la

d)

-

1] - e

[N(Z,)

l]

(51)

• • •

with

In (S/ L) + (o +

Z,,

=

-

jo

•

2)t

(52)

• • • • •

19.4.2

The Fractional

•

Bonus Option

•

1 for a call. This A variant of lookback options has a strike price equal to ¢L, for ¢> corresponds and to a fractional Viswanathan and (1991) option is described in Conze is The formula bonus option. =¢

ce with

e

"

N(Z,

+Aa

)-e-*¢N(Z,)

(53)

•

•

FOR DISCUSSION

Give an example of a standard lookback call. Give an example of a standard lookback put. Give an example of an in-the-money standard lookback call. Give an example of an in-the-money standard lookback put. What is a floating strike lookback? What is a fixed strike lookback? What are the main lookback strategies? What is the pay-off of a call on an extreme value? What is the pay-off of a put on an extreme value? What is the pay-off of a limited risk call? What is the pay-off of a limited risk put? What is the pay-off of a partial lookback call? What is the pay-off of a partial lookback put? What is the strike bonus option? What is the relationship between a strike bonus option, a lookhack and a standard option? What is a fiactional bonus option?

20 Asian and Flexible Asian Options

CHAPTER

OUTLINE

This chapter is organized as follows: 1. Section 20.1 reviews the important results regarding the valuation and hedging of Asian options. It presents also simulation results of option values, since the approaches used for the valuation of Asian options are also used in the valuation of basket options. 2. Section 20.2 presents the framework for the valuation of these options. 3. In Section 20.3, the concept of flexible geometric Asian options is extended to flexible arithmetic Asian options. Therefore, the general context for the analysis and valuation of flexible arithmetic and geometric Asian options is presented. An analytic approximation is provided for nexible arithmetic Asian options. The solution is based on an extension of a method used to approximate standard arithmetic Asian options with their corresponding geometric Asian options. It is convenient to note that most studies approximating arithmetic Asian options with geometric Asian options use either an arbitrarily fixed number of moments (Levy, 1990; Turnbull and Wakeman, 1991) or a reduced effective strike price (Vorst, 1992). The study of Zhang (1995) is based on an efficient approximation for an arithmetic mean based on a Taylor series approximation method to the corresponding geometric mean.

INTRODUCTION The deregulation of financial services, the competition pressure, the absence of talents for financial products and the ease with which banks can mimic their competitors' ranges of products irreversibly push fmancial institutions into an endless and vital quest for the ultimate option package, hybrid security or swap for their customers. An increased emphasis has been placed on the ability to design new products which solutions bring more effective to increasingly complex financial problems. Lookback, contingent, compound, knock-out and balloon options now belong to the ever-increasing lexicon of exotic assets.

OPTIONS,

410

FUTURES

AND EXOTIC

DERIVATIVES

a new class of derivative products has been added to this already impressive list, namely the Asian options, also called average rate options. These options are issued with a special feature: their pay-offs depend on the average price of the underlying asset over a fixed period, leading up to the maturity date. These options appear in a straightforward form or may be implicit in a bond contract. Average price options or Asian options, like many other exotic options, are gaining in popularity in the foreign exchange market, interest rate and commodity markets. Asian options and their final pay-off is a function of the average options are path-dependent values of the underlying asset in the past. These financial innovations are traded in over-the-counter markets and enable investors to accomplish several hedging strategies. Asian options allow the hedging of a series of cash flows. The averaging process gives the underlying asset price on a of overcoming the problem of manipulating means a of price manipulation of reduction value contribute the options to particular day. Average the underlying asset at the maturity date. bond contracts and average Examples of these options include commodity-linked right to the holder to receive currency options. Commodity-linked bond contracts give the period or the nominal value certain commodity the average value of the underlying over a bond has a of the bond, whichever is higher. Hence, the holder of a commodity-linked position in a straightforward bond and an option on the average value of the commodity. The strike price is given by the bond's nominal value. An average currency call on sterling, for example, gives the right to its holder to of two quantities: zero or the difference receive at the maturity date the maximum averaged rate exchange arithmetic the daily over some specified period and the between

Quite recently,

strike price. of Asian options is When the average is based on the geometric mean, the valuation simple and closed-form solutions of the B-S type are easily obtained. This is due to the fact that the product of log-normal prices is itself log-normal. However, when the average is based on the arithmetic mean, the valuation of Asian options is rather complex since the sum of log-normal components has no explicit representation. Therefore, it is rather difScult to get a closed-form solution as in the B-S case. with exotic options has given rise to basket options, which are Risk management simply options on a basket of assets or currencies. The basket option is often cheaper than the total value of options on individual assets. That is why an option on the basket is more cost-effective than a basket of single options. The main differences between standard and non-standard options lie not only in the final pay-off but also in the path by which the settlement prices are attained, i.e. from above, from below or zigzag. Asian options, based on average values of the underlying asset, are actively traded in the over-the-counter market. The pay-off of these options depends on the way the average is calculated and on a prespecified observation frequency. When the average is arithmetic, the option is an arithmetic Asian option. When the average is geometric, the option is a geometric Asian option. asset is log-normally distributed and this makes It is often assumed that the underlying and geometric Asian options. In fact, even arithmetic an important difference between arithmetic have the same weighting scheme, the options Asian geometric and though the while the arithmetic average is not. Theregeometric average is log-normally distributed

ASIAN AND FLEXIBLE ASIAN OPTIONS

41 I

fore, it is easy to get the price of a geometric average option and it is hard to obtain an equivalent result for an arithmetic Asian option. Flexible Asian options are extensions of standard Asian options which are more nexiblewith regard to the weighting scheme. For example, for a travel company, an exporter or any economic agent facing a seasonal exchange rate risk, leading him to attribute heavier weights for weeks or months with greater cash flows, he is better off (in hedgmg) with flexible Asian options than with standard Asians. In fact, in the latter case the weighting is equal while in the former a heavier weight can be assigned for periods with higher cash flows. In the same way as for standard Asian options, there are two kinds of flexible Asian options: arithmetic and geometric. The word refers to the flexibility m giving weights to a series of observations. 'fiexible'

20.1

THE AVERAGE VALUATION

PRICE OPTIONS:

ANALYSIS

AND

Asian options are path-dependent options whose pay-off is based on an average. In some cases, the underlying asset of the option is an average (see,for example, Bankers Trust's average rate option on commodities). In other cases, the strike price is a floating one: it is computed as an average of the underlying asset prices (see, for instance, Lazard Frères' money-back gold note). In this latter instance, the average is calculated over the entire lifetime of the option and the option is known vanilla' as a average option; or, over a shorter time slot, close to the option's expiration date, the option is named starting' by Midland. The 'Asian contagion' has reached not only the currency and commodity options markets; it has spilled over mto the equity market. Although the original use of averaging was in the commodity sector to prevent any market squeeze just before expiry, corporate managers quickly figured out the kind of poison pills that could result from an astute use of averaging. Next to staggered board elections, supermajority provisions and dual class recapitalization, one can now find the Asian warrant. The French investment bank Compagnie Financière Indosuez and the French construction company Bouygues, for example, have successfully issued such warrants to protect themselves against potentially unfriendly bidders. Both companies issued a vast amount of warrants, mostly held by friendly investors. Bond markets also experience the 'Asian fever'. For instance, Oranje Nassau issued a bond with a maturity of eight years, at which the bond holder is deemed to receive the value, face plus the difference, if positive, between the average price of 10.5 barrels of Brent blend oil over the last contract year and face value. Oranje Nassau bond holders are thus more closely associated to the firm's prospects since these are linked to the oil price. Moreover, bond holders are protected against oil price manipulation because of the average feature. Such bonds that make interest and/or principal payments linked to a specific mdex or commodity, such as the price of oil or silver, are obviously attractive to institutions that are not permitted to invest directly in commodity options, and they can serve as a hedge for an issuer who is a producer of the commodity, Asian options are the appropriate hedging instruments for traders who transact continuously over finite time horizons. Indeed, Asian options provide an effective way of cappmg costs or placing fioors on aggregate profits. 'plain

'forward

AND EXOTIC

FUTURES

OPTIONS,

412

ASIAN AND FLEXIBLE ASIAN OPTIONS

DERIVATIVES

hedging. Companies are increasAverage rate options are also used in balance-sheet hedge their exposures. Hedgmg year-end to than rather rates ingly turning to average value for average rate fmancial accounting exposure, for instance, is an area of substantial products. issues. First, these options Valuing and hedging options on average values raises some of the underlymg asset the history pay-offs upon depend their are path-dependent since underlying is distributed logasset prices over the averaging period. Second, if the maturity date option's the if Third, will log-normal. not be normally, the arithmetic mean of the option can change properties hedging the period, averaging does not match the significantly. possible to get analytic solutions for If the geometric mean is used for averaging, it is arithmetic with the mean, no analytic solutions yet options on average values. However, n this section

five difhferentapproachaesdto

e

o9fan averaage value obptionaan

valuation

20.1.1

(1991), that

C-V

Viswanathan, Wakeham, T-W (1991), that of Conze and and Crouhy Briys (1994). Bouaziz, (1992),and that of

The put-call p[A(t),

K, t]

parity condition can be written for t,

=

r

-

p[A(t),

20.1.2

d,

S

for t,,

(N + 1)(1 -

er")

K, t]

tu - t. When t < to, this relationship

=

c[A(t), K, t]+ e'"

=

The Valuation

>

S(t)ez"°

becomes

eg(to-t+H(N+1))

-

(N + 1)(1

-

-

egH

K

(4)

Approaches

ssetnarnost opt onndprcying problems,

it is assumed that the dynamics of the underlying µS(t)dt + US(t)dW

=

<

ts

So_i exp

=

[(µ -

ja2)H

(5)

+ a ÑY,]

c[S(t), K, t]

" =

e

E*[max {M(ts)

- K, 0}

max {A(ts)

K, t]

=

0} max {K - A(ts),

K*, 0}]

(7)

t,: ti

=

to + iH for

where operator conditioned on [A(t) - S(t)] at time t under the risk-adjusted density function. This means that µ is replaced by r - d in the dynamics of the underlying asset price, i.e. dS(t)=(r-d)S(t)dt+oS(t)dW

(8)

Hence, to value the average option, we have to determine the distribution of M(ty). denote the density function by f*(.), then the average call value is

c[S(t), K, t]=

=

-

E e is the expectation

to

c[A(t), K, t]

(6)

.

e "E*[max

K*, 0}]=

[M(t)-

K*]f*(w)dw

K*

(9)

Since the function f*(.) is non-standard, numerical methods and Monte-Carlo techniques must be used to value this integral. The main drawback of these techniques is that they are time-consuming. An alternative approach is proposed by Levy and Turnbull (1992).They propose the following analytic approximation:

(2)

options to get the value of the other, smce In general, it is sufficient to value one of these between average parity relationship put-call is there options, a as with standard European

If we

oc {M(tN)-

=

p[A(t),

eg(N-m)H

where Y, follows a normal distribution with a zero mean and unit standard deviation. a U sing teh risk-neutral approach, the average option value is given by its expected payoff discounted to the present:

average defined as follows:

S(t,)

N + 1

[l -

µ and o are the drift and the volatility parameters. Under this assumption, S(t,) can be expressed in terms of S(tyi) as

at different points in the interval, N. -to)

l

e""

where

0 for t< to. and A(0) of N + 1 prices. These pricesdare In this context, A(ts) is the arithmetic average tx]. The time interval may be a ay, considered at equal intervals oftime Hbetween [to, prices. of This is the basic form average a week or a month. respectively The pay-off of the average call and put option is

with 0 mmsN

t - t., and r

=

dS(t)

Tis the option's maturity date, t is the current time, d is the continuous yield on the underlying asset, which the average is calculated. [to, ts] is the time interval over

m+

t,se as

of Curran

We will use the following notation.

=

ts

(3) where g

So

A(t)

<

(m+ 1)A(t)

c[A(t), K, t] + e

=

Analysis

The avera e is often calculated ...,N,whereH=(t i=0,l,2 We denote by A(t) the running

4g

E*[max {M(ts)

K*, 0}]

-

=

exp

[a + (82]N(di)

-

K*N(d2)

(10)

where

di

=

a

-

In(K*) + 32,

d2

=

di - S

(11)

OPTIONS,

414

If M(t) is distributed as its corresponding

G(t)

=

[S(ti)S(t2)

ASIAN AND FLEXtBLE AS1AN OPTIONS

AND EXOTICOERIYATIVES

FUTURES

The C-YApproach

geometric average, i.e. ...

S(tN)]"

St,)

415

"

(12)

...

then a and v correspond to the mean and standard deviation of log G(t) which are not easily calculated. The approaches presented here are attempts to approximate the risk-neutral probability distribution of the arithmetic mean price. We now present the different approaches to the valuation of an average value option: Kemma and Vorst, K-V (1990),Turnbull and Wakeman, T-W (1991),Conze and Viswanathan, C-V (1991)and Curran (1992)

Following Conze and Viswanathan (1991),let Mr,,, be the asset's average value in the and r the riskless rate of interest. When T, < T, the pay-off at maturity of a European average call is

interval ( Ti, T), o its volatility

K, 0]

max [Mr

¡Mrsr

or

- K)*

(16)

The pay-off of a European average put in the same context is K - M,

(17)

If we denote by Z and Ï the following quantities: The K-Y Approach Z= Using arbitrage arguments, Kemma and Vorst (1990) presented a dynamic hedging strategy, from which the value of the average option can be obtained. They showed that the average option's price is always less than or equal to the price of standard European option. Using Monte-Carlo simulation methods, they calculated the a price of an arithmetic average option and presented the following formula for the option value when the geometric mean is used:

E.max {G(T) - K, 0}

=

E{G(T)

G(T)

K}

>

{G(T) - K.prob

>

M

S

exp

r

72

72

2(T - T,)

(T - T,)2

(18) y

a

(19)

then, using standard arbitrage arguments, the average call value when

C

K}

ZN(di)

=

"N(d2)

Ke

-

Ti

<

0 is

(20)

with =

e/S(To)N(d)

-

KN(d

(T - To))

- o

d*=)(rd

The average put

2XT-T

In [S(To)/k]

(r

=

=

value

(15)

let Z'

Z' and =

ZN(-di)

=

=

=

0,

Soexp

.

.,

*

+ Ke

N(-d2)

(21)

(22)

o'

-r -

2

Ï' .

Ïv'Ì

Ï' be the following expressions:

where

S( T,) is the stock price at different instants T,, for i G(T) is the geometric average for S, G(T) [[I

di -

=

is P

In the same outext,

d2

,

(14)

+ a2)(T - To)

+ rT +)Ï2T

ln(Z/K)

di

12

(23) (24)

=

n,

S(T,)]MM

When Ti

>

0 the average call value is C'

The T-W Approach of European Turnbull and Wakeman (1991)presented an algorithm for the valuation arithmetic options. When testing it against Monte-Carlo methods, they found it to be accurate and not time-consuming. solutions for the pricing of European geometric options. They also derived closed-form results the of Kemma and Vorst Contrary to (1990), they proved the following result: when the option's maturity is less than the averagmg penod, the price of an average value option can be greater than that of a standard European option.

Z'N(d')

=

- Ke

*

N(d')

(25)

.

with

d'

ln(Z'/K)+rT+ =

Z'2T d'

,

=

41 -

(26)

and the average put value is y, Table 20.1 simulates

_

Z'N(-d')

+ Ke

"

N(-d')

option prices using the C-V formulas.

(27)

TABLE 100, r

=

20.1 0.I,

AND EXOTIC

FUTURES

OPTIONS,

416

Prices: C-V Option 90 days, Ti 0.2, T

«

ASIAN AND FLEXIBLE ASIAN OPTIONS

The Bouaziz-Briys-Crouhy

K 0

=

=

=

=

DERIVATIVES

417

Approach 'closed-form

Average

Asset Price

Standard

catt

call

80 85 90 95 I00 105 I 10 I 15 120

0.07 0.35 i.ii 2.70 5.29 8.82 13.04

0.00 0.007 0.1I 0.80 2.87 6.5 \ I I.08 15.95 20.88

17.68 22.54

Ordinary Put

Average Put

17.60 12.88 8.64 5.23 2.82 1.35

18.59 13.66 8.84 4.59 i.73 0.44 0.07

0.57 0.2\ 0.07

.

Bouaziz, Briys and Crouhy (1994) presented a solution' for a European vam11a Asian option whose strike price is an average. The formula applies to average options (those for which the time interval taken into account for the strike average calculation is the life of the option) and forward-starting average options. The formula relics upon a slight linear approximation. Although some previous contributions in the literature already use approximation techniques, this approach derives a formal upper bound to the approximation error. It seems that the closed-form solution perforrns quite well and is obviously computationally efficient. In this model, the Asian option is written on a generic underlying asset with a maturity date T The option is forward-starting since its strike is computed as an arithmetic average of the underlying asset prices over the period [T - A, T]: 'plam

.

.

.

0.00 0.00

K The Curran Approach

=

S(u)du

-

A

(35)

,

where A is a time instant after date 0 of issuance of the option. The case of a plain vanilla Asian option corresponds to T - A 0. The price of the underlying asset is generated by the familiar equation: =

in the valuation of average Curran (1992)proposed a technique known as is performed across all the integration technique, this options. When implementing possible geometric mean prices for which the probability density is simple followmg Using the expected pay-off, conditional on that geometric mean price, the option: formula is presented for an average Asian 'conditioning'

"r

P

-

In(K*)

µ (µ> +¼a )N -+---

e xp

e

-

o

µ KN

o,

o,

e

C"(S, K, t)

(28)

ln(So) +

=

µ,

o µ

So +

=

a

= "

q2

K

=

2K

a'[ti

'" =

"E[(S,

e

|F,] - K)

(37)

where E is the conditional expectation operator with respect to the transformed riskneutral probability measure Q which replaces P, and K is defined as in equation (35). Two time windows A, A]. In the time window are considered: [0, T - A] and [T prior to T A, applying the iterated conditional expectations (see Duffie, 1988) to (37)

ti +

a

exp

y

µ

(i (n -

(31)

2

i(i - 1) At 2n . 1)(2n - 1) At 6n

(32)

1)

(lnK - µ)

-

gives

(30)

-

(r - d - jo2)

a2

--

=

(29)

(r - d - Ja2)t + (i 1)At]

(33) U

C (S, K, t)

'a =

"E{E[(S,

e

Fr_a] F,}

- K)

(38)

At this level Taylor expansions and a linearization are used. After some computations, the closed-form solution to the pricing of the Asian option over [0, T A] is found to be

v/37v/Ã+

Ai

C(S,K,t)=S,e'^

-N 2

2A -e 6x

2a

^U"'

forts

T-A

(39)

with r

(34) In the time window

is the time of first averaging point and At is n is the number of averaging points, t the time between averaging pomts. It seems that the geometric conditioning method gives better results than the previous methods.

(36)

-

with

where

µ dt + o dz

=

The price of the forward-starting Asian option at time t, C (S, K, t), is given by

ln(K*)

-

-

dS -S

=

r -

2

Ao2

AF

-J2

m

'

=

-

'

2

v

=

-

3

posterior to T - A, the strike price is gradually revealed.and and M, which is equal to x, S,]

(40) investors

know the sequence [S,

.,

S(u)du '

T

^

(41)

FUTURES

OPTIONS,

4 18

S(u)du=

A

,_,

=

"'

e

S(u)du

A

,_,

"E

S,

Using Taylor expansions and the same linearization =

S, e

'"

"

mN

+

\F,

S(u)du

cr

(43)

procedure gives the desired formula: "'726

C(S, K, t)

(42)

S(u)du

a,S

=

, -

K

which is also equal to

M, A

-

419

At the option's maturity date, the basket pay-off is given by

as follows:

The pricing problem can be reformulated

C (S, K, t)

ASIAN AND FLEXIBLE ASIAN OPTIONS

as of time t:

The strike price can be split into components A

AND EXOTICOERWATIYE

for t > T - A

e

b,S

=

cr

K*

-

a,F o

(49)

where

(44)

b

=

(50)

a,F with t)2

(T-

T-1 m=l--+r(T-t)-î

2A

A

2

3

sothatLb

land

-

2

-M=o

(T

t)+

A

3A3

The corresponding

a,F

=

-

rA

1 S, + C(S,, K, t)

for tsT

- A

(46)

At time 0, the basket option value can be obtained by taking its ekted under the appropriate probabihty measure, or

or e_gr

P(S,, K, t)

20.2

=

put formula is given by

P(S,, K, t)

=

K*

Sr

(45)

M, e

ANALYSIS OPTIONS

'

-n +

rA

co

,,

K, t) - 1 S, + C(S,,

AND VALUATION

for t > T - A

e "E

=

b;S

,

K

-

a,Fio

(52)

(47)

OF BASKET

valuation of basket Rubinstein (1994)presented a bivariate binomial lattice for the methodology The can be underlying assets. options where the basket comprises two the However, underlying assets. with valuation options basket of many extended to the of basket book time to manage a method is time-consuming and can not be used in real

Since the distribution of [Lb,S ,] is unknown, average: [G¿S K']. Finally, the basket option formula is given by co

=

e

TE

it is approximated

K'

S

of basket options. His Gentle (1993)presents an alternative method for the pricing used in Vorst (1992)for the pricing of an that similar technique to model is based on a average rate option. by: Following Gentle (1993),we denote respectively

by the geometric

a,Fi.o

(53)

EL b,S$r

(54)

avhere K'

=

K

E

S"

options.

S,,: the spot price of asset i at time t, F,,: the forward price of asset i at time I, of assets, at: the weight of asset i in the basket volatility of asset i, o,: the coefficient between assets i and j. p,,: the correlation

discounted value

Aner some tedious calculations, the basket option co

=

e

"

a,F o[a exp

h *

'

'

a

=

=

(v

value

is given by

)N(vv'Ì- h) -

In

vy exp

(ja

K'N(-A

1

(55) (56)

T)

(57)

410

v2

K'

OPTIONS,

FUTURES

Li

p,,b,bpp

S"

E

ba +

=

K + E

=

AND EXOTIC

DERIVATIVES

(58) b,S

(59)

The above formula was simulated by Gentle (1993)for prices of standard sterling and yen call options relative to forward rates as the basket of options. When compared with results of Rubinstein's model, simulations show that the formula is accurate and the basket option is cheaper than the pair of single options.

OF FLEXIBLE

ANALYSIS AND VALUATION ASIAN OPTIONS

20.3

ASIAN AND FLEXIBLE ASIAN OPTIONS

FAA(n)

20.3.1

Flexible

In this section we use the following notation: T:

the option's maturity

t: r:

the current time,

t:

for i

W(n, a, i)P(i)

=

1, 2,

=

.

.

.,

n

(60)

n:

h:

(n 1)h: [r -(A -

i)}hn

ps:

with

W(n, a, i)

for i

=

=

1, 2,

n

...,

(61)

where

The main advantage of the GWA is its flexibility. The flexible geometrië average, which is a simple extension of the standard geometric average, is given by FGA(n)

When j= 0, the averaging period does not start and r>(n When j 0, n, I - l)h. and the option is at its maturity date. When 1 « j< n, the option lies in the averaging period. At the option's maturity date, the pay-off of the option based on the flexible geometric average is given by =

=

max

(K, {graa(n) -

0}

=

(64)

Under the assumption of log-normality, the value of the flexible geometric Asian option is given by its expected pay-off discounted to the present under the appropriate probability. The general formula given in Zhang (1995)is

C

(62)

S'"'

=

date,

the time remaining to maturity, a binary indicator which is 1 (-1) for a call (a put), refer respectively to flexible, geometric and arithmetic, the number of observations specified in the option contract, the observation frequency or the time interval between two consecetive observations, the number of observations already done, the length of the averaging period, the starting time of the averaging period, the correlation coefficient between assets i and j.

P''

a is a weighting positive parameter, P(i) is the ith observation.

(63)

Asian Options

f: GWA(n, a)

w(i)S,

=

where all the parameters are as defined above.

f, g, a Before valuing flexible Asian options, we introduce the flexible weighted average and flexible geometric and arithmetic averages. Following Zhang (1995),the general weighted average (GWA) is given by

421

=

(SA'(j)N($d _

+

) - (Ke

ga

"

N($d

_,)

(65)

)]

(66)

where with w(i)

n

=

W(n, a, i) as given in (61)for i

is thhenumbber co

=

1, 2,

...,

n, where

A'(j)

=

B'(j)exp[-r(r-

T

)

servations,

B'(0) Note that the log of the FGA corresponds underlying asset prices. The flexible arithmetic average, which average, is given by

to a flexible arithmetic is a simple extension

=

1, Bf(j)

average of the log of the of the standard arithmetic

ds

=

ln (S/K) =

S[r -

+

(n

-

i)h]

a2)

(r a

a2(Tr

yr

for I sjsn M [B'U)

(67) (68)

OPTIONS,

422

FUTURES

w2(i)[r

=

i)h] + 2

(n -

-

w(i)w(k)[r i

DERIYATIVES

ASIAN AND FLEXIBLE ASIAN OPTIONS

k)h]

Note that this formula can be obtained directly from the preceding formula by the current asset price by the approximation factor Kr

i-I

«

T

AND EXOTIC

(n

-

-

j+2 k-j+1

(70)

In the above formula (65), the weighted average of the returns of the already passed observations is B'(j). The term T _, is assimilated to the mean function. The term Ts is regarded as the effective variance function. This formula reduces to that in Kemma and Vorst (1990)and Turnbull and Wakeman 1/n for all i 1 to n. {1991)for a 0 and w(i) =

20.3.2

=

=

Approximating

Flexible

Arithmetic

Asian Options

20.3.3

Some

yr

=

{$'"(n)-

max

(K, 0}

(71)

Using the approximations given above and discounting the option pay-off, the flexible arithmetic Asian option price is given bv

C

Af(j)N((di_

(SK

=

ga

(Ke

)-

"

N((d _

)

(72)

replacing

Properties

When compared to standard options and standard Asian options, flexible arithmetic Asian options have some special characteristics with respect to the Greek-letter risk measures: the delta, gamma, vega, theta, rho, charm, etc. For standard Asian options the delta is fixed given the number of observations and the observation frequency. For the flexible arithmetic Asian options, the delta and the other Greeks depend on the weighting scheme used. For example, when the observation numbers 0.5, S 10%, 100, K 100, r 4 and the weight parameters a are v 10% and r « 1 year, and the observation frequency is 1 month, the values of the 10.066 and 10.057 and flexible arithmetic and geometric Asian options are respectively the option's delta is 0.85. The flexible arithmetic Asian option's value is greater than that of the flexible geometric Asian option. prices) are accurate and show a It seems that the formulas presented (approximated small under-valuation bias when compared to prices obtained by the Monte-Carlo method. The approximation formulas provide accurate results and are not time consuming. =

=

At the maturity date, the pay-off of the option based on the flexible arithmetic average is given by

423

=

=

=

=

=

with

SUMMARY In(SKL/K)

d*

(r

.=

where x is approximated

)

a

_,

+WFUM

(73)

by =

1 + (E[¢

]+

¼[(E[¢

])2+

var

(74)

(¢ )]

In this chapter, Asian options are analyzed and valued. First, we propose the different approaches for the valuation of arithmetic and geometric Asian options. Simulation results are given. Second, we present the general context for the analysis and valuation of basket options. Finally, we introduce the concept of flexible arithmetic and geometric mean to the valuation of flexible arithmetic and geometric Asian options.

with

E[¢

]

=

v var

iw(i)[1 - w(i)] - 2

oh

[i|w(i)]+

iw(i)

w(j)

(75)

POINTS •

where E[¢

] is the

mean function

with

flexible

weights w(i) for i

=

1 to n, with

• •

var

[i|w(i)]

=

(i -

M)2w(i),

M

=

(76)

iw(i)

• -a2)

v

=

•

h(r -

(77)

2

• • •

var(¢í)

and

=

2v2 var[i|w(i)]{E[¢

]-

iv'var

[i w(i)]}

+

4Û2hQ-

(E[¢

])2

•

FOR DISCUSSION

What is an average option? What is the pay-off of an average call option? What is the pay-off of an average put option? Describe briefly the different valuation approaches for Asian options. What is a basket option? What is a flexible Asian option? What is a flexible arithmetic Asian option? What is a flexible geometric Asian option? What are the main properties of Asian options?

-

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in d ex

of

Absolute priority rule 213 AFFI252 AlG Combined Risks 227 Alcar Group 6 All-or-nothing option 361 AM International 8-9 American call options on spread 322 valuation175 Arnerican calls early exercise with continuous

distributions 159 early exercise with discrete

distributions 159-61 early exercise without distributions 159 on dividend paying stocks 282-3 with dividends 290-1 American cap options, simulation 178 American cap prices 178, 179 American commodity options 163-5 American futures options 166-8 American options 157-79 applications 176 early exercise 170 m French market 267-72 numerical pricing methods 281-95 pricing 158 pricing models 158 valuation 158-63 with discrete cash distributions 170-6 with dividends 175 American puts early exercise with continuous distributions 162 early exercise with discrete earlyde erreb ni hl -d stributions 161-2 paying stocks 284-5 dividend on price with dividends 266-8 with dividends 291-2 Apple Computer 2-3 Arbitrage arguments 4l4

between futures 101-2

Arbitrage

index

options and

Arbitrage portfolio, and Itõ's differential 42-3 Arrow-Debreu security 50 Asian options 409-20 average price options 411-18 B-B-C approach 417-18 Curran approach 416 C-Vapproach415 K-Vapproach 414 T-W approach 414 valuation 413-18 Asset-liability management (ALM) 219 Asset-or-nothing options 369-70 Asset prices 15, 60 continuous time processes for 28-34 dynamics 27-47, 262, 265 in complete markets 49-71, 73-86 At-maturity trigger forward contract 367 At-the-money down-and-in calls 386 At-the-money option 253 ATT 4-5 Auction markets 14, 15

Bacheller call values 89 Bachelier formula 88-9 Backward difference 57 Bank deposit rates 185-6 Bankruptcy 9, 212, 214 Bankruptcy-tnggermg mechamsm Barrier options 84-6, 359 90 analysis 361-8 categories 360 characteristics 368 bvaal atl edgin373f 3886-8 Basket options 418-20 valuation 418-20 153 BAW model l14-20, Bearish put spread 23 Binary barrier options 359-90

204

441

INDEX

Binary barrier options (cont.) analysis 361-8 valuation of368-71 Binomial model 330 Binomial option pricing model 43 Binomial process 161 Binomial pyramid 320-3 Binomial tree 264 Bivariate cumulative normal density function 175 Black and Scholes. See B-S Black's model 105-7, 152 Bond futures traded on LIFFE, tions on 109-10 Bond options 103-4 Rabinovitch's model 189 Bondholders' final pay-off 200 Boness formula 90 Boolean digitals 364 Boundary binary option 363 Boundary conditions 58, 61, 76-8, 94, 16 Box spread strategy 25-6 Brennen and Schwartz model 284 Briys-de Varenne model 207-24 Brownian bridges l 14 Brownian motion 28-31, 81 -3, 85, 88, 199, 205 and martingale 30-l geometric 40 mathematical definiton 30 p-dimensional 40 standard 31 standard independent 39 see also Wiener process B-S call option 56 B-S economy 76-9, 81, 83 B-S equation 78 B-S hedge portfolio 42 B-S method 39, 50, 5l 151 B-S mode192-l04, applications 99-104 precursors 88-92 B-S partial differential equation 42 B-S pricing of options 53 B-S process 39 B-S solution 83

B-S theory 88 Bullish call spread 22

CAC40 index 268, 314, 402 Calendar spread 16, 17 California Earthquake Authority 227 Call on maximum of two assets 316 Call on minimum Call options 77

of two assets 314

16

Continuous-time model 229 Continuous-time processes for asset

model 92-5 sensitivity parameters for 130-3 Call ratio strategy 23 Call sensitivity parameters 151-3 Call spread 22-3 Call values, with jump-difTusion process 24 Call values function 98 Call's delta 125 Call's elasticity 128-9 Call's gamma 126 Calls on the maximum 324 Calls on the minimum 323 Call's Rho 128 Call's theta 127 Call's vega 127 Capital Asset Pricing Model 43, 106, 302 Capitalization factor 75 over time 75 CAPM theory 227 Capped options 348 50 Capped variable loan commitments 168-70

prices 28-34

Continuous-time stochastic process 46 Contracts 8 Conversions

50

numerical solution 287-8 simulations 288 specificities 285-9 valuation 286-7 Convertible call and put prices 293-5

Convertibles 13 Corporate asset process 199 Corporate Corporate

1

Complex binaries 361-4 Complex chooser options 307 Nelken's approach 309-10 Rubinstein's approach 308-9 Compound digitals 360 Compound option 171-4 generalization

173

Conditional expectation 46 7 Constant elasticity of variance (CEV) difths process 243-4 Constant trigger level 230 Contingent claims 50, 56, 274-5 modeling 198- 202 Continuous auction systems 14 Continuous barrier options 382 3 Continuous quote-driven system 13-14 Continuous strike option 383-4 Continuous strike range options 384

24-5

Convertible bonds 285-8

Cash flow 6 Cash-or-nothing options 369 Charm 123, 124 Chen's correction to Rabinovitch's model 190 Chicago Board Options Exchange 14, 26, 192 Choice date 307 Chooser options 307-10 Chrysler 8 Closed-form pricing formula 233 Collar 349 Color 123 Commodity futures options 120 Commodity movements 10 Commodity options 120 Complete markets 49-71, 74-5, 199 asset pricing in 49 71, 73-86 characterizations

443

INDEX

øn

asset volatility 216

bonds

interest rate elasticity of 219-24 pricing of 197-224 valuation 197-224 Corporate default 203 Corporate option bonds 326-7 Corporate spreads pricing of 202 term structure of 203-5 valuation of 212-18 Corporate zero-coupon bond 212 risky 222 Corridor options 381-2 Corridors 364 Cost-of-carry formula 78, 8l Coupon-paying bonds 104 Cox, Ingersoll and Ross model 193 Cox, Ross and Rubinstein (CRR) model 259, 330 Crank-Nicholson numerical scheme 70 Creditor claims, write-down of 204 Critical stock price 176 Cumulative normal density function 95 Cumulative normal distribution, approximation 68 Cumulative normal distribution function 53 Cumulative standard normal distribution 77 Currency bonds, pricing of 325-6 Currency call formula lll Currency forwards, options on 109 Currency futures, options on 109 Currency options 111, l14, 341-57 Currency put formula 112 Dealer markets 14, 15 Default-free zero-coupon bond 206, 207 Default spreads, term structure of 202 Delivery options 328-31 Delta 124 6 monitoring and managing

134-9

Delta measures 123 Delta-neutral hedging 137 Density function 30 Derivative assets 50 pricing 49-71 Deterministic functions 75, 81, 208 Diagonal spread 16, 17 Diffusion processes 45-6 Dirac measure 44 Discount functions 272, 273 Discrete dividends 264-5 Dividends 264 Dow Jones 6 Down-and-in call 367 Down-and-out call 365-6 Down-and-out call options 379 Dual-strike options 327 Duration as function of time-to maturity 222, 223 Duration of insurance linked bonds 234-1 as function of interest rate 236, 237

Earnings Per Share (EPS) 5-6 Earnings-window dressing 6 Earthquake Risk Bond 227 i Economic level ll Elasticity 123, 128-9 Embedded digitals 359 Equilibrium option pricing theory 43 Equity-linked foreign exchange call 344-5 Equity-linked foreign exchange options 342-7 Equity options 99-100 Rabinovitch's model 186-9 Equivalent probability 63 Error terms 43-4 European call option 76, 79, 82, 83, 88, 201 European call price 94, l15 European calls 59-61, 86 algorithm 70-1 European cap prices 178, 179 European option values 98 European Options Exchange (EOE) 14, 15 European put price l 15 with dividends 266, 267 Exchange offer 301 Exchange options 298-301 Exotic options 360, 379-82 Exotic timing options analytic formula 401-3 characteristics 403 simulations 403 valuation of402 Explicit difference scheme 58-9 Extended Vasicek model 278 Extendible bonds 339-40

†NDEX

444

on security with no income l17 timing option in 329

Extendible calls 334-5 Extendible options 333-40 pricing of 333 -7 simple writer 337-9 valuation of 333-4 Extendible puts 336-7 Extendible warrants 340

margined American options 107, 263 price 105-7 prices l16

Futures Futures Futures Futures

FIFO 6 Filtration 46, 76 Finance level 11- 13 Fmance theory 2, 3 Fmancial engmeermg 11 of 14 Fmancial markets, internationalization Fmancial risk management . implementation 9 13 policies 2 Fmancial stability 8 Finite difference methods 57 First passage time distribution computation 240 Fisher option call values with uncertain exercise 303 Fisher option put values with uncertain exercise 304 Fixed exchange rate foreign equity options Jamshidian's approach 346-7 Reiner's approach 345-6 Fixed strike lookbacks 394 Flexible arithmetic Asian options 422-3 Flexible Asian options 420-3 analysis and valuation 420 -3 Floating strike lookback 393 4 equation 44 Fokker-Planck Foreign currency, futures contracts l18 Foreign currency options. See Currency optums Foreign equity option struck in domestic currency 343-4 struck in foreign currency 343 Foreign exchange 10 Forward contracts 77, 78, 105-6, l 16-19 Forward currency options, pricing of l 13 Forward difference 58 Forward extra contract 367 Forward functions 272 Forward option 78 Forward prices l16 Forward rate contracts l1-12 Forward start options 304 Fractional bonus option 406 Frankfurt Stock Exchange 14 Futures contracts 12 on commodities l16-19 on foreign currencies l l8 on security with discrete income l18-19 on security with known income l17 .

-

Gamma 123, 124, 126

options 109-10

.

and managmg 139-43 nn K7o0hlhagen model 110-11,

l

Gaussian diffusion process 75 Gaussian interest rate uncertainty 75 Gaussian law 54 Gaussian variables, simulation 62 General Electric 6 General weighted average (GWA) 420 .

.

.

-

.

Generalized payoff segment 353 Geske's approach for a call on a call 171-3 GG1sd Shaechr 5C 2330 .

an

Grabbe model 110-11 Greek-letter risk measures 121, 123-4, Grillil5B l

445

Hybrid securities 341-57, 354-6 Hybrids 12

monitoring moa

INDEX

a0n6o 226

Heat transfer equation 66, 69 Heath, Jarrow and Morton (HJM) model 193-5 Hedgeability of lookback options 404 Hedged portfolio 106, 137, 183 Hedged position 138 Hedging 7-9 at-the-money down-and-in calls 386 in-the-money down-and-in calls 387-8 of strike bonus options 407 out-of-the-money down-and-in calls 386-7 relationship between parameters 154-5 Hedging error 43 Hedging ratio of strike bonus option 406 Heston's model 248-9 Ho and Lee model deficiency in 276-7 for contingent claims 274-5 for interest rates and bond prices 272-3 term structure 273-4 Hoffman, Platen and Schweizer (HPS) diffusion model 248 -50 Hokey-Cokey option 363 Hull and White model 244-5, 277-9 Hybrid foreign currency options pricing of 351-4 tailoring 351

Implicit difference scheme 57-8 'In' barrier options 374 Incomplete information 254 Index futures, options on 108-9 onndss100-2 d ed Indexed notes 349-50 Information costs, option valuation with 251-5 Insolvency 200 Instantaneous interest rates under certainty 190-1 under uncertainty 191-2 Instantaneous standard deviation 201 Instantaneous volatility 83 Insurance industry 228 Insurance linked bonds duration of 234-7 pricing of 225-40 structure 228-9 time-series properties of237-9 valuation of 229-39 Insurance linked spreads as function of time-to-maturity 234 valuation of 232-4 Insurance risks 225, 226-8 Integral differentiation, Leibniz's rule for 71 Interest rate derivative assets 272-9 Interest rate elasticity measure 235 Interest rate elasticity of corporate bonds 219-24 Interest rate models 192-5 Interest rate options 190-5 Interest rate risk ll Interest rate theorem 113 Interest rate uncertainty 204 Internationalization of financial markets 14 In-the-money down-and-in calls 387-8 In-the-money lookback calls 393 In-the-money lookback puts 393 Investment banks 227 Iterative procedure 174 Itõ's differential and arbitrage portfolio 42-3 and replication portfolio 41-2 Itõ's formula application 38 example 37 generalized 39 40 mathematical expression 40 mathematical form 36-9 multidimensional 39 Itôs lemma 34-40, 187, 201, 22I, 346

example 36

implementation 43 intuitive form 35 Itô's process 32-3 Jump-diffusion

model 242-3

Kellogg's 8 Knock-out range options 382 Knock-out wall option 363 Known dividend yield 264 Kolmogorov equation backward 44 forward 44

.

Lambda 123, 124 Lattice approach 259-80 for bond prices 275 for put price 276 model for options on spot asset without payouts 161-3 survey 260-1 Lebesgue measure 30 Leibniz's rule for integral differentiation 71 LIBOR 349 LIFFE, options on bond futures traded on 109-10 .

LIFO 6 Limit binary option 363 Limited liability 199-201 Limited risk options 397-8 Linear tax schedule 8 Liquidation date 269 Loan covenants 9 Log-normal distribution 33 Log-normal property 34 London International Stock Exchange Long calls 18-19, 24 Long puts 20 Long straddle 21 Long-term bonds, short-term options on 102-3, 186 Long-term planning 3 Long-term view 6 Longstaff-Schwartz model 205-7 Lookback call 392 Lookback options 391-407 analysis 392-5 analytic formulas 395-400 hedgeability of404 simulations 399-400 standard 392-3, 395-6 Lookback put 392-3

13

446

Lookback Lufthansa

M4DEK

strategies 394-5

Options 12 B-S pricing of 53

9

combinations on on on on

Macaulay duration 222 Mandarin collar 362 Marché à Règlement Mensuel 267 Margin account 300-1 Market imperfection 8 Markov diffusion process 44 Markovian process 277 Martingale, and Brownian motion 30-1 Martingale method 32, 50, 52-6, 74, 81 Mega-premium option 362 Merton's model 153, 157 applications 185-6 derivation 182-6 Mini-premium option 363 Modified lattice 269 Modigliani-Miller theorem 199 MONEP 252, 268 Monte-Carlo method 56, 61-2

of 16

bond futures traded on LIFFE 109-10 currency forwards 109 currency futures 109

extrema on the maximum 396-7 on the minimum 397 on index futures 108-9 with uncertain exercise prices 301-3 Options markets 15 Order-driven system 13-14 Ornstein-Uhlenbeck process 187, 193, 247, 250 'Out' barrier options 375-6 Out-of-the-money 255 Outside barrier options 377-8 Over-the counter (OTC) markets 109, 114, 391, 403 Over-the-counter (OTC) options 99

Multi-currency bonds 326

NASDAQ14 Nationwide Mutual Insurance Company 227 Natural catastrophes 228 Natural hazards 226-8 New York Stock Exchange 14, 15, 329 Non-dividend paying stocks 59-61 Numeraire, change of 78-81 Numerical analysis 56-62, 69-70 Numerical methods for American option pricing 281-95 Numerical solutions 59-61 OEX wildcard options 329 30 One-touch all-or-nothing option 361 Open outcry system 15 Opening Automated Report System (OARS) 15 Option contracts 105-7 trading in 15-26 Option implicit volatility 17 Option markets 14 Option pay-offs 15 Option positions monitoring and management 123-55 in real time 129-49 Option price sensitivities 124-9 simulation and analysis 129-30 Option prices 54, 60, 107, 176 Option pricing models 87-122 Option strategies 15-19 Option valuation with information costs2¾I

5

Pan Am 8 Paribas call values 270 Paribas put values 271 Paris Bourse 14, 267, 269, 314 Partial differential equation 50-3, 57,ß9, 73-4, 76, 81, 82, 94, l15 resolution 64-8 Partial lookback options 398 Path-dependent binary barrier options 370--i Path-independent binary barrier options 369-70 Pay later options 305 -6 Pay later premium values 306 Pay-out ratio 304 PepsiCo 5 Performance incentive fee 300 Perls 354-6 valuation of355-6 Perturbation functions 272 Philadelphia Stock Exchange 114 Poisson process 243, 257 Portfolio options 327 Pricing of bonds 190-5, 192-5 of corporate bonds 197-224 limits of traditional approach 203 5 of corporate debt 200 of corporate spreads 202 of currency bonds 325-6 of extendible options 333-7 of forward currency options l13 of hybrid foreign currency options 351 of insurance linked bonds 225-40

INDEX

447

of options 192-5 Pricing models, American options 158 Private placements 229 Proactive risk management 7 Probabilistic method 74 Probability adjusted ratio 221 Probability measure (Q)74-5 Property Claims Services (PCS) 230 put on minimum (maximum)of two assets 316-17 on the minimum 324 with bound payoff 352 with disappearing deductable 352 with proportional coverage 351 Put-call parity condition 413 Put-call parity relationship 95-6, 107 Put options 19, 86 sensitivity parameters for 134 Put sensitivity parameters 151-3 Put spread 23 Put values function 99 Put's delta 125-6 Put's elasticity 129 Put's gamma 126 Put's Rho 128 Put's theta 127 Put's vega 128

212, 218 Q-economy 75, 79, 231 Q-martingale 240 Q-probability options 380 Qualitative Qualityoptions with Ndeliverable assets 328-9

options 380 Quantitative Quantooptions 342-7

Random variable 46 Range binary 361 Range forward contract 348-9 Range options 361-4 Range structures 359 Rate of return, distribution 34 Ratio spread 16 R&D 6, 9-10

Real-valued random variable 79 Rebate range binary 362 Reflection principle 85 Regular chooser 307 Reinsurance industry 228 Replicating portfolio 77 Replication portfolio, and Itô's differentia141-2 Report mechanism 268- 9 Reversals 24-5 Rho 123, 128 Risk evaluation 73 Risk exposure 8 Risk-free bond 116

bond 201 Risk-neutral economy 233 Risk-neutral probability (Q)75 Risk premium 4 Risk-return relationship 18 Riskless zero-coupon bond 2l3, 221 Risky asset, dynamics 75-6 Risky corporate bond 207 Risky corporate zero-coupon bond 222 Risky discount bond 207 Risky zero-coupon bonds 20l default before maturity 210- 12 no default before maturity 2l0 RM market 268

Risk-free zero-coupon

Jamshidian's approach 346-7 Reiner's approach 345-6 ratio 233 Quasi-debt-to-firm-value

Safety covenant 204 Sallie Mae straight bond 355 Samuelson formula 90-2 Scapegoat hypothesis 6

Rabinovitch call values 189

Securities 50

Schwartz model 282 Rabinovitch's model bond options 189 Chen's correction to 190 equity options 186-9

Radon-Nikodym derivative 79 Rainbow options 313-31 delivering the best of two assets and cash 317-19

discrete approach 320 of several on minimum (maximum) assets 319--20 valuation 314--23

Securities markets, trading mechamsms in 13-15 Securitization 229 Self-financing portfolio 42 Sensitivity parameters for call options 130-3 for put options 134 Share value 2 Shareholder stake 200 Shareholder value 3 Shareholders, final pay-off200 Short calls 18- 19

448

Short puts 20, 24 Short straddle 21 Short strangle 22 Short-term effects 2 Short-term gains 5 Short-term options on long-term bonds 102-3, 186 Short termism vs. long termism 6 Simple chooser option values 308 Simple writer extendible calls 337 Simple writer extendible puts 337-9 Simulation methods 56-62 Simulations 176, 323-5 American cap options 178 convertible bonds 288 exotic timing options 403 lookback options 399-400 Smile effect 250 for bond and currency options 250-1 in stock and index options 250 Soft barrier options 384-6 Solvency 200 S&Pl00 index 314, 329-30 Speed 123, 124 Spread options 327 Spreads 22-3 Sprenkle formula 89-90 Standard binary options 361 Standard options 379 Static hedging of barrier options 386-8 Stein and Stein's model 245-7 Stochastic component 31 Stochastic differential equation 75, 81, 206 Stochastic economy 74 Stochastic interest rates 181-96 extension 240 Stochastic process 45, 73, 230 governing loss index 240 having no memory 46 Stochastic revision strategy 43-4 Stochastic volatiles 247-50 Stock index options 100 Stock prices 5

dynamics 32-3 Stock value 5 Straddles 20-2 buying 20-1

INDEK Strike price bias 250 Structured barrier options 364-8 Structured products 359 Suppa, Enrico 226 Swaps l2 Swiss Option and Financial Futures Exchange (SOFFEX)

14

Switch options 380 Synthetic contracts 16 Synthetic forward contract 17 Synthetic positions 15-16

Taxes 8 Taylor series 35, 47 and Itõ's differential 40-1 Technically default-free bond 232 Technically default-prone bond 232 Term structure of corporate spreads 203-5 of default spreads 202 Theta 123, 124, 127 monitoring and managing 143-5 Time change 81- 3 Time-series properties of insurance linked bonds 237-9 Timing option in futures contracts 329 Tokyo Stock Exchange's Computer Assisted Routing and Execution System (CORES) 14 Toronto Stock Exchange's Computer Assisted Trading System (CATS) 14

Trading in option contracts

15-26 underlying asset 17 Trading calls 18 Trading mechanisms in securities markets 13-15 Trading ratios 23 Treasury bill yields 185-6

Treasury bond futures 329 Treasury bonds 227, 229 Trigger forward contract 367 Trigger point 233

Strategic level 10 Strict priority rule deviations 204

Uncertain exercise price option call values 301-3 Underlying asset, trading 17 Underlying asset price and time to maturity 178 Underlying asset price volatility and time to

hedging of 407 hedging ratio of 406 Strike option 405 Strike price 15-18, 20, 21, 26, 83, 94

Unique associated random variable 46 Up-and-in put 365 Up-and-out call 366 Up-and-out put 365

selling 21-2 Strangles 20-2

INDEX

Valuation exotic timing options 402 of Asian options 413-18 of barrier options 373-8 of basket options 418-20 of binary barrier options 368-71 of convertible bonds 286-7 of corporate spreads 212-18 of extendible options 333-4 of flexible Asian options 420-3 of insurance linked bonds 229 -39 of insurance linked spreads 232-4 of perls 355-6 of rainbow options 314-23 of wildcard options 330 Valuation formula 175 Vasicek formula 221 Vasicek framework 222 Vasicek model 192 Vasicek term structure 230 Vector of variables 46 Vega 123, 124, 127-8 monitoring and managing 146-9 Vertical bulls spread 16 Vertical spread 16 Volatility function 194 Volatility parameter 137 Volatility smiles empirical evidence 256 theory 250-1 Volatility spread 16 characteristics 149

449

Volatility strategies

17

Wall option 363

Warrants 13

au Weighting schemes 100 Westinghouse 6 Wiener process 28-31, 298 example 29 generalized 31 when is very small 30 with drift 240 see also Brownian motion Wildcard options 328-31 analysis 329-30 OEX 329-30 valuation of 330 Write-down of creditor claims 204

ot

Yield spread as function of asset volatility 219, 220 as function of initial quasi-debt ratio 216 as function of time-to-maturity 214, 215, 217, 218

Zaccaria, Benedetto 226 Zero-coupon bonds 104, 192, 198, 201, 206, 210-12 riskless 213, 221 risky corporate 222