Engineering BGM

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Engineering BGM

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CHAPMAN & HALL/CRC Financial Mathematics Series Aims and scope: The field of financial mathematics forms an ever-expanding slice of the financial sector. This series aims to capture new developments and summarize what is known over the whole spectrum of this field. It will include a broad range of textbooks, reference works and handbooks that are meant to appeal to both academics and practitioners. The inclusion of numerical code and concrete real-world examples is highly encouraged.

Series Editors M.A.H. Dempster Centre for Financial Research Judge Business School University of Cambridge

Dilip B. Madan Robert H. Smith School of Business University of Maryland

Rama Cont Center for Financial Engineering Columbia University New York

Published Titles American-Style Derivatives; Valuation and Computation, Jerome Detemple Financial Modelling with Jump Processes, Rama Cont and Peter Tankov An Introduction to Credit Risk Modeling, Christian Bluhm, Ludger Overbeck, and Christoph Wagner Portfolio Optimization and Performance Analysis, Jean-Luc Prigent Robust Libor Modelling and Pricing of Derivative Products, John Schoenmakers Structured Credit Portfolio Analysis, Baskets & CDOs, Christian Bluhm and Ludger Overbeck Numerical Methods for Finance, John A. D. Appleby, David C. Edelman, and John J. H. Miller Understanding Risk: The Theory and Practice of Financial Risk Management, David Murphy Engineering BGM, Alan Brace

Proposals for the series should be submitted to one of the series editors above or directly to: CRC Press, Taylor and Francis Group 24-25 Blades Court Deodar Road London SW15 2NU UK

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CHAPMAN & HALL/CRC FINANCIAL MATHEMATICS SERIES

Engineering BGM

Alan Brace

Boca Raton London New York

Chapman & Hall/CRC is an imprint of the Taylor & Francis Group, an informa business

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Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-58488-968-7 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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Dedicated to the memory of my father George James Brace

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Contents

Preface

xiii

1 Introduction 1.1 Background HJM . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The first ‘correct’ Black caplet . . . . . . . . . . . . . . . . . 1.3 Forward BGM construction . . . . . . . . . . . . . . . . . . . 2 Bond and Swap Basics 2.1 Zero coupon bonds - drifts and volatilities . . 2.2 Swaps and swap notation . . . . . . . . . . . . 2.2.1 Forward over several periods . . . . . . 2.2.2 Current time . . . . . . . . . . . . . . .

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11 11 14 18 19

3 Shifted BGM 3.1 Definition of shifted model . . . . . . . . . . . . . . . . . . . 3.1.1 Several points worth noting . . . . . . . . . . . . . . . 3.2 Backward construction . . . . . . . . . . . . . . . . . . . . .

21 21 22 24

4 Swaprate Dynamics 4.1 Splitting the swaprate . . . 4.2 The shift part . . . . . . . 4.3 The stochastic part . . . . 4.4 Swaption values . . . . . . 4.4.1 Multi-period caplets 4.5 Swaprate models . . . . . .

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27 28 29 31 34 35 36

5 Properties of Measures 5.1 Changes among forward and swaprate measures . . 5.2 Terminal measure . . . . . . . . . . . . . . . . . . . 5.3 Spot Libor measure . . . . . . . . . . . . . . . . . . 5.3.1 Jumping measure . . . . . . . . . . . . . . . .

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39 40 41 42 44

6 Historical Correlation and Volatility 6.1 Flat and shifted BGM oﬀ forwards . . . . . . . . . . . . . . . 6.2 Gaussian HJM oﬀ yield-to-maturity . . . . . . . . . . . . . . 6.3 Flat and shifted BGM oﬀ swaprates . . . . . . . . . . . . . .

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vii

viii 7 Calibration Techniques 7.1 Fitting the skew . . . . . . . . . . . . . 7.2 Maturity only fit . . . . . . . . . . . . . 7.3 Homogeneous spines . . . . . . . . . . . 7.3.1 Piecewise linear . . . . . . . . . . 7.3.2 Rebonato’s function . . . . . . . 7.3.3 Bi-exponential function . . . . . 7.3.4 Sum of exponentials . . . . . . . 7.4 Separable one-factor fit . . . . . . . . . 7.5 Separable multi-factor fit . . . . . . . . 7.5.1 Alternatively . . . . . . . . . . . 7.6 Pedersen’s method . . . . . . . . . . . . 7.7 Cascade fit . . . . . . . . . . . . . . . . 7.7.1 Extension . . . . . . . . . . . . . 7.8 Exact fit with semidefinite programming

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55 57 58 59 59 60 60 60 61 63 65 66 69 71 71

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75 75 76 77 78

9 Simulation 9.1 Glasserman type simulation . . . . . . . . . . . . . . . 9.1.1 Under the terminal measure Pn . . . . . . . . . 9.1.2 Under the spot measure P0 . . . . . . . . . . . 9.2 Big-step simulation . . . . . . . . . . . . . . . . . . . 9.2.1 Volatility approximation . . . . . . . . . . . . . 9.2.2 Drift approximation . . . . . . . . . . . . . . . 9.2.3 Big-stepping under the terminal measure Pn . . 9.2.4 Big-stepping under a tailored spot measure P0

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79 79 80 80 81 81 82 84 84

10 Timeslicers 10.1 Terminal measure timeslicer . . . . . . . 10.2 Intermediate measure timeslicer . . . . . 10.3 A spot measure timeslicer is problematical 10.4 Some technical points . . . . . . . . . . . 10.4.1 Node placement . . . . . . . . . . 10.4.2 Cubics against Gaussian density . 10.4.3 Splining the integrand . . . . . . . 10.4.4 Alternative spline . . . . . . . . . . 10.5 Two-dimensional timeslicer . . . . . . . .

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87 88 89 90 91 91 92 92 93 93

8 Interpolating Between Nodes 8.1 Interpolating forwards . . . . . . 8.2 Dead forwards . . . . . . . . . . 8.3 Interpolation of discount factors 8.4 Consistent volatility . . . . . . .

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ix 11 Pathwise Deltas 11.1 Partial derivatives of forwards . . . . 11.2 Partial derivatives of zeros and swaps 11.3 Diﬀerentiating option payoﬀs . . . . . 11.4 Vanilla caplets and swaptions . . . . . 11.5 Barrier caps and floors . . . . . . . .

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95 96 97 98 99 100

12 Bermudans 12.1 Backward recursion . . . . . . . . . . . . . . . . . . . 12.1.1 Alternative backward recursion . . . . . . . . . 12.2 The Longstaﬀ-Schwartz lower bound technique . . . . 12.2.1 When to exercise . . . . . . . . . . . . . . . . . 12.2.2 Regression technique . . . . . . . . . . . . . . . 12.2.3 Comments on the Longstaﬀ-Schwartz technique 12.3 Upper bounds . . . . . . . . . . . . . . . . . . . . . . 12.4 Bermudan deltas . . . . . . . . . . . . . . . . . . . . .

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103 104 106 106 107 108 109 110 111

13 Vega and Shift Hedging 13.1 When calibrated to coterminal swaptions 13.1.1 The shift part . . . . . . . . . . . . 13.1.2 The volatility part . . . . . . . . . 13.2 When calibrated to liquid swaptions . . .

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113 114 115 116 118

14 Cross-Economy BGM 14.1 Cross-economy HJM . . . . . . . . . . . . . 14.2 Forward FX contracts . . . . . . . . . . . . 14.2.1 In the HJM framework . . . . . . . . 14.2.2 In the BGM framework . . . . . . . 14.3 Cross-economy models . . . . . . . . . . . 14.4 Model with the spot volatility deterministic 14.5 Cross-economy correlation . . . . . . . . . 14.6 Pedersen type cross-economy calibration .

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121 121 123 124 125 127 128 131 135

15 Inflation 15.1 TIPS and the CPI . . . . . . . . . . . . 15.2 Dynamics of the forward inflation curve 15.2.1 Futures contracts . . . . . . . . . 15.2.2 The CME futures contract . . . .

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141 141 143 145 146

16 Stochastic Volatility BGM 16.1 Construction . . . . . . . . . . . . . . . 16.2 Swaprate dynamics . . . . . . . . . . . 16.3 Shifted Heston options . . . . . . . . . 16.3.1 Characteristic function . . . . . . 16.3.2 Option price as a Fourier integral

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x 16.4 Simulation . . . . . . . . . . . . . . 16.4.1 Simulating V (t) . . . . . . . 16.5 Interpolation, Greeks and calibration 16.5.1 Interpolation . . . . . . . . . 16.5.2 Greeks . . . . . . . . . . . . . 16.5.3 Caplet calibration . . . . . . 16.5.4 Swaption calibration . . . . .

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165 165 166 168 168 169 169 170 172 172

A Notation and Formulae A.1 Swap notation . . . . . . . . . . . . . . . . . . . . . A.2 Gaussian distributions . . . . . . . . . . . . . . . . . A.2.1 Conditional expectations . . . . . . . . . . . A.2.2 Density shift . . . . . . . . . . . . . . . . . . A.2.3 Black formula . . . . . . . . . . . . . . . . . . A.2.4 Gaussian density derivatives . . . . . . . . . . A.2.5 Gamma and vega connection . . . . . . . . . A.2.6 Bivariate distribution . . . . . . . . . . . . . A.2.7 Ratio of cumulative and density distributions A.2.8 Expected values of normals . . . . . . . . . . A.3 Stochastic calculus . . . . . . . . . . . . . . . . . . . A.3.1 Multi-dimensional Ito . . . . . . . . . . . . . A.3.2 Brownian bridge . . . . . . . . . . . . . . . . A.3.3 Product and quotient processes . . . . . . . . A.3.4 Conditional change of measure . . . . . . . . A.3.5 Girsanov theorem . . . . . . . . . . . . . . . A.3.6 One-dimensional Ornstein Uhlenbeck process A.3.7 Generalized multi-dimensional OU process . . A.3.8 SDE of a discounted variable . . . . . . . . . A.3.9 Ito-Venttsel formula . . . . . . . . . . . . . . A.4 Linear Algebra . . . . . . . . . . . . . . . . . . . . . A.4.1 Cholesky decomposition . . . . . . . . . . . . A.4.2 Singular value decomposition . . . . . . . . . A.4.3 Semidefinite programming (SDP) . . . . . . . A.5 Some Fourier transform technicalities . . . . . . . .

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175 175 176 176 176 178 179 181 182 182 183 185 185 185 185 186 186 188 188 188 189 189 189 190 192 195

17 Options in Brazil 17.1 Overnight DI . . . . . . . . . . . . . . 17.2 Pre-DI swaps and swaptions . . . . . 17.2.1 In the HJM framework . . . . . 17.2.2 In the BGM framework . . . . 17.3 DI index options . . . . . . . . . . . . 17.3.1 In the HJM framework . . . . . 17.4 DI futures contracts . . . . . . . . . . 17.4.1 Hedging with futures contracts 17.5 DI futures options . . . . . . . . . . .

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xi A.6 The chi-squared distribution . . A.7 Miscellaneous . . . . . . . . . . A.7.1 Futures contracts . . . . . A.7.2 Random variables from an A.7.3 Copula methodology . . . References

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Preface

Over the past several years the author has found himself frequently asked to give explanatory talks on BGM, some of which extended into one- or two-week workshops with detailed head-to-head technology transfer. The main interest came from small groups of quants either in banks or in software companies wanting to implement the model without wasting too much time decoding papers to find a suitable approach, and also academics and students wanting to get into the subject. This book is therefore naturally targeted at such people, who generally have several years experience around finance and a good grounding in the relevant mathematics. The stimulus to begin writing was an invitation to join the Quantitative Finance Research Centre (QFRC) at the University of Technology Sydney (UTS) as an Adjunct Professor, and give a series of lectures on BGM for an audience of academics, students and industry quants over the course of a couple of semesters during 2006. This book grew out of those lectures, but the starting point was some eleven years of notes on various aspects of BGM, that were all prepared either for implementers writing production code, or as formal documentation to accompany production code, or in response to consulting tasks. Thus most of the techniques and methods described in this book originate in practical problems needing a solution and address real requirements. Moreover, many of them have been implemented, tried and tested either in an R&D environment like MatLab, or in production code. A reader from a mathematics, physics or engineering background (or the quantitative end of another science) with a decent knowledge of analysis, optimization, probability and stochastic calculus (that is, familiar with Ito and Girsanov at the very least) should find this book fairly self-contained and thus hopefully a suitable resource and guide to implementing some version of the model. Indeed, part of the reason why the author has tried to keep the book relatively short is to make it easy to slip into one’s briefcase and use as a ready reference; the other part is a pathological fear of catching blitherer’s disease, which in extreme cases seems to dilute ideas to one per page! The book starts with the standard lognormal flat BGM, and then focuses on the shifted (or displaced diﬀusion) version to develop basic ideas about construction, change of measure, correlation, calibration, simulation, timeslicing (like lattices), pricing, delta hedging, vega hedging, callable exotics and barriers. Further chapters cover cross-economy BGM, adaption of the HJM inflation model to the BGM framework, a simple tractable stochastic volatility version of BGM, and financial instruments in Brazil, which have evolved

xiii

xiv in a unique way and are amenable to BGM analysis. Because shifted BGM can fit a cap or swaption implied volatility skew (but not a smile) and has the advantage of being just as tractable as flat BGM, it seems the right framework to present basic techniques. The stochastic volatility version aims to add a measure of convexity to the skew version, but we do not go so far as trying to calibrate to a full smile, which is a complex task appropriate to a cutting edge specialist. Overall the author can’t help feeling that shifted BGM with the stochastic volatility extension as described here is about right for both the Mortgage Backed Security world, and also second tier banks wanting a robust framework in which to manage structured products sold into their customer base, without having to worry too much about being arbitraged. To sum up, the reader is presented with several, progressively more sophisticated, versions of BGM, and a range of methods and recipes that (after some expansion and articulation) can be programmed into production code, and is free to choose an implementation to suit his requirements. Thus the book attempts to be an implementer’s handbook oﬀering straightforward models suitable for more conservative institutions who want a robust, safe and stable environment for calibrating, simulating, pricing and hedging interest rate instruments. Advanced versions for market makers, hedge funds or leading international banks are left to their top quants, though their newer quants might conveniently learn about market models from this book and then do better. Many people contributed in some way to this book. In particular, it was a pleasure working with Marek Musiela through the early ‘90s at Citibank, where Mike Hawker in Sydney and Pratap Sondhi in Hong Kong provided support and a framework to do much of the original work. Since then, innumerable conversations with colleagues, reading and decoding many excellent papers, attendance at wide ranging professional conferences and some foolish mistakes have added enormously to the author’s basic knowledge. In direct preparation of this manuscript Chapman and Hall were patient and encouraging, Marek Rutkowski gave me a copy of his extensive bibliography greatly simplifying the task of preparing references, and my thanks to Carl Ang, Peter Buchen, Andrew Campbell, Daniel Campos, Tim Glass, Ben Goldys, Ivan Guo, Steve McCarthy, Frank Merino, Paul O’Brien, Erik Schlogl and Rob Womersley for helping check diﬀerent parts of the book. Further thanks are due to both National Australia Bank1 and UTS for their material support in terms of time and infrastructure over the past couple of years, and also to MY for encouragement at some diﬃcult moments. A word on the title ‘Engineering BGM’. The background is that Miltersen, Sandmann and Sondermann (MSS), see [78], were the first to get a ‘kosher’ 1 All views expressed in this book are the author’s and in no way reflect NAB policy, philosophy or technology.

xv Black caplet formula out of HJM, but unfortunately they did not establish existence, which is an essential feature of a model (along with, the author feels, the technology to price complex options). We, that is Brace, Gatarek and Musiela (BGM), see [30], grasped the intuition behind the model, proved existence, derived swaption formulae, calibrated to the market and constructed simulation technology for pricing. So generally speaking the model has more-or-less become known as ‘BGM’ in the industry and the ‘Libor Market Model’ in academic circles. My preference for the title ‘Engineering BGM’ over the alternative ‘Engineering the Libor Market Model’, is partly because this book is aimed at industry quants and traders and partly because it is shorter and more punchy. But unequivocally, MSS made the first breakthrough in this area, and we referenced their work in our paper [30] describing it as a ‘key piece of information’. Finally, if that nightmare for a single author ‘the bad stupid mistake’ should materialize, it is soley the author’s fault and he apologizes in advance. Of course, all information about any, hopefully more minor, mistakes found by readers would be gratefully received (at any one of the author’s email addressess on the title page), as would any suggestions for inclusions, exclusions and better ways of doing things (in case there should ever be a second edition of this book).

Alan Brace (Sydney 25 September 2007)

Chapter 1 Introduction

Modern interest rate modelling began1 with Ho & Lee’s (HL) important 1986 paper [54], and matured into the Heath, Jarrow and Morton (HJM) model [52], which was circulating in 1988, and which became the standard framework for interest rates in the early ‘90s. Initial work on the market models was done within that framework, so to set the scene, the single-currency domestic version of HJM is reviewed in Section-1.1. When the volatility function is deterministic, HJM is Gaussian, extremely tractable, and includes versions like Hull and White [58] and many other models. But until the advent of the market models [30], [66], [78] and [79] around 1994-97, the market’s use of the Black caplet and Black swaption formulae (which priced assuming that forwards and swaprates were lognormal) was regarded as an aberration which could not be reconciled with HJM. A further problem was that HJM exploded when the instantaneous forward rates were made lognormal. The author can recall comments at conferences in the early ‘90s along the lines that ‘the market is foolish and should adopt some arbitrage free Gaussian HJM model as a standard’. To avoid explosions, attention shifted to modelling the cash forwards, and in 1994 Miltersen, Sandmann and Sondermann [78] found a PDE method, described in Section-1.2 below, to derive the Black caplet formula within the arbitrage free HJM framework. Knowing that was possible, and that the Black caplet formula was not an aberration, was a key piece of information. The author’s main contribution to events was to grasp the intuition, described in Section-1.3 below, that the cash forwards want to be lognormal, but under the forward measure at the end of their interval. With that realization, the derivation of the Black caplet formula became trivial, and led to the so called forward construction of BGM detailed in [30], by Musiela, Gatarek and the author, which established existence of the model, derived approximate analytic swaption formulae, calibrated to the market, and provided suitable simulation technology for pricing exotics.

1 Though

intriguingly, the previous long standing actuarial practice of hedging bonds by matching duration turned out to be equivalent to delta hedging within the HL model.

1

2

Engineering BGM

1.1

Background HJM

REMARK 1.1 Before beginning, a word on our ‘∗’ notation for transposes. Throughout this book we will generally be dealing with multi-factor models involving an n-dimensional vector volatility function, say ξ : R → Rn and a corresponding multi-dimensional Brownian motion W (t) ∈ Rn . Usually they (or similar expressions as in (1.3) below) appear together as inner products, so we use the ‘∗’ notation to indicate transpose and write ξ ∗ (t) dW (t) ≡ hξ (t) , dW (t)i for that inner product. Of course, in single factor models ξ (t) dW (t) would simply mean the product of two scalor quantities. Note that many authors today adopt the practice (which is beginning to appeal to the author) of simply writing ξ (t) dW (t) and leaving the reader to work out from the context if an inner product is implied. The ingredients of the HJM domestic interest rate model are: 1. An instantaneous at t forward rate f (t, T ) for maturity T , with SDE df (t, T ) = α (t, T ) dt + σ∗ (t, T ) dW0 (t)

(1.1)

where the stochastic driving variable W0 (t) is multi-dimensional Brownian motion (BM) under the arbitrage-free measure P0 , and σ (t, T ) is a possibly stochastic vector volatility function for f (t, T ). 2. A spot rate r (t) = f (t, t) and numeraire bank account to accumulate it β (t) = exp

µZ

t

0

¶ r (s) ds ,

3. Assets in the form of a spectrum of time T maturing zero coupon bonds ! Ã Z T

B (t, T ) = exp −

f (t, u) du ,

t

paying 1 at their maturity T . To be arbitrage free, the zeros discounted by the bank account as numeraire Ã Z ! Z T t B (t, T ) = exp − r (s) ds − f (t, u) du , (1.2) Z (t, T ) = β (t) 0 t

Introduction

3

must be P0 -martingales for all T . Because d

Z

Z

T

T

f (t, u) du = df (t, u) du − f (t, t) dt, t t ! ÃZ ! ÃZ

= −r (t) dt +

T

α (t, u) du

T

σ ∗ (t, u) du dW0 (t) ,

dt +

t

t

applying Ito to (1.2), the SDE for Z (t, T ) is ³R ´   T  −r (t) dt + r (t) dt − t α (t, u) du dt dZ (t, T )  ¯2 ¯R ³R ´ , =  − T σ ∗ (t, u) du dW0 (t) + 1 ¯¯ T σ (t, u) du¯¯ dt  Z (t, T ) 2 t t  · ¯ ¸  ¯  − R T α (t, u) du − 1 ¯R T σ (t, u) du¯2 dt  ¯ 2 ¯ t t = . RT ∗   − t σ (t, u) du dW0 (t)

For this to be a P0 martingale the drift must vanish, so α (t, T ) = σ ∗ (t, T )

Z

T

σ (t, u) du,

t

and the SDE for the instantaneous forwards is Z T σ (t, u) du dt + σ ∗ (t, T ) dW0 (t) . df (t, T ) = σ ∗ (t, T )

(1.3)

t

Diﬀerentiating B (t, T ) = β (t) Z (t, T ), the corresponding SDE for the zero coupon bond is dB (t, T ) = r (t) dt − B (t, T ) REMARK 1.2

Z

T

σ ∗ (t, u) dudW0 (t) .

(1.4)

t

The HJM approach therefore implies that the volatility b (t, T ) = −

Z

T

σ (t, u) du,

(1.5)

t

of each zero coupon bond B (t, T ) is continuous in T , a restriction ruling out piecewise constant bond volatilities. Because assets discounted by the bank account numeraire are P0 -martingales, the present value of a cashflow X (T ) occurring at time T is ¯ ¶ µ ¯ β (t) X (T )¯¯ Ft , (1.6) X (t) = E0 β (T )

4

Engineering BGM

where E0 is expectation under P0 , and Ft is the underlying filtration (total accumulated information up to t). In particular, because a zero coupon pays 1 at maturity !¯ ! Ã Ã Z ¯ ¶ µ ¯ T β (t) ¯¯ ¯ 1¯ Ft = E0 exp − r (s) ds ¯ Ft . (1.7) B (t, T ) = E0 ¯ β (T ) t

A forward contract FT (t, T1 ) on a zero-coupon bond B (t, T1 ) maturing at T1 , exchanges at time T the zero coupon B (T, T1 ) for FT (t, T1 ). The present value of the exchange must be zero, hence FT (t, T1 ) must satisfy ¯ ¾ ½ ¯ β (t) [FT (t, T1 ) − B (T, T1 )]¯¯ Ft = 0 E0 β (T ) giving the following model free result for forward contracts FT (t, T1 ) =

B (t, T1 ) . B (t, T )

(1.8)

When T1 = T + δ, the cash forward K (t, T ) over the interval (T, T1 ] is defined in terms of the forward contract FT (t, T1 ) by FT (t, T1 ) =

1 B (t, T1 ) = . B (t, T ) 1 + δK (t, T )

(1.9)

REMARK 1.3 In the following equation (1.10), please note that the one variable Radon-Nikodym derivative Z (t) = E0 { Z (T )| Ft } is not the two ) variable discounted zero coupon function Z (t, T ) = B(t,T β(t) . Being a strictly positive process, the bank account β (t) induces a forward measure PT (expectation ET ) at any maturity T through PT = ZT P0 Z (T ) =

or

ET {·} = E {· ZT }

(1.10)

1 . β (T ) B (0, T )

It follows, from the conditional change of measure result of Appendix-A.3.5, that ¯ ´ ³ ¯ β(t) E X (T ) ¯ Ft 0 β(T ) E0 ( X (T ) Z (T )| Ft ) X (t) ¯ ³ ´ = , = ET ( X (T )| Ft ) = β(t) ¯ E0 ( Z (T )| Ft ) B (t, T ) E0 β(T ) ¯ Ft

which simplifies the present value equation (1.6) to ¯ ¶ µ ¯ β (t) X (T )¯¯ Ft = B (t, T ) ET ( X (T )| Ft ) . X (t) = E0 β (T )

(1.11)

Introduction

5

Also X (t) discounted by B (t, T ) is a martingale under the forward measure PT because for s < t ¯ ¶ µ X (t) ¯¯ X (s) ET . Fs = ET ( ET ( X (T )| Ft )| Fs ) = ET ( X (T )| Fs ) = B (t, T ) ¯ B (s, T ) Integrating (1.4) over [0, T ] identifies Z (T ) because Ã Z Z ! T T ∗ B (T, T ) = 1 = B (0, T ) β (T ) E − σ (t, u) du dW0 (t) , ( Z Z (T ) = E −

⇒

0

0

T

Z

t

T ∗

)

σ (t, u) du dW0 (t) , t

showing, from the Girsanov Theorem of Section-A.3.5, that WT (t), given by dWT (t) = dW0 (t) +

Z

T

σ (t, u) du dt,

(1.12)

t

is PT -BM. Subtracting from a similar expression for WT1 (t), a PT1 -BM, dWT1 (t) = dWT (t) +

Z

T1

σ (t, u) du dt.

(1.13)

T

From equations (1.4), (1.9) and the result in the Appendix A.3.3, the SDE for the forward contract FT (t, T1 ) is ( ) r (t) dt − r (t) dt dFT (t, T1 ) h i RT RT = , − T 1 σ ∗ (t, u) du dW0 (t) + t σ (t, u) du dt FT (t, T1 ) =−

ZT1

σ ∗ (t, u) du dWT (t) ,

(1.14)

T

while the SDE for its reciprocal is ³ ´ ( ) 1 d FT (t,T r (t) dt − r (t) dt 1) h i RT RT ³ ´ = , + T 1 σ ∗ (t, u) du dW0 (t) + t 1 σ (t, u) du dt 1 FT (t,T1 )

=

ZT1

σ ∗ (t, u) du dWT1 (t) .

(1.15)

T

Hence FT (t, T1 ) is a PT -martingale while, more importantly as we will see, 1 is a PT1 -martingale. its reciprocal FT (t,T 1)

6

Engineering BGM

1.2

The first ‘correct’ Black caplet

Miltersen, Sandmann and Sondermann [78] started with the assumption that under the T -forward measure PT the cash forward K (t, T ) over [T, T1 ] was of lognormal type with deterministic volatility γ (which we here set constant for easy exposition), that is, they assumed the SDE for K (t, T ) has form dK (t, T ) = (drift) dt + K (t, T ) γ dWT (t) ,

(1.16)

and then worked with the corresponding forward contract FT (t, T1 ) (because it is a PT -martingale). Diﬀerentiating (1.9) using (1.14), and then comparing the stochastic term with that of (1.16), gives an SDE for FT (t, T1 ): 1 dK (t, T ) = d δ

µ

¶ 1 −1 FT (t, T1 )

= (drift) dt + ⇒ ⇒

Z

1 δFT (t, T1 )

T1

T

(1.17) Z

T1

σ (t, u) du dWT (t)

T

σ (t, u) du = K (t, T ) γδFT (t, T1 ) = [1 − FT (t, T1 )] γ

dFT (t, T1 ) = −FT (t, T1 ) [1 − FT (t, T1 )] γ dWT (t) .

The time t value of a Black caplet struck at κ, fixed at T and paid at T1 , is ¯ ¾ 1 + ¯¯ δ [K (T, T ) − κ] ¯ Ft cpl (t) = E0 β (T1 ) ¯ ¾ ½ B (T, T1 ) + ¯¯ δ [K (T, T ) − κ] ¯ Ft , = E0 β (T ) ( ¸+ ¯¯ ) · 1 ¯ − 1 − δκ ¯ Ft , = B (t, T ) ET FT (T, T1 ) ¯ FT (T, T1 ) ¯ o n ¯ = B (t, T ) ET [1 − (1 + δκ) FT (T, T1 )]+ ¯ Ft . ½

Applying Ito, Miltersen et al then set to zero the drift of the PT -martingale v (t, FT (t, T1 )) =

cpl (t) , B (t, T )

so that cpl (t) is given by the solution v (t, FT (0, T1 )) to the non-linear PDE 2 ∂v 1 2 2 2 ∂ v + 2 γ x (1 − x) =0 ∂t ∂x2

with

+

v (T, x) = [1 − (1 + δκ) x] .

Introduction

7

This converts to a heat equation problem with the transformations x , 1−x s 1 ∂2u e− 8 ∂u = v (t, x) = z z u (s, z) ⇒ − ∂s 2 ∂z 2 e2 + e 2 · ¸+ ª © z (1 + δκ) − z2 2 1− with u (0, z) = e + e , 1 + e−z s = γ 2 (T − t) ,

z = ln

(1.18)

which has the solution (substitute in the PDE and integrate by parts) Z ∞ ¡ √ ¢ u (s, z) = u 0, z + υ s N1 (υ) dυ −∞

=

ZΥ

−∞

£ ¡ £ ¡ £ √ ¤¢ √ ¤¢¤ exp − 12 z + υ s − δκ exp 12 z + υ s N1 (υ) dυ,

µ µ √ ¶ √ ¶ ³z ³ z s´ s´ s s − δκ exp + , = exp − + N Υ+ N Υ− 2 8 2 2 8 2 1 in which Υ = − √ (z + ln δκ) . s Inverting the transforms (1.18) to go from u (s, z) back to v (t, x) ½ ³ ´¾ √ [1 − x] v (t, x) = x N (h) − δκN h − 12 γ T − t , x ½ µ ¶ ¾ 1−x 1 1 ln h= √ + 12 γ 2 (T − t) x δκ γ T −t the caplet price cpl (t) follows from v (t, x) on using 1−x δK (t, T ) = , x ³ ´ √ cpl (t) = B (t, T ) v (t, x) = δB (t, T1 ) B K (t, T ) , κ, γ T − t . x = FT (t, T1 ) =

B (t, T1 ) , B (t, T )

where B (·) is the Black formula, see Appendix-A.2.3. A probabilistic proof of this result obtained by the author while trying to articulate the insight of MSS, runs as follows. Simplify notation by setting PT = P, FT (t, T1 ) = Ft , Kt = K (t, T ) and WT (t) = Wt . From the SDE (1.17) for Ft , if Ft 1 , then or Ft = Zt = ln 1 − Ft 1 + exp (−Zt ) ¢ ¤ £ ¡ dZt = −γ dWt − 12 γ tanh 12 Zt dt , exp (−Zt ) = δKt .

8

Engineering BGM

ft according to Change measure between P with BM Wt , and Q with BM W ¢ ¡ ft + 1 γ tanh 1 Zt dt dWt = dW 2 2 ³ ´ 2 ft , hZi = γ t and ZT − Zt = −γ W fT − W ft , ⇒ dZt = −γ W t

which, from Girsanov’s theorem (A.3.5), means ( Z ) (Z T ¢ ¡ 1 1 f P=E − Q =E 2 γ tanh 2 Zt dWt 0

T

0

¢ ¡ ¢ cosh 12 ZT ¡ 1 2 ¢ Q ¡ = exp − 8 γ [T − t] cosh 12 Zt

1 2

tanh

¡1

2 Zt

¢

dZt

)

Q

because

E

(Z

¢¤ £ ¡ d ln cosh 12 Zt = T

t

1 2

tanh

¡1

2 Zs

¢

dZs

1 2

tanh )

¡1

2 Zt

= exp

¢

(Z

dZt + T

t

1 8

sech2

¡1

¢

d hZit ⇒ ) Z £ ¡ 1 ¢¤ 1 T d ln cosh 2 Zs − 8 d hZis . 2 Zt

t

Hence, using (A.3.4) and (A.2.3), the time-t value of the option is ¯ o n ¯ cpl (t) = B (t, T ) EP [1 − (1 + δκ) FT ]+ ¯ Ft , ½ i+ ¯¯ ¾ ¡ 1 2 ¢ cosh( 12 ZT ) h (1+δκ) 1 − 1+exp(−ZT ) ¯¯ Ft , = B (t, T ) EQ exp − 8 γ [T − t] cosh 1 Z ( 2 t) (· ¡ 1 ¢ ¸+ ¯¯ ) 1 2 exp (−Zt ) exp − [Z − Z ] − γ [T − t] ¯ B(t,T ) T t 2 8 ¢ ¡ = [1+exp(−Zt )] EQ ¯ Ft , −δκ exp 12 [ZT − Zt ] − 18 γ 2 [T − t] ¯  ³ ³ ´´ + ¯  ¯  fT − W ft  Kt E 1 γ W ¯ ³ 2 ³ ´´  ¯ Ft , = δB (t, T ) FT (t, T1 ) EQ  ¯   −κE − 1 γ W fT − W ft ¯ 2 ³ √ ´ = δB (t, T1 ) B K (t, T ) , κ, γ T .

1.3

Forward BGM construction

The intuition behind BGM is that the forward K (t, T ) over the interval (T, T1 ] wants to be lognormal, but under the forward measure PT1 located at its payoﬀ T1 at the end of the interval. Specifically, recall (1.9) that the cash forward K (t, T ) over (T, T1 ] with coverage δ = |(T, T1 ]| is related to the reciprocal of the forward contract by 1 + δK (t, T ) =

1 . FT (t, T1 )

Introduction

9

Diﬀerentiating this equation using (1.15) gives dK (t, T ) δ = [1 + δK (t, T )]

ZT1

σ ∗ (t, u) du dWT1 (t) .

T

So if the HJM volatility function σ (t, T ) is made stochastic and chosen to satisfy ZT1 δK (t, T ) ξ (t, T ) , (1.19) σ (t, u) du = 1 + δK (t, T ) T

where ξ (t, T ) is a deterministic vector function, then the forward K (t, T ) becomes lognormal under the PT1 -forward measure dK (t, T ) = K (t, T ) ξ ∗ (t, T ) dWT1 (t) . That produces the Black formula (A.2.3) for the time t value cpl (t) of a caplet struck at κ, setting at T and paying at T1 , because ¯ ¾ ½ β (t) + ¯¯ δ [K (T, T ) − κ] ¯ Ft , (1.20) cpl (t) = cpl (t, κ, T, T1 ) = E0 β (T1 ) " ¯ ! #+ ¯  ÃZ  T ¯  = δB (t, T1 ) ET1 ξ ∗ (s, T ) dWT1 (s) − κ ¯¯ Ft , K (t, T ) E  t ¯    s Z T = δB (t, T1 ) B K (t, T ) , κ, |ξ (s, T )|2 ds . t

In general, for t > Tj the forwards K (t, Tj ) are dead and equal to K (Tj , Tj ), while for t ≤ Tj the forwards K (t, Tj ) are alive and lognormal under PTj+1 . By repeatedly using (1.13) and (1.19) for t ≤ Tj SDEs for any K (t, Tj ) under one fixed terminal measure Pn at Tn (n > j) are therefore dK (t, Tj ) − = K (t, Tj )

Pn−1

δK (t, Tk ) ∗ ξ (t, Tk ) ξ (t, Tj ) dt 1 + δK (t, Tk ) ∗ +ξ (t, Tj ) dWn (t) .

k=j+1

(1.21)

That provides a framework for simulation because Pn is a unique measure under which the random number driving the simulation can be regarded as working. Establishing existence of this forward constructed BGM model, however, is not straightforward because the HJM volatility function σ (t, T ) cannot be recovered from (1.19) in terms of the variables t, T and f (t, T ), making it impossible to return to (1.3) to query existence. Thus a direct proof of the existence of a solution to the system of equations (1.21) is required, and in the original BGM paper [30] it was pushed

10

Engineering BGM

through by letting all coverages be equal at δ (thereby creating a continuous spectrum of forwards K (t, T ) for all maturities), setting σ (t, T ) = 0 for t ≤ T ≤ t + δ, and then working in the setting of infinite dimensional SDEs. REMARK 1.4 An alternative way to construct BGM is suggested by the technique for jointly simulating the SDEs (1.21) under Pn . The expression for the drift in the SDE for K (t, Tj ) contains only later maturity forwards K (t, Tj+1 ),...,K (t, Tn−1 ), so one must work backwards j = n − 2, n − 3, ... obtaining successive incremented forwards K (t + ∆t, Tj ) from the later maturing forwards. The intuition that capacity to program is close to proof by construction, then suggests the backward construction approach of Section3.2.

Chapter 2 Bond and Swap Basics

The most basic of interest rate instruments are bonds and swaps, and in one form or another they underlie most activity in interest rate trading. The notion of a zero coupon bond, an asset paying 1 at its maturity, was introduced in Section-1.1. The more usual coupon bond, which pays a series of usually semiannual coupons before returning the principal 1 at maturity, can of course be expressed as a linear combination of zero-coupon bonds, but they are not really within the scope of this book, and we do not consider them. The basic object modelled in HJM - the instantaneous forward rate - is not observed in the market, so working with it requires abstraction through at least one intermediate layer to reach real world objects like bonds or options. Thus, for example, calibrating HJM to options, that is, fitting the instantaneous forward rate volatility function σ (t, T ) to option implied volatilities, can be unnecessarily complicated. Moreover, see Remark-1.2, HJM bond volatilities must be continuous, which is restrictive. For these reasons, in the first section, we develop some arbitrage free methods based on zero-coupon bonds and forward measures to tackle construction of shifted BGM by backward induction. The other fundamental interest rate instrument is the swap, in which interest rate risk is exchanged between two parties by swapping a sequence of floating Libor rates against a series of fixed coupons (which contrarily may actually be complex and variable in some exotic swaps). Thus, in the second section, we derive some fundamental model free results for swaps, while establishing a standard swap notation, summarized in Section-A.1, that we will adopt (as far as possible) throughout this book.

2.1

Zero coupon bonds - drifts and volatilities

We will assume that zero coupon bonds B (t, T ) follow Ito processes like dB (t, T ) = f (t) dt + b∗ (t, T ) dW (t) B (t, T )

t ≤ T,

(2.1)

11

12

Engineering BGM

under some fixed reference measure P (like the real world measure). Note that while the bond volatility vector b (t, T ) is maturity dependent and in general diﬀerent for each bond, the drift f (t) is assumed maturity independent and the same for all bonds. To justify that suppose the contrary were true and dB (t, T ) = f (t, T ) dt + b∗ (t, T ) dW (t) . B (t, T ) Presuming just a 2-factor model (the following argument works equally well for n-factors), arbitrarily select 3 bonds maturing at diﬀerent T1 , T2 and T3 , and eliminate the risky, that is, stochastic, components between them. That produces the following expression involving two determinants ¯ ¯ ¯ dB(t,T1 ) dB(t,T2 ) dB(t,T3 ) ¯ ¯ B(t,T1 ) B(t,T2 ) B(t,T3 ) ¯ ¯ ¯ (1) (2.2) ¯ b (t, T1 ) b(1) (t, T2 ) b(1) (t, T3 ) ¯ ¯ ¯ (2) ¯ b (t, T1 ) b(2) (t, T2 ) b(2) (t, T3 ) ¯ ¯ ¯ ¯ f (t, T1 ) f (t, T2 ) f (t, T3 ) ¯ ¯ ¯ (1) = ¯¯ b (t, T1 ) b(1) (t, T2 ) b(1) (t, T3 ) ¯¯ dt. ¯ b(2) (t, T1 ) b(2) (t, T2 ) b(2) (t, T3 ) ¯

The right-hand side of equation (2.2) is risk free, implying that if we took fixed at time t constant amounts ¯ (1) ¯ ¯ b (t, Tj ) b(1) (t, Tk ) ¯ 1 ¯ ¯ of bond B (t, Ti ) , Ai = B (t, Ti ) ¯ b(2) (t, Tj ) b(2) (t, Tk ) ¯ for

(i, j, k) = (1, 2, 3) ,

(2, 3, 1)

and

(3, 1, 2)

the resulting portfolio would instantaneously be at some risk free rate f (t). In other words, the left-hand of (2.2) would have form A1 dB (t, T1 ) + A2 dB (t, T2 ) + A3 dB (t, T3 ) = d [A1 B (t, T1 ) + A2 B (t, T2 ) + A3 B (t, T3 )] = [A1 B (t, T1 ) + A2 B (t, T2 ) + A3 B (t, T3 )] f (t) dt. Substituting back into the left hand determinant in (2.2), yields ¯ ¯ ¯ ¯ ¯ ¯ ¯ f (t, T1 ) f (t, T2 ) f (t, T3 ) ¯ 1 1 ¯ ¯ ¯ (1) 1 ¯ ¯ b (t, T1 ) b(1) (t, T2 ) b(1) (t, T3 ) ¯ f (t) = ¯ b(1) (t, T1 ) b(1) (t, T2 ) b(1) (t, T3 ) ¯ , ¯ ¯ ¯ ¯ ¯ b(2) (t, T1 ) b(2) (t, T2 ) b(2) (t, T3 ) ¯ ¯ b(2) (t, T1 ) b(2) (t, T2 ) b(2) (t, T3 ) ¯ or, combining everything on the right hand side ¯ ¯ ¯ f (t, T1 ) − f (t) f (t, T2 ) − f (t) f (t, T3 ) − f (t) ¯ ¯ ¯ ¯ b(1) (t, T1 ) b(1) (t, T2 ) b(1) (t, T3 ) ¯¯ = 0. ¯ ¯ b(2) (t, T1 ) b(2) (t, T2 ) b(2) (t, T3 ) ¯

Bond and Swap Basics

13

For this equation to hold for arbitrary T1 , T2 and T3 , the first row must be some time t dependent linear combination of the second and third rows, that ´∗ ³ is, there exists a function λ (t) = λ(2) (t) , λ(2) (t) such that for any T f (t, T ) − f (t) = λ(1) (t) b(1) (t, T ) + λ(2) (t) b(2) (t, T ) ⇒ f (t, T ) = f (t) + λ∗ (t) b (t, T ) , ⇒

(2.3)

dB (t, T ) = f (t) dt + b∗ (t, T ) [dW (t) + λ (t) dt] . B (t, T )

A change of measure then yields a model with drift independent of maturity. In an arbitrage free setting we know that there is a measure equivalent to P under which asset prices divided by the T -maturing zero coupon bond B (t, T ) as numeraire, are martingales, and in Section-1.1 that measure is identified as the forward measure PT . A better alternative is to simply define the forward measure PT (with corresponding BM WT (t)) as the martingale measure induced by B (t, T ) as numeraire. Thus for any zero coupon bond B (t, S) with an earlier maturity S (so t ≤ S < T ) we have d

³

B(t,S) B(t,T )

´. ³

B(t,S) B(t,T )

´

= [b (t, S) − b (t, T )]∗ [dW (t) − b (t, T ) dt]

(2.4)

∗

= b (t, S, T ) dWT (t) ,

where the BM WT (t) under PT , and the bond volatility diﬀerence b (t, S, T ) are specified by dWT (t) = dW (t)−b (t, T ) dt

and

b (t, S, T ) = b (t, S)−b (t, T ) . (2.5)

Observe that if S runs through a discrete set of nodes the resulting system of SDEs will clearly have a solution if, for example, the bond diﬀerences b (·, ·, T ) are bounded. REMARK 2.1 In future please distinguish between the two variable bond volatility functions b (t, T ) and the three variable bond volatility diﬀerence functions b (t, T, T1 ). For T + δ = T1 (replace S = T and T = T1 in the above) the measure change between PT and PT1 will be given by dWT1 (t) = dWT (t) + b (t, T, T1 ) dt.

(2.6)

Substituting zero coupons for the forward contract FT (t, T1 ) using the model free result (1.9), and applying (A.3.3), SDEs for FT (t, T1 ) and its

14

Engineering BGM

reciprocal are then given by ³ ´ 1) d B(t,T B(t,T ) 1) FT (t, T1 ) = B(t,T = [b (t, T1 ) − b (t, T )]∗ [dW (t) − b (t, T ) dt] , B(t,T ) B(t,T ) 1

B(t,T )

1 = FT (t, T1 )

d

B(t,T ) B(t,T1 )

³

B(t,T ) B(t,T1 )

B(t,T ) B(t,T1 )

´

∗

= [b (t, T ) − b (t, T1 )] [dW (t) − b (t, T1 ) dt] .

That is, for t ≤ T , their respective SDEs are dFT (t,T1 ) FT (t,T1 )

∗

= −b (t, T, T1 ) dWT (t) , ³ ´. ³ ´ ∗ 1 1 d FT (t,T FT (t,T1 ) = b (t, T, T1 ) dWT1 (t) , 1)

(2.7)

showing, as in the HJM framework, that the forward contract FT (t, T1 ) is a martingale under the forward measure PT at the start of its interval [T, T1 ], while its reciprocal is a martingale under the forward measure PT1 at the end. REMARK 2.2 The above formulation clearly includes HJM, with the bond volatility b (t, T ), and bond volatility diﬀerence or forward contract volatility b (t, T, T1 ) defined by b (t, T ) = −

Z

T

σ (t, u) du and b (t, T, T1 ) =

t

ZT1

σ (t, u) du

T

respectively.

2.2

Swaps and swap notation

An interest rate swap mediates risk between two parties by exchanging a sequence of floating Libor rates against a series of fixed coupons. On the floating side we assume fixings and cashflows take place at the floating side nodes Tj (j = 0, 1, 2, ..., n), where Libor L (t, Tj ), which is at a margin µj to the cash forward K (t, Tj ) L (t, Tj ) = K (t, Tj ) + µj , is fixed to L (Tj , Tj ) at node Tj and paid in arrears at node Tj+1 . The tenor intervals (Tj , Tj+1 ] (j = 0, 1, ..) thus defined by the sequence of floating time nodes Tj (j = 0, 1, 2, ..., n) are then assumed to be open on the

Bond and Swap Basics

15

left and closed on the right because cash usually flows at the end of an interval (after some action during that interval), and tracking cash is after all what is most important in finance! That means thinking of Libor L (t, Tj ) being fixed at the end of one interval (Tj−1 , Tj ], and then accruing over the next interval (Tj , Tj+1 ] to be paid at its end. Intervals are indexed by the unique time node they contain so that (Tj−1 , Tj ] is the j th floating interval for j = 1, 2, .... The index 0 is reserved for the time T0 = 0 unique point to which cashflows are present valued. Indexed quantities should be stored accordingly when programming; for example, Kj = K (0, Tj ) should be stored as the j th component of an initial cash forward vector K, while Bj+1 = B (0, Tj+1 ) should be stored as the (j + 1)th component of an initial discount vector B. This scheme turns out to suit C++ in which arrays start at 0, but in MatLab, where arrays start at 1, an extra number is necessary for the zero component. In practice we are only interested in a finite number of tenor intervals (for example, enough to include all the cashflows of the instruments in some portfolio) so the terminal node Tn is assumed to be at a time greater than all other relevant times. The notation does lead to some contortion (any notation will), for example the j th coverage δ j (period of accrual) of Libor L (t, Tj ) is actually the width ∆j+1 of the (j + 1)th interval (Tj , Tj+1 ]. We react by thinking of the coverage δ j as being associated with Libor L (t, Tj ) and hence the node Tj , that is δ j = δ (Tj ) = ∆j+1 = |(Tj , Tj+1 ]| = Tj+1 − Tj . But if the width of the interval (Tj , Tj+1 ] was specifically required, for example in an integration routine, we would use ∆j+1 . There is little confusion in practice. Similarly on the fixed side, fixings and cashflows are assumed to take place ¡ ¢ at the fixed side time nodes T i i = (0, 1, 2, ...), with coupon κi = κ T i determined at T i , paid at T i+1 and having the fixed side coverage ¯¡ ¤¯ ¡ ¢ δ i = δ T i = ∆i+1 = ¯ T i , T i+1 ¯ = T i+1 − T i .

Notation for expressions indexed by maturity, like forward measures PT (expectation ET ) and BM WT under PT , may be simplified in the obvious way when the maturities themselves are indexed, just put PTj = Pj

ETj = Ej

WTj = Wj .

With the above notation the time t value of a forward starting payer swap pSwap (t), in which N floating Libor rates are fixed at Tj0 , Tj1 , ..., TjN−1 and received in arrears against M ≤ N constant coupon κ payments fixed at

16

Engineering BGM

T i0 , T i1 , ..., T iM −1 and paid in arrears, is pSwap (t) = pSwap (t, M, N ) ¢ ¡ = pSwap t, κ, Tj0 , .., TjN , T i0 , .., T iM , M, N ¯   P ¯  β (t)    δ L (T , T )  j∈{j0,j1,...,jN −1}  j j j ¯¯ β (Tj+1 ) ¯ Ft , = E0 P ¯ β (t)   − i∈{i0,i1,...,iM −1} ¡  ¢ δ i κ ¯¯    β T i+1 =

jN−1 X j=j0

iM −1 X ©£ ¤¯ ª ¢ ¡ ¯ δ j B (t, Tj+1 ) Ej+1 K (Tj , Tj ) + µj Ft − κ δ i B t, T i+1 . i=i0

Hence, because K (t, Tj ) is a PTj+1 -martingale, pSwap (t) = pSwap (t, M, N ) =

jN −1 X j=j0

(2.8)

δ j B (t, Tj+1 ) L (t, Tj ) − κ

= B (t, Tj0 ) − B (t, TjN ) +

jN −1 X j=j0

iM −1 X i=i0

¢ ¡ δ i B t, T i+1 ,

¢ ¡ δ j µj B (t, Tj+1 ) − κ level t, T i0 , T iM ,

in terms of the level function ¡ ¢ ¡ ¢ level (t) = level t, T i0 , T iM = level t, T i0 , T iM , δ i0 , .., δ iM −1 =

iM −1 X i=i0

(2.9)

¢ ¡ δ i B t, T i+1 .

Note that payer in payer swap pSwap (t) means the coupon is paid, in contrast to a receiver swap rSwap (t) in which the coupon is received. This use of payer and receiver is standard and always refers to the coupon. In these expressions the relevant floating and fixed nodes are respectively {Tj0 , Tj1 , ..., TjN }

and

ª © T i0 , T i1 , ..., T iM ,

and if we are dealing concurrently with several swaps we will retain this notation. But if just one swap is under consideration, there is no loss of mathematical clarity and some simplification in modifying the above notation and letting the floating and fixed nodes be respectively {T0 , T1 , ..., TN }

and

ª © T 0 , T 1 , ..., T M ,

so that indices are simplified and the payer swap written in the standard swap

Bond and Swap Basics

17

form pSwap (t) =

N −1 X j=0

δ j B (t, Tj+1 ) L (t, Tj ) − κ

= B (t, T0 ) − B (t, TN ) +

N−1 X j=0

M−1 X i=0

¢ ¡ δ i B t, T i+1 ,

(2.10)

¢ ¡ δ j µj B (t, Tj+1 ) − κ level t, T 0 , T M .

Because indices begin at 0 (as opposed to say 1) in such formulae, they are easy to program; simply add j0 or i0 throughout to the respective floating or fixed side indices. Note also, that when we are using this standard form our modification of notation requires that the nodes {T0 , T1 , ..., TN } be regarded as a finite set tailored to a specific swap, with T0 not necessarily zero. Continuing this modification of notation, when the meaning is clear we often drop the 0 suﬃx setting, for example, T0 = T and δ 0 = δ. Thus if focused on just one Libor rate or cash forward fixing at T0 and paying at T1 , we would probably refer to L (t, T ) or K (t, T ) meaning that the rate is set at T = T0 and paid at T1 = T0 + δ 0 = T + δ. Similarly, in the next equation (2.11), we might set the common maturity T0 of the forwards to T , and write FT0 (t, Tj+1 ) = FT (t, Tj+1 ). Introducing the corresponding forward swap rate, a weighted average of Libor defined as that value of κ which makes the swap value zero PN −1 j=0 δ j B (t, Tj+1 ) L (t, Tj ) (2.11) ω (t) = ω (t, M, N ) = ¢ ¡ PM−1 i=0 δ i B t, T i+1 PN−1 j=0 δ j FT0 (t, Tj+1 ) L (t, Tj ) = , ¡ ¢ PM−1 i=0 δ i FT0 t, T i+1 B (t, T0 ) − B (t, TN ) ¢ ¡ = if µj = 0 j = 0, .., N − 1 level t, T 0 , T M the payer swap can be rewritten as the product of the level function and the diﬀerence between the swaprate ω (t) and coupon κ like (M−1 ) X ¢ ¡ δ i B t, T i+1 [ω (t) − κ] . pSwap (t) = (2.12) i=0

= level (t) [ω (t) − κ] .

When the swap starts now at time t = 0 with the first fixing at T0 = 0, (2.11) generates the current swaprate, a value that is used as the strike for most traded vanilla swaps. Note that if the margins µj are set zero and the level function eliminated between (2.10) and (2.12), the swap can be expressed as ¸ · κ pSwap (t) = {B (t, T0 ) − B (t, TN )} 1 − ω (t)

18

Engineering BGM

in terms of just three variables, the swaprate ω (t) and the zero coupons B (t, T0 ) and B (t, TN ) maturing at the start and end of the swap respectively. To tackle the diﬀerent fixed and floating schedules and retain clarity of exposition, we often make the further simplifying assumption that each fixed side T i coincides with some floating side Tj , rolling at regular multiples r called the roll - of the floating side nodes (for example, r = 1 for quarterly fixed side coupon, r = 2 for semiannual coupon and r = 4 for annual coupon), and in particular T 0 = T0 = 0

T i0 = Tj0

T iM = TjN .

That way quantities associated with the fixed side can also be easily indexed to the floating side, and if we know the floating side schedule {T0 , ªT1 , ...} and © the roll r, then we can work out the fixed side schedule T 0 , T 1 , ... and fixed © ª side coverages δ 0 , δ 1 , ... in terms of the floating nodes and coverages. To help map fixed side indices to corresponding floating side indices we introduce the index mapping function J : {i0, i1, .., iM } → {j0, j1, .., jN } , where

J (i) = j

⇔

(2.13)

Tj = T i ,

which identifies the floating side node Tj corresponding to some fixed side node T i .

2.2.1

Forward over several periods

Libors over several periods usually have diﬀerent margins over the cash forwards, so let the N -period Libor L(N ) (t, T0 ) over (T0 , TN ] be at margin µ(N) to the cash forward K (N ) (t, T0 ). We have L(N ) (t, T0 ) = K (N ) (t, T0 ) + µ(N ) , (N )

1 + δ 0 K (N ) (t, T0 ) =

N −1 Y j=0

[1 + δ j K (t, Tj )] =

1 , FT0 (t, TN )

(N )

δ0

=

N −1 X

δj ,

j=0

showing that K (N) (t, T0 ) and L(N) (t, T0 ) are PTN -martingales. If L(N) (t, T0 ) is fixed at T0 and exchanged against κ at time TN (where TN = T M and (N ) δ 0 = δ 0 ), the present value of the resulting swap is ½ ´¯¯ ¾ β (t) (N) ³ (N ) δ0 (T0 , T0 ) − κ ¯¯ Ft , L pSwap (t) = E0 β (TN ) ³ ´ (N ) = δ 0 B (t, TN ) L(N) (t, T0 ) − κ , =

N −1 X j=0

h i (N) δ j B (t, Tj+1 ) K (t, Tj ) − δ 0 B (t, TN ) κ − µ(N ) .

Bond and Swap Basics

19

In this case the swaprate, which is a martingale under PTN , is ω (t) = K

2.2.2

(N)

(t, T0 ) =

PN −1 j=0

δ j B (t, Tj+1 ) K (t, Tj ) (N) δ0 B

+ µ(N) .

(t, TN )

Current time

To locate the current time t among the floating side nodes define the att function @ (·) by ª © @ (t) = inf ∈ Z+ : t ≤ T . Thus @ (t) = j, or simply @ = j, means that the present time t is in the j th floating interval (Tj−1 , Tj ], and we can write without confusion T@ = Tj , T@−1 = Tj−1 , T@+k = Tj+k , ... etc t ∈ (T@−1 , T@ ] = (Tj−1 , Tj ] = J (t) = J and refer to J as the current interval. Because neither Libor L (t, T ) nor the cash forward K (t, T ) live past their maturity T , we make the convention that all maturity dependent functions h (t, T ) die at T , that is h (t, T ) = h (T, T )

for t > T.

So during the current interval J • functions h (t, T@ ) are alive until the end of J, • functions h (t, T ) for T ∈ {T@+1 , T@+2 , ...} continue to live on subsequent intervals, • and functions h (t, T ) for T ∈ {T0 , T1 , ..., T@−1 } have died at the ends of previous intervals. These notions will be most useful in developing the spot Libor measure, see Section-5.3.

Chapter 3 Shifted BGM

Shifted (or displaced diﬀusion) BGM generalizes flat BGM by making the combination of a cash forward K (t, T ) plus a shift a (T ) lognormal under PT +δ with deterministic volatility ξ (t, T ). By varying the shift a (T ) the resulting model can be made to move between flat BGM (a (T ) = 0) and Gaussian HJM (a (T ) = 1δ ). That sort of flexibility is viewed as a plus by those traders who see the market either behaving normally or at least something less than lognormally. A disadvantage (acceptable in other models) is that rates can go negative. Shifted BGM, similarly to the CEV model, generates cap or swaption implied volatility skews but not smiles. Nevertheless for modest shift values, for example, 0.05 < a (T ) < 0.20, it can bestfit a cap or swaption smile quite well. So shifted BGM is a good start for the simple stochastic volatility version of BGM, introduced in Chapter-16, in which ξ (t, T ) is made stochastic, with the shift part fitting the skew and the stochastic volatility adding curvature to the wings. The author acknowledges the CEV model, as popularized by Andersen & Andreasen [4], is a viable alternative way of introducing a skew in BGM and has the advantage of retaining positive forwards. But the relative complexity of CEV computations compared to the shifted model, inclines the author to favour the latter for a basic standard skew BGM model.

3.1

Definition of shifted model

In shifted BGM the shifted forward H (t,³Tj ) which is´the driver, cash forward K (t, Tj ), Libor and shift a (Tj ) = aj 0 ≤ aj ≤ δ1j are related by H (t, Tj ) = K (t, Tj ) + a (Tj ) = L (t, Tj ) − µj + a (Tj ) ,

(3.1)

21

22

Engineering BGM

where H (t, Tj ) is lognormal with deterministic volatility ξ (t, T ) under PTj+1

⇒

dH (t, Tj ) = ξ ∗ (t, Tj ) dWTj+1 (t) H (t, Tj ) ½Z t ¾ H (t, Tj ) =E ξ ∗ (s, Tj ) dWTj+1 (s) . H (0, Tj ) 0

(3.2)

Critically, because the shift a (Tj ) is time t independent dH (t, Tj ) = dK (t, Tj ). From the zero coupon bond SDEs (2.1) and (2.4) ¶ µ 1 B (t, Tj ) , dH (t, Tj ) = H (t, Tj ) ξ ∗ (t, Tj ) dWTj+1 (t) = dK (t, Tj ) = d δj B (t, Tj+1 ) ¶ µ 1 B (t, Tj ) ∗ = b (t, Tj Tj+1 ) dWTj+1 (t) . δ j B (t, Tj+1 ) so the bond volatility diﬀerence and corresponding measure change (2.6) are b (t, Tj , Tj+1 ) = ⇒

δ j H (t, Tj ) ξ (t, Tj ) = hj (t) ξ (t, Tj ) , [1 + δ j K (t, Tj )]

(3.3)

dWTj+1 (t) = dWTj (t) + hj (t) ξ (t, Tj ) dt,

where the j th drift term hj (t) is a martingale under PTj dhj (t) = hj (t) [1 − hj (t)] ξ ∗ (t, Tj ) dWTj (t) .

(3.4)

Similarly to the flat caplet formula (1.20), in shifted BGM the time t value of a Black caplet struck at κj , setting at Tj and paying at Tj+1 , is ¯ ¾ ½ δ j β (t) + ¯¯ [L (Tj , Tj ) − κj ] ¯ Ft , (3.5) Cpl (t) = Cpl (t, κ, Tj ) = E0 β (Tj+1 ) " ¯  ³R ´ #+ ¯ T  ¯  H (t, Tj ) E t j ξ ∗ (t, Tj ) dWTj+1 (t) ¯ Ft , = δ j B (t, Tj+1 ) ETj+1 ¡ ¢ ¯   − κj − µj + a (Tj ) ¯   s · ¸ · ¸ T j R L (t, T κ − µ ) − µ 2 j j j = δ j B (t, Tj+1 ) B  , , |ξ (s, Tj )| ds . +a (Tj ) +a (Tj ) t That is, the formula for the price of a caplet in shifted BGM is like that in the flat case, except that the shift is added to both forward and strike.

3.1.1

Several points worth noting

1. The constant elasticity of variance (CEV) model is similar to shifted BGM in that both generate skews but not smiles. Comparing their SDEs in a one-factor framework dL (t, T ) = [L (t, T ) + a (T )] ξ (t, T ) dWT1 (t) , dL (t, T ) = Lβ (t, T ) γ (t, T ) dWT1 (t) ,

Shifted BGM

23

and asking that their instantaneous volatilities initially have the same value and slope gives a rule-of-thumb relationship between β and a (T ) a (T ) =

(1 − β) L (0, T ) . β

2. For a (Tj ) < δ1j clearly 0 < hj (t) < 1 and so the bond volatility diﬀerences b (t, Tj , Tj+1 ) will be positive and bounded. Moreover, from (3.4), the stochastic component of the volatility of hj (·) lies in (0, 1) suggesting the initial value hj (0) as a possible approximation for stochastic hj (t). 3. Setting the shift equal to the inverse of the coverage a (Tj ) = δ1j (about 4, which is quite large), makes the bond volatility diﬀerence (3.3) deterministic h i δ j K (t, Tj ) + δ1j ξ (t, Tj ) = ξ (t, Tj ) , b (t, Tj , Tj+1 ) = [1 + δ j K (t, Tj )] and the the model Gaussian. The shift can be negative, but that produces a skew increasing with strike which is not observed in the market. Hence the practical operating range for the shift is 0 ≤ aj ≤ δ1j ; that is, shifted BGM lives between flat lognormal and Gaussian skew extremes. 4. The magnitude of the volatility ξ (·) varies with the shift and can be quite diﬀerent from flat BGM volatility. From (3.3), a rule of thumb conversion between shifted models ha1 (T ) , ξ 1 (t, T )i and ha2 (T ) , ξ 2 (t, T )i is [K (t, T ) + a1 (T )] ξ 1 (t, T ) ∼ = [K (t, T ) + a2 (T )] ξ 2 (t, T ) . Hence a flat (that is, zero shift) lognormal volatility of 70% converts to 46% at shift a (T ) = 2%, and to 0.7% in the Gaussian case at shift a (T ) = 400%. 5. An alternative way of setting up shifted BGM is to write it in the aﬃne form dK (t, T ) = {β (T ) K (t, T ) + [1 − β (T )] K (0, T )} ξ ∗ (t, T ) dWT1 (t) , ¾ ½ [1 − β (T )] K (0, T ) β (T ) ξ ∗ (t, T ) dWT1 (t) . = K (t, T ) + β (T ) That stabilizes the magnitude of ξ (t, T ) as β (T ) changes because, from the previous rule of thumb, for diﬀerent hβ (T ) , ξ (t, T )i regimes [β 1 (T ) K (t, T ) + [1 − β 1 (T )] K (0, T )] ξ 1 (t, T ) ∼ = [β 2 (T ) K (t, T ) + [1 − β 2 (T )] K (0, T )] ξ 2 (t, T ) , and the [β (T ) K (t, T ) + [1 − β (T )] K (0, T )] terms on each side will tend to be similar.

24

Engineering BGM

3.2

Backward construction

Using the bond basics of Section-2.1 and some of the techniques introduced by Musiela and Rutkowski in [79] and [80] for flat BGM, we can now implement a backward construction of shifted BGM. Temporarily simplify notation by indexing variables by maturity and dropping time t, for example, set BT = B (t, T )

WT = WT (t)

bT = b (t, T ) .

For 0 ≤ t ≤ R < S < T < U we will illustrate the method by moving back through the intervals (T, U ], (S, T ] and (R, S] successively defining the bond volatility diﬀerences [bT − bU ], [bS − bT ] and [bR − bS ] in the process. Our basic assumption and starting point is that the deterministic forward volatilities ξ R , ξ S and ξ T are specified exogenously and are bounded, and that KT is a martingale under PU . Hence for the three intervals we have BT −1 HT = KT + aT BU dHT = HT ξ ∗T dWU , BS δ S KS = −1 HS = KS + aS δ S = |(S, T ]| BT dHS = HS [µS dt + ξ ∗S dWU ] , BR δ R KR = −1 HR = KR + aR δ R = |(R, S]| BS dHR = HR [µR dt + ξ ∗R dWU ] . δ T = |(T, U ]|

δ T KT =

On the first interval (T, U ], comparison of volatility terms in δ T dHT = δ T HT ξ ∗T dWU = δ T dKT = d

µ

BT BU

¶

=

µ

BT BU

¶

∗

[bT − bU ] dWU

identifies the diﬀerence in the bond volatilities [bT − bU ] =

δ T HT ξ , 1 + δ T KT T

which must be bounded (assuming δ T aT < 1) because the SDE for HT has a positive solution, and that defines a change of measure from PU to PT by dWT = dWU − [bT − bU ] dt = dWU −

δ T HT ξ dt. 1 + δ T KT T

Shifted BGM

25

On the second interval (S, T ], in the equation ¶ µ Á ¶ µ BS BT BS =d δ S dHS = δ S HS [µS dt + ξ ∗S dWU ] = δ S dKS = d BT BU BU ¶ µ BS [bS − bT ]∗ {dWU − [bT − bU ] dt} , = BT comparison of volatility and drift terms yields ¶ ¶ µ µ BS BS ∗ [bS − bT ] , δ S HS µS = − [bS − bT ] [bT − bU ] ⇒ δ S HS ξ S = BT BT µS = −ξ ∗S [bT − bU ] ⇒ dHS = HS ξ ∗S [dWU − [bT − bU ] dt] = HS ξ ∗S dWT . The SDE for HS has a solution which is strictly positive, which in turn defines the bond volatility diﬀerence [bS − bT ] =

δ S HS ξ , 1 + δ S KS S

and a change of measure from PU to PS by dWS = dWU − [bS − bU ] dt = dWU − [(bS − bT ) + (bT − bU )] dt ¾ ½ δ S HS δ T HT dt. ξ + ξ = dWU − 1 + δ S KS S 1 + δ T KT T On the third interval (R, S] proceed as on the second. δ R dHR = δ R HR [µR dt + ξ ∗R dWU ] ¶ µ BR ∗ [bR − bS ] {dWU − [bS − bU ] dt} = BS ⇒ dHR = HR ξ ∗R [dWU − [bS − bU ] dt] = HR ξ ∗R dWS δ R HR ⇒ [bR − bS ] = ξ . 1 + δ R KR R As before, the SDE for HR has a strictly positive solution which defines [bR − bS ] and the next measure change from PU to PR by dWR = dWU − [bR − bU ] dt, ¾ ½ δ S HS δ T HT δ R HR dt. ξ + ξ + ξ = dWU − 1 + δ S KS S 1 + δ T KT T 1 + δ R KR R The measures PR , PS and PT thus defined are clearly the forward ³ measures ´ B at R, S and T . For example, for any bond BQ (Q < T ) the ratio BQ must T be a PT -martingale because ¶ µ Á ¶ µ ¶ µ BQ BT BQ √ BQ =d = V [bQ − bT ]∗ dWT . d BT BU BU BT

26

Engineering BGM

In general, starting with the terminal measure Pn at the terminal node Tn and working backwards in the above fashion constructs the system of equations (1.21) with each forward K (t, Tj ) an exponential martingale under the corresponding forward measure Pj+1 . REMARK 3.1 The system {K (·, Tj ) : 0, 1, .., n − 1} we have constructed is finite and discrete with variable coverages (an essential feature to fit holidays and business conventions). To make the model operational, however, we often need K (t, T ) for maturities T strictly within the tenor intervals; that requires intelligent interpolation methods and is the topic of a later chapter.

Chapter 4 Swaprate Dynamics

This chapter is devoted to analyzing swaprate dynamics, finding suitable deterministic approximations to swaprate volatilities, and deriving corresponding swaption formulae. In the shifted-BGM model constructed in the last chapter the forwards plus time independent shifts were dynamically lognormal under the appropriate forward measures. We now show, following [31] (and others - it’s the sort of result many implementers have probably obtained independently) that swaprates in this framework exhibit the same sort of behavior as forwards in that swaprates plus shifts that are nearly time independent are dynamically almost lognormal under new measures, equivalent to the forward measures, that we call swaprate measures. Thus in the first Section-4.1 we separate the swaprate into its shift and stochastic parts, and then in Section-4.2 and Section-4.3, analyze each separately and justify our approximations. A consequence obtained in Section-4.4, is that in shifted-BGM swaptions can be priced accurately with Black type formulae very similar to the ones (3.5) used to price caplets. This is a standard outcome in the BGM framework whatever method prices caplets, also usually fairly accurately prices swaptions after some necessary adjustments and approximations. For example, see Chapter-16 on the stochastic volatility version of BGM for a similar outcome, or [9] and [95] for generic methods using Markovian projection. When the shift in BGM is zero the swaprate SDEs (4.13) permit, see Section-4.5, the easy derivation of Jamshidian’s [66] swaprate model in which the swaprates of coterminal swaps can be made jointly lognormal under a collection of appropriate swaprate measures (rather similar to the forwards under the collection of forward measures in BGM). But more than that, it’s possible to construct many other market models in which the swaprates of any set of swaps with a strictly increasing total tenor structure, which may include forwards, can be made jointly lognormal under appropriate measures. When the shift is non-zero, however, further work is needed to properly construct a Jamshidian type swaprate model that behaves like the forwards in shifted BGM with an exact closed formula for swaptions corresponding to (3.5) for caplets. Among all the market models, the relative algebraic simplicity of shiftedBGM combined with its ability to handle swap dynamics and calibrate to

27

28

Engineering BGM

both caps and swaptions, makes it a central interest rate model.

4.1

Splitting the swaprate

From (2.11) and (3.1), and letting T = T0 = T 0 the swaprate in shifted BGM is PN−1 j=0 δ j FT (t, Tj+1 ) H (t, Tj ) (4.1) ω (t) = ¡ ¢ PM−1 i=0 δ i FT t, T i+1 ¤ £ PN −1 j=0 δ j FT (t, Tj+1 ) a (Tj ) − µj − , ¡ ¢ PM−1 i=0 δ i FT t, T i+1

and we will show that its stochastic part ω1 (t) is almost lognormal PN −1 j=0 δ j FT (t, Tj+1 ) H (t, Tj ) ω1 (t) = ¡ ¢ PM−1 i=0 δ i FT t, T i+1 =

N −1 X

uj (t) H (t, Tj ) =

j=0

N −1 X

vj (t) ,

j=0

while what we will call its shift part ω2 (t) is almost constant ¤ £ PN −1 j=0 δ j FT (t, Tj+1 ) a (Tj ) − µj ω2 (t) = ¡ ¢ PM−1 i=0 δ i FT t, T i+1 =

N−1 X j=0

(4.2)

(4.3)

¤ £ uj (t) a (Tj ) − µj .

The following lemma helps analyze components of the decomposition. LEMMA 4.1 If Y (t) is an Ito process and X0 , .., XM−1 are exponential martingales under some reference measure P dY (t) = µ (t) dt+ξ ∗ (t) dW (t) , Y (t)

dXi (t) = σ ∗i (t) dW (t) Xi (t)

e is measure equivalent to P defined by and P PM−1 Xi (t) σi (t) f (t) = dW (t) − i=0 dW dt, PM−1 i=0 Xi (t)

(i = 0, .., M − 1) ,

then

(4.4)

Swaprate Dynamics

29

¶

µ

Y (t) " #∗ PM−1 d PM−1 X (t) X (t) σ (t) i i i i=0 f (t) . µ ¶ = µ (t) dt + ξ (t) − i=0 dW PM−1 Y (t) X (t) i i=0 PM−1 i=0

Hence

S1 Xi

(4.5)

Xi (t)

e is a P-martingale, and so is

SY Xi

when Y is a P-martingale.

PROOF Bearing in mind that each Xi (t) is strictly positive, an SDE for PM −1 i=0 Xi (t) is PM −1 PM−1 Xi (t) σ ∗i (t) d i=0 Xi (t) = i=0 dW (t) . PM−1 PM−1 i=0 Xi (t) i=0 Xi (t)

The result follows on applying (A.3.3) to Y (t) and

PM−1 i=0

Xi (t)

¶ µ i∗ h SM−1 Y (t) d PM−1 i=0 Xi (t)σ i (t) S ξ (t) − M−1 Xi (t) i=0 i=0 Xi (t) i . µ ¶ = µ (t) dt + h SM−1 Xi (t)σ i (t) i=0 Y (t) S × dW (t) − dt M−1 PM−1 i=0 Xi (t) i=0

Xi (t)

e is determined solely by the denominator PM−1 Xi (t). Note that P i=0

4.2

The shift part

Recalling the weights uj (t) arise as the coeﬃcients of H (t, Tj ) and a (Tj ) in the formulae (4.2) and (4.3), and introducing ui (t) and f (t), for j = 0, .., N −1 and i = 0, .., M − 1 we have f (t) = PM−1 i=0

uj (t) = δ j FT (t, Tj+1 ) f (t) ,

1

¡ ¢, δ i FT t, T i+1

¡ ¢ ui (t) = δ i FT t, T i+1 f (t) .

(4.6)

¡ ¢ In Lemma 4.1 set Xi (t) = δ i FT t, T i+1 , Y (t) = δ j FT (t, Tj+1 ) with SDEs d [δ j FT (t, Tj+1 )] = −b (t, T, Tj+1 )∗ dWT (t) , [δ j FT (t, Tj+1 )] ¡ ¢¤ £ ¢∗ ¡ d δ i FT t, T i+1 £ ¡ ¢¤ = −b t, T, T i+1 dWT (t) . δ i FT t, T i+1

30

Engineering BGM

e eT equivIf the corresponding P-measure is defined be the swaprate measure P f alent to PT induced by the BM WT (t) where fT (t) = dWT (t) + dW

M −1 X i=0

¢ ¡ ui (t) b t, T, T i+1 dt

then

(4.7)

M −1 X ¡ ¢ df (t) fT (t) , ui (t) b∗ t, T, T i+1 dW = f (t) i=0 " #∗ M−1 X ¢ ¡ duj (t) fT (t) , = −b (t, T, Tj+1 ) + ui (t) b t, T, T i+1 dW uj (t) i=0 " #∗ M−1 ¢ X ¢ ¡ ¡ dui (t) fT (t) , = −b t, T, T i+1 + ui (t) b t, T, T i+1 dW ui (t) i=0

e is that it is a sort of average The best intuition for the swaprate measure P of the M forward measures Pi (i = 1, ..M ) over the tenor of the underlying forward swap in the sense that fT (t) = dW

M−1 X

ui (t) dWT i+1 (t)

i=0

¢ ¡ (get this result by substituting dWT i+1 (t) = dWT (t)+b t, T, T i+1 dt in (4.7) and using the fact that the ui (t) are weights summing to 1). eT -martingales with small volatilities because the Both uj (t) and ui (t) are P terms they contain will tend to cancel. Their initial values uj (0) and ui (0) ought therefore to be good approximations to subsequent values. Hence the following approximation to the shift part (4.3) of the shifted BGM swaprate. ¤ £ PN −1 j=0 δ j FT (0, Tj+1 ) a (Tj ) − µj ∼ (4.8) ω2 (t) = ω2 (0) = ¡ ¢ PM−1 i=0 δ i FT 0, T i+1 =

N −1 X j=0

¤ £ uj (0) a (Tj ) − µj .

The drift term appearing in the measure change (4.7), can be rewritten by using (3.3) and changing the order of summation M −1 X i=0

=

N −1 X j=0

X ¢ M−1 ¡ ui (t) b t, T, T i+1 = ui (t)

− → u j (t)

i=0

j X =0

h (t) ξ (t, T ) =



N−1 X j=0

J(i+1)−1

X

h (t) ξ (t, T )

=0

h (t)  j

N−1 X =j

− → u (t)

  

ξ (t, Tj )

Swaprate Dynamics

31

→ where the vector − u (t) is simply the vector u (t) with zeros added so as to be able to sum over the floating side j rather than the fixed side i. Specifically ½ ui (t) when j = J (i + 1) − 1 (i = 0, ..., M − 1) → − u j (t) = , (4.9) 0 when j ∈ / {J (i + 1) − 1 : i = 0, 1, ..., M − 1} → → u (t) = u (t). that is, all the − u (t) are 0 except for the − j

4.3

i

J(i+1)−1

The stochastic part

The second formula concerns the j th term in the stochastic part of the swaprate (4.2) δ j FT (t, Tj+1 ) H (t, Tj ) vj (t) = PM−1 ¡ ¢ i=0 δ i FT t, T i+1

j = 0, .., N − 1,

(4.10)

An SDE for the top comes from (3.2), (2.7) and (A.3.3) dH (t, Tj ) = ξ ∗ (t, Tj ) dWTj+1 (t) = ξ ∗ (t, Tj ) [b (t, T, Tj+1 ) dt + dWT (t)] H (t, Tj ) dFT (t, Tj+1 ) and = −b (t, T, Tj+1 )∗ dWT (t) ⇒ FT (t, Tj+1 ) d [δ j FT (t, Tj+1 ) H (t, Tj )] = {ξ (t, Tj ) − b (t, T, Tj+1 )}∗ dWT (t) . (4.11) [δ j FT (t, Tj+1 ) H (t, Tj )] To apply Lemma (4.1) set Y (t) = δ j FT (t, Tj+1 ) H (t, Tj ), and then ¸∗ · ξ (t, T ) − b (t, T, Tj+1 ) dvj (t) fT (t) , ¢ dW ¡ PM−1j = + i=0 ui (t) b t, T, T i+1 vj (t)

(4.12)

establishing that vj (t) is an exponential martingale with approximate volatility ξ (t, Tj ). From this equation (4.12) and (4.2), and introducing the weights δ j FT (t, Tj+1 ) H (t, Tj ) wj (t) = PN−1 j=0 δ j FT (t, Tj+1 ) H (t, Tj )

j = 0, .., N −1

⇒

N −1 X

wj (t) = 1,

j=0

an exact (no approximations involved) SDE for the stochastic part ω1 (t) of the swaprate is therefore PN−1 dω1 (t) j=0 dvj (t) fT (t) where = PN −1 (4.13) = σ ∗ (t) dW ω1 (t) v (t) j j=0 " # N−1 M−1 X X ¢ ¡ wj (t) ξ (t, Tj ) − b (t, T, Tj+1 ) + ui (t) b t, T, T i+1 . σ (t) = j=0

i=0

32

Engineering BGM

Substituting for the bond volatility diﬀerences b (·) from (3.3) and changing the order of summation using the index mapping function J (2.13) and the → − u j (·)-vector (4.9), the swaprate volatility σ (t) can be expressed as a linear combination of the forward volatilities ξ (t, Tj ) as follows   J(i) j N−1 M−1 X X X X wj (t) ξ (t, Tj ) − h (t) ξ (t, T ) + ui (t) h (t) ξ (t, T ) σ (t) = j=0

=

N−1 X j=0

=

wj (t) ξ (t, Tj ) − 



N−1 X j=0

i=0

=0

N −1 X

+

j=0

N −1 N −1 X X

wj (t) h (t) ξ (t, T )

=0 j=



wj (t)

w (t) − hj (t)  j

=0

N −1 X

− → u k (t)

k=0

N −1 X =j

k X

h (t) ξ (t, T ) ,

=0

  − (w (t) − → u (t)) ξ (t, Tj ) . 

Within this expression for σ (t) the weights wj (t) dominate. Further, using (4.11) and Lemma 4.1, SDEs for the weights wj (t) are · ¸∗ ξ (t, Tj ) − b (t, T, Tj+1 ) dwj (t) fT• (t) PN −1 = dW − j=0 wj (t) {ξ (t, Tj ) − b (t, T, Tj+1 )} wj (t) fT• (t) = dWT (t) − dW

N −1 X j=0

wj (t) {ξ (t, Tj ) − b (t, T, Tj+1 )} dt,

e• induced by showing they are low variance martingales under a measure P T • f WT (t) and may be approximated by their initial values wj (t) ∼ = wj (0). That permits a deterministic approximation for the swaprate volatility σ (t) ∼ =

N−1 X

wj (0) ξ (t, Tj )

j=0

that is suitable, for example, for correlation analysis, see Chapter-6. Moreover, → u (t) can be approximated by their initial values, it recalling that hj (t) and − is tempting to use the deterministic approximation   N −1  N −1  X X → wj (0) − hj (0) (w (0) − − u (0)) ξ (t, Tj ) . σ (t) ∼ =   j=0

=j

This approximation is accurate and satisfactory in unshifted BGM, but as was pointed out to the author1 , for large shifts and diﬀerent fixed and floating 1 My

thanks to Marc-Olivier Seguin for this critical bit of information.

Swaprate Dynamics

33

schedules (for example, swaps of quarterly Libor against semi-annual coupon), this formula fails and a diﬀerent approximation is required. Start by considering how changes in vj (t) arise. We have vj (t) = uj (t) H (t, Tj ) = uj (t) [K (t, Tj ) + a (Tj )] , where a (Tj ) may be (much) larger than K (t, Tj ) and is time independent, while the uj (t) are virtually constant. So the uj (t) a (Tj ) term varies little in contrast to uj (t) K (t, Tj ), but is much larger, suggesting it should be immediately isolated. That leads to the approximation vj (t) ∼ = uj (t) K (t, Tj ) + uj (0) a (Tj ) d [uj (t) K (t, Tj )] d [uj (t) K (t, Tj )] K (t, Tj ) dvj (t) = = vj (t) uj (t) H (t, Tj ) uj (t) K (t, Tj ) H (t, Tj )

⇒ j = 0, .., N − 1.

eT -martingales, and Because both vj (t) and uj (t) are P

d [uj (t) K (t, Tj )] = dvj (t) − a (Tj ) duj (t) ,

eT -martingale. Also, because the PT clearly uj (t) K (t, Tj ) must also be a P j e forward and P-swaprate measures are equivalent, we can write dH (t, Tj ) H (t, Tj ) H (t, Tj ) ∗ dK (t, Tj ) fT (t) . = = (drift) dt + ξ (t, Tj ) dW K (t, Tj ) H (t, Tj ) K (t, Tj ) K (t, Tj )

Hence, using the SDE (4.7) for uj (t), the SDE for uj (t) K (t, Tj ) must be " #∗ H(t,Tj ) d [uj (t) K (t, Tj )] ξ (t, T ) − b (t, T, T ) j j+1 ) fT (t) , = K(t,T dW ¢ ¡ PjM−1 uj (t) K (t, Tj ) + i=0 ui (t) b t, T, T i+1

and it follows that an approximate SDE for vj (t) is ∗  ξ (t, Tj ) ¶ µ dvj (t) ∼  fT (t) .  dW −b (t, T, Tj+1 ) = K(t,T ) ¢ ¡ PM−1 + H(t,Tjj ) vj (t) + i=0 ui (t) b t, T, T i+1

(4.14)

Note that the two SDEs (4.12) and (4.14) for the kosher and approximate vj (t) both coincide in the flat case. Hence the following approximate SDE for the stochastic part (4.2) of the swaprate dω1 (t) fT (t) , = σ ∗ (t) dW ω1 (t) ¶¸ µ · N−1 X K (t, Tj ) −b (t, T, Tj+1 ) ∼ ¢ ¡ P . wj (t) ξ (t, Tj ) + σ (t) = M−1 H (t, Tj ) + i=0 ui (t) b t, T, T i+1 j=0

34

Engineering BGM

Substituting for the b (·) and changing the order of summation as above, then expresses σ (t) as a linear combination of the ξ (t, Tj ) σ (t) ∼ =



N−1 X j=0

w (t) − hj (t)  j

N −1 X =j

 K (t, T )  w (t)  ξ (t, Tj ) .  H (t, T ) PN −1  − k) −→ u (t) k=0 K(t,T H(t,Tk ) wk (t) 

Putting everything together yields the following lognormal SDE approximation to the stochastic part (4.2) of the shifted BGM swaprate NP −1 dω1 (t) fT (t) , = σ ∗ (t) dW σ (t) = Aj ξ (t, Tj ) ω1 (t) j=0   ¶ N−1 N −1  Xµ K X K → , λ= w −− u λ w Aj = wj − hj   H H =j

wj = wj (0)

4.4

hj = hj (0)

(4.15)

=0

K = K (0, T )

H = H (0, T )

− → → u =− u (0) .

Swaption values

A payer swaption maturing at time T (= T0 = T 0 ) with strike κ is an option to acquire at T a swap with coupon κ. So its time t value is ¯ ) ¯ ¢ ¡ ¯ δ i B T, T i+1 [ω (T ) − κ]+ ¯ Ft , pSwpn (t) = B (t, T ) ET ¯ i=0 (M −1 ) ¯ o n X ¢ ¡ e T [ω (T ) − κ]+ ¯¯ Ft , δ i B t, T i+1 E = ( M−1 X

i=0

eT using Sectionchanging between forward measure PT and swaprate measure P 5.1 of the next chapter  ¯  (M −1 ) ¯   X ¯ ¡ ¢ f (T ) eT n ¯ Ft . o δ i FT t, T i+1 E ET { f (T )| Ft } = ¡ ¢ P ¯  M−1  ¯ δ F T, T i=0

i=0

i T

i+1

But from (4.1), (4.8) and (4.15) the swaprate

ω (T ) = ω1 (0) E

 Z 

0

−1 T N X j=0

 −1  NX ¤ £ ∗ f Aj ξ (t, Tj ) dWT (t) − uj (0) a (Tj ) − µj .  j=0

Swaprate Dynamics

35

Hence the present value of a swaption is approximately pSwpn (0) ∼ = 2

ζ =

Z

0

(M−1 X i=0

) ¢ ¡ δ i B 0, T i+1 B {ω (0) + α, κ + α, ζ} ,

(4.16)

¯ ¯2 Z T ¯ NP −1 NP −1 ¯ ¯ Aj ξ (t, Tj )¯ dt = Aj1 Aj2 ξ ∗ (t, Tj1 ) ξ (t, Tj2 ) dt, ¯ ¯ j=0 ¯ j1=0 j2=0 0

T ¯N −1 P

α=

N −1 X j=0

Aj = wj − hj

N−1 Xµ =j

¤ £ uj (0) a (Tj ) − µj ,

¶ K → − w − u λ , H

λ=

N −1 X =0

K w. H

Reiterating in part, some terminology for parts of formula (4.16) for future use, includes: 1. The level function level (t) =

M−1 X i=0

¢ ¡ δ i B t, T i+1 ,

level (0) =

M −1 X i=0

¢ ¡ δ i B 0, T i+1 ,

2. The swaption shift α (t) =

N −1 X j=0

¤ £ uj (t) a (Tj ) − µj

α=

N−1 X j=0

¤ £ uj (0) a (Tj ) − µj ,

3. The swaption zeta v uZ u √ ζ = ζ (T ) = β (T, TN ) T = t

0

¯ ¯2 ¯ ¯ ¯ Aj ξ (t, Tj )¯ , ¯ ¯ j=0 ¯

T ¯N−1 P

in terms of the swaption implied volatility β (T, TN ).

4.4.1

Multi-period caplets

Further to Section-2.2.1 which was about simple forwards over many intervals, a multi-period caplet is simply a swaption with just one coupon payment → at its final maturity. From the definition (4.9) of the − u vector, we have − → u =0

= 0, 1, .., N − 2

and

− → u N −1 = 1,

36

Engineering BGM

so substituting in (4.16), the value of a multi-period caplet in shifted BGM is o ¢ n (N) ¡ Cpl(N) (0) ∼ = δ 0 B t, T M B K (N) (t, T0 ) + α, κ + α, ζ , ¯2 Z T ¯¯N−1 N−1 ¯ i X h ¯P ¯ 2 (N) α= uj a (Tj ) − µ Aj ξ (t, Tj )¯ dt, , ζ = ¯ ¯ ¯ j=0 0 j=0

Aj = wj + hj

j−1 X K =0

4.5

H

w.

Swaprate models

For simplicity we will illustrate construction of Jamshidian’s swaprate models with two examples, rather than formal proofs; the reader might like to consult either [66] or [80] for more thorough expositions. The first example is coterminal swaptions, the second is a set of swaps with a strictly increasing total tenor structure. If shifts and margins are zero (that is, a (Tj ) = 0 and µj = 0 for all j), then from (4.13) an accurate SDE for the swaprate ω (t) is dω (t) fT (t) = σ ∗ (t) dW where ω (t)   N−1 N −1  X X → wj (t) − hj (t) σ (t) = (w (t) − − u (t)) ξ (t, Tj )   j=0

=

N−1 X

=j

Aj (t) ξ (t, Tj ) ,

j=0

in which, we emphasize, the Aj (t) are stochastic. To obtain BGM we assumed ξ (t, T ) was deterministic, but there is nothing to stop us letting it be stochastic (other than the backward construction technique of Section-3.2 requiring a modification that we do not address here). Our modus-operandi will therefore be to choose stochastic ξ (t, Tj ) in a way that makes swaprate volatilities become deterministic. In our two examples we will assume that fixed and floating sides are quarterly, and the coverage is a uniform δ.

Swaprate Dynamics

37

Example 4.1 Start with a set of coterminal swaps, the last three of which are pSwapN −1 (t) = δ B (t, TN ) [ω N−1 (t) − κ] , ( N−1 ) X δ j B (t, Tj+1 ) [ω N −2 (t) − κ] , pSwapN −2 (t) = i=N −2

pSwapN −3 (t) =

(

N−1 X

i=N −3

)

δ j B (t, Tj+1 ) [ω N −3 (t) − κ] .

Note that the total tenor structure is increasing, that is, the first swap tenor is over (TN −1 , TN ], the second is over (TN −2 , TN ], and the third over (TN−3 , TN ]. Now make the swaprates for these three forward swaps lognormal as follows: [1] The swaprate ω N −1 (t) for pSwapN −1 (t) is simply the forward K (t, TN −1 ) which is lognormal under the forward measure PTN when ξ (t, TN −1 ) is deterministic. [2] The swaprate ω N−2 (t) for pSwapN−2 (t) has volatility (N −2)

(N −2)

σN −2 (t) = AN −2 (t) ξ (t, TN−2 ) + AN −1 (t) ξ (t, TN −1 ) in which ξ (t, TN −1 ) is already determined. Make ω N−2 (t) lognormal under its eN −2 by setting σ N −2 (t) to whatever deterministic unique swaprate measure P value is required (for example, to fit a swaption implied volatility) by setting ξ (t, TN −2 ) =

(N −2)

σ N −2 (t) − AN −1 (t) ξ (t, TN−1 ) (N −2)

.

AN−2 (t)

[3] The swaprate ω N−3 (t) for pSwapN−3 (t) has volatility (N −3)

(N−3)

σ N −3 (t) = AN −3 (t) ξ (t, TN −3 ) + AN −2 (t) ξ (t, TN−2 ) (N −3)

+AN −1 (t) ξ (t, TN−1 ) in which ξ (t, TN −1 ) and ξ (t, TN −2 ) are already determined. Make ω N −3 (t) eN −3 by letting σ N −2 (t) be lognormal under its unique swaprate measure P deterministic and setting ξ (t, TN −3 ) =

(N −3)

(N −3)

σ N −3 (t) − AN −2 (t) ξ (t, TN−2 ) − AN −1 (t) ξ (t, TN −1 ) (N −3)

.

AN −3 (t)

Continuing in this way, the swaprates ω N−1 (t), ω N−2 (t), ω N−3 (t),.... of the coterminal swaps can be made jointly lognormal under the corresponding eN −2 , P eN −3 ,... forward and swaprate measures PTN , P

38

Engineering BGM

Example 4.2 We now show how to make a set of two forwards and the swaprates of two swaps jointly lognormal. Start with the swaps pSwap1 (t) = δB (t, T2 ) [K (t, T1 ) − κ] , pSwap2 (t) = δB (t, T3 ) [K (t, T2 ) − κ] , pSwap1,3 (t) = δ {B (t, T2 ) + B (t, T3 ) + B (t, T4 )} [ω 1,3 (t) − κ] , pSwap3,4 (t) = δ {B (t, T4 ) + B (t, T5 )} [ω 3,4 (t) − κ] , and notice that they have a strictly increasing total tenor structure. Proceed as follows: [1] Make the forwards K (t, T1 ) and K (t, T2 ) lognormal under the forward measures PT2 and PT3 by letting ξ (t, T1 ) and ξ (t, T2 ) be deterministic. [2] The swaprate ω 1,3 (t) for pSwap1,3 (t) has volatility (1,3)

σ1,3 (t) = A1

(1,3)

(t) ξ (t, T1 ) + A2

(1,3)

(t) ξ (t, T2 ) + A3

(t) ξ (t, T3 )

in which ξ (t, T1 ) and ξ (t, T2 ) are already determined. Make ω 1,3 (t) lognormal e1,3 by letting σ 1,3 (t) be deterministic and under its unique swaprate measure P setting (1,3)

ξ (t, T3 ) =

σ 1,3 (t) − A1

(1,3)

(t) ξ (t, T1 ) − A2

(t) ξ (t, T2 )

(1,3) A3

.

[3] The swaprate ω 3,4 (t) for pSwap3,4 (t) has volatility (3,4)

σ 3,4 (t) = A3

(3,4)

(t) ξ (t, T3 ) + A4

(t) ξ (t, T4 )

in which ξ (t, T3 ) is already determined. Make ω 3,4 (t) lognormal under its e3,4 by letting σ 3,4 (t) be deterministic and setting unique swaprate measure P (3,4)

ξ (t, T3 ) =

σ3,4 (t) − A3

(t) ξ (t, T3 )

(1,3) A4

.

One can continue in this way so long as for some Tj there is always a new ξ (t, Tj ) available to make the volatility of the next swaprate deterministic, that is, so long as the total tenor structure of the underlying swaps is strictly increasing.

Chapter 5 Properties of Measures

The previous chapters show that BGM, swaprate models and market models in general are in many ways plays on measures; just find the right measures to make the variables of interest lognormal! In this chapter we try to capture the essentials of the diﬀerent measures available. First note from Geman et al [43] that in general measure changes are governed by: LEMMA 5.1 If P and Q are equivalent measures induced by the numeraires Mt and Nt respectively, then for 0 < t ≤ T and f (·) measurable FT EQ { f (T )| Ft } = EP PROOF

½

¯ ¾ ¯ NT / Nt f (T )¯¯ Ft . MT / Mt

For g (·) measurable FT we have

¯ ¾ ¯ ¾ ½ g (t) g (t) g (T ) ¯¯ g (T ) ¯¯ , EP , EQ Ft = Ft = ¯ ¯ NT Nt MT Mt ¯ ¯ ½ ¾ ½ ¾ ¯ g (T ) ¯¯ 1/ Nt ¯ Ft . ⇒ EQ = E g (T ) F t P ¯ NT ¯ MT / Mt ½

Now set g (T ) = f (T ) NT and the result follows.

In Section-5.1 we discuss changes between forward and swaprate measures as used in the derivation of the swaption formula of Section-4.4. The two most practical measures for pricing either by simulation or timeslicer are the terminal measure Pn , for which the zero coupon bond B (t, Tn ) is numeraire, and the spot Libor measure P0 , for which the pseudo-bank account consisting of rolled up zeros (an analogue of the HJM bank account) is numeraire. Using the terminal measure for simulation can lead to blowouts in the sample standard deviation, see Example-5.1, something that does not occur when using Spot Libor. But the terminal measure is technically much easier to use than Spot Libor in timeslicers, see Section-10.

39

40

5.1

Engineering BGM

Changes among forward and swaprate measures

Let 0 < t ≤ T ∗ ≤ T < T1 with f (·) measurable FT ∗ . Change between the forward measures PT and PT1 is defined by (2.6) dWT1 (t) = dWT (t) + b (t, T, T1 ) dt ⇒ ) ( (Z ) T ∗ ∗ ∗ ET {f (T ) |Ft } = ET1 E b (s, T, T1 ) dWT1 (s) f (T ) |Ft , = ET1

½

t

FT (t, T1 ) f (T ∗ ) |Ft FT (T, T1 )

¾

= FT (t, T1 ) ET1

½

f (T ∗ ) |Ft FT (T ∗ , T1 )

¾

,

using Girsanov’s theorem of Section-A.3.5, and nested conditional expections on the reciprocal of the forward which (integrating the SDE (2.7)) is a PT1 martingale 1 1 = E FT (T, T1 ) FT (t, T1 )

(Z

T ∗

)

b (s, T, T1 ) dWT1 (s) .

t

Note that if T and T1 are adjacent nodes where T1 = T + δ, then ET {f (T ∗ ) |Ft } =

1 ET {f (T ∗ ) [1 + δK (T ∗ , T )] |Ft } . 1 + δK (t, T ) 1

From (4.7), the change between the forward measure PT and the swaprate eT is defined by measure P fT (t) = dWT (t) + dW

M−1 X i=0

¢ ¡ ui (t) b t, T, T i+1 dt

ET { X (T ∗ )| Ft } ¯ ) ) ( (Z M−1 ¯ T X ¡ ¢ ∗ ∗ fT (s) X (T )¯¯ Ft , eT E ui (s) b s, T, T i+1 dW =E ¯ t i=0 ¯ ) ( PM−1 ¡ ¢ ¯ e T P i=0 δ i FT ¡ t, T i+1 ¢ X (T ∗ )¯¯ Ft , =E M−1 ¯ i=0 δ i FT T, T i+1  ¯  ) (M −1 ¯   ∗ X ¯ ¡ ¢ X (T ) eT n ¯ Ft , o δ i FT t, T i+1 E = ¡ ¢ P ¯  M−1  ∗ ¯ i=0 i=0 δ i FT T , T i+1 ⇒

eT using Girsanov Section-A.3.5 and integrating (4.6) for f (t) (which is a P

Properties of Measures

41

martingale) f (t) = PM−1 i=0

f (T ) =E f (t)

(Z

t

1 ¡ ¢, δ i FT t, T i+1

T M−1 X i=0

) ¡ ¢ f ui (s) b s, T, T i+1 dWT (s) . ∗

It follows from Lemma 5.1 that the swaprate numeraire is simply M−1 X i=0

5.2

¢ ¡ δ i B t, T i+1 .

Terminal measure

Under the terminal measure Pn located at Tn dWj+1 (t) = dWn (t) − b (t, Tj+1 , Tn ) dt = dWn (t) − = dWn (t) −  n−1 X dH (t, Tj ) = − H (t, Tj )

=j+1

H (t, Tj ) = H (0, Tj ) E

n−1 X

δ H(t,T ) [1+δ K(t,T )]

b (t, T , T

−

ξ (t, T ) dt,

⇒

(5.1)



ξ ∗ (t, T ) ξ (t, Tj ) dt + ξ ∗ (t, Tj ) dWn (t) ,

R t hPn−1 0

+1 ) dt,

=j+1

=j+1

δ H(t,T ) [1+δ K(t,T )]

(

n−1 X

i

δ H(s,T ) ∗ =j+1 1+δ K(s,T ) ξ (s, T ) Rt + 0 ξ ∗ (s, Tj ) dWn (s)

ξ (s, Tj ) ds

)

.

(5.2)

Because any asset divided by B (t, Tn ) as numeraire is a Pn -martingale, the time t (≤ T ∗ ≤ T < Tn ) value X (t) of a cashflow X (T ∗ ) determined at T ∗ and made at T , is given by ¯ ¾ ½ X (T ∗ ) ¯¯ X (t) = En Ft B (t, Tn ) B (T, Tn ) ¯ ¯ ¾ ¯ ¾ ½ ½ X (T ∗ ) ¯¯ X (T ∗ ) ¯¯ = E = En F Ft . t n FT (T, Tn ) ¯ FT (T ∗ , Tn ) ¯ In the case when the cashflow is at a node T = Tj ¯   ¯  n−1  Y ¯ X (t) ∗ = En X (T ) [1 + δ K (Tj , T )]¯¯ Ft .  B (t, Tn ) ¯  =j

(5.3)

42

Engineering BGM

Present valuing a cashflow X (Tj ) at Tj by simulation using (5.3), requires averaging the term inside the expectation over all trajectories. But the cashflow X (Tj ) for each trajectory must be forward valued to Tn by multiplying by the corresponding unbounded n−1 Y

[1 + δ K (Tj , T )]

=j

before averaging, and that can create a problem when occasional trajectories produce very large forward values. Example 5.1 Suppose we are simulating a caplet exercising at 5-years in flat BGM under the terminal measure P10 at 10 years with constant forward volatility, of ξ (t, T ) = 40% and an intial yieldcurve of 5%. With a quarterly tenor structure = 0, 1, 2... (making 5-years = T20 and 10-years = T40 ) T = .25

cpl (0) = .25 B (0, T40 ) E40

(

+

[K (T20 , T20 ) − .05]

39 Y

=21

)

[1 + .25K (T21 , T )] .

With 100k simulations a positive 4-standard deviation value for the BM driver W5 will occasionally occur because [1 − N (4)] 105 > 3, and from (5.2) that produces forwards K (T20 , Tj ) j = 20, 21, .., 39 of the order of 100% with a forward value factor from T21 of approximately 19

[1 + .25 × 1]

∼ = 70.

The result is to blow out the sample standard deviation, which can only be reduced by making further simulations, which increases computation time. The forward value factor is an inherent deficiency with the terminal measure that makes the author prefer to work with the spot Libor measure for simulation.

5.3

Spot Libor measure

For the spot Libor measure P0 the numeraire is the pseudo-bank account M (t) started at time t = 0 with an investment of 1 in the zero coupon maturing at T1 , which is then rolled over at time t = T1 into the zero coupon

Properties of Measures

43

maturing at T2 , and so on. Hence M (t) = B (t, T@ )

@−1 Y =0

1 B (T , T

M (Tj ) =

⇒

j−1 Y

+1 )

= B (t, T@ )

@−1 Y

[1 + δ K (T , T )]

(5.4)

=0

[1 + δ K (T , T )]

j = 1, 2, ...

=0

On the current interval, the active part of M (t) is the zero coupon B (t, T@ ) Q@−1 1 maturing at time T@ , in which we invested =0 B(T ,T +1 ) at time T@−1 , and so the dynamics of M (t) on the current interval are therefore solely determined by B (t, T@ ). Thus if Z (t, Tj ) is the zero B (t, Tj ) discounted by the numeraire Z (t, Tj ) =

B (t, Tj ) , M (t)

t ≤ Tj ,

then

(5.5)

dZ (t, Tj ) = − [b∗ (t, T@ ) − b∗ (t, Tj )] {dW (t) − b (t, T@ ) dt} , Z (t, Tj ) = −b∗ (t, T@ , Tj ) dW0 (t) , and so Z (t, Tj ) will be a martingale under the spot Libor measure P0 , determined by the Brownian motion W0 (t) where making dW0 (t) = dW (t) − b (t, T@ ) dt, dWj+1 (t) = dW0 (t) + b (t, T@ , Tj+1 ) dt.

(5.6)

Substituting into the SDE for H (t, Tj ), under P0 for j = @ (t) , .., n − 1, the shifted forward H (t, Tj ) satisfies dH (t, Tj ) = ξ ∗ (t, Tj ) dWTj+1 (t) = ξ ∗ (t, Tj ) [b (t, T@ , Tj+1 ) dt + dW0 (t)] , H (t, Tj ) =

j X

b∗ (t, T , T

+1 ) ξ (t, Tj ) dt

+ ξ ∗ (t, Tj ) dW0 (t) ,

(5.7)

=@

) δ H (t, T ) ∗ ξ (t, T ) ξ (t, Tj ) dt + ξ ∗ (t, Tj ) dW0 (t) , = [1 + δ K (t, T )] =@ ¸  ·  δ H (s, T ) ∗  R t Pj  ξ (s, T ) ξ (s, T ) ds j =@(s) 0 . H (t, Tj ) = H (0, Tj ) E 1 +R δ K (s, T )   t + 0 ξ ∗ (s, Tj ) dW0 (s) (

j X

Because any asset divided by M (t) as numeraire is a P0 -martingale, the time t (≤ T ∗ ≤ T < Tn ) value X (t) of a cashflow X (T ∗ ) determined at T ∗ and made at T , is given by

44

Engineering BGM X (t) = E0 M (t)

½

¯ ¾ ¾ ½ X (T ∗ ) ¯¯ B (T ∗ , T ) X (T ∗ ) . = E Ft 0 M (T ) ¯ M (T ∗ )

In the case when the cashflow is at the node T = Tj , by ¯ ) ( ¯ X (t) X (T ∗ ) ¯ = E0 Qj−1 ¯ Ft . ¯ M (t) [1 + δ K (T , T )] =0

Qj−1 Note that the unbounded factor =0 [1 + δ K (T , T )] > 1 is now in the denominator where it cannot cause blowouts in the payoﬀ.

5.3.1

Jumping measure

Working under the spot Libor measure P0 is equivalent to working under the successive forward measures P1 , ..., Pj , Pj+1 , ..., Pn−1 , jumping from Pj to Pj+1 at time Tj . To see that, suppose @ (t) = j, that is T@ = Tj , and concentrate on the current interval t ∈ (Tj−1 , Tj ] = (T@−1 , T@ ] = J. Relative to the reference measure P, Brownian motions under the spot measure P0 and the forward measure PTj are respectively given by dWj (t) = dW (t) − b (t, Tj ) dt, dW0 (t) = dW (t) − b (t, T@ ) dt, ⇒ dW0 (t) = dWj (t) . Moreover, the shifted forwards H (t, T ) for ≥ @ (t) under the spot measure P0 and the forward measure PTj are the same dH (t, T ) = ξ ∗ (t, T ) [b (t, T@ , T +1 ) dt + dW0 (t)] , H (t, T ) = ξ ∗ (t, T ) [b (t, Tj , T +1 ) dt + dWj (t)] . Thus for t ∈ J all defining variables are identical under both P0 and Pj . For this reason P0 is sometimes called the jumping measure because it always coincides with the forward measure P@ at the end of the current interval J.

Chapter 6 Historical Correlation and Volatility

If there are no market instruments implying future correlation, backward looking analysis of historical data is the only reasonable source of information. Hence the topic of this chapter is estimating historical correlation within a shifted BGM framework. That said, increasingly in some markets like EUR, there is implied correlation information available in the form of options on differences in short-term and long-term swap rates (CMS spread options), and in the future that will probably become the dominant source of correlation information. Once determined, correlation can either be input directly into the model becoming part of the parameter set, or it can be used as a desirable target for optimization routines that bestfit cap and swaption prices. Our historical data will be assumed to consist of day-by-day quarterly (coverage a uniform δ = 14 year) readings at maturities jδ (j = 0, 1, .., N − 1) oﬀ either yield-to-maturity, forward or swaprate curves. To get quarterly readings, one might use any of the standard curves readily available in banks, designed to fit relevant data and produce discount functions for all maturities. A practical diﬃculty is that the (relative) lack of smoothness in these standard curves tends to create phantom principal components. Thus special curves designed to be super-smooth and bestfit data are required to give the best results for correlation; for example, use a forward curve specified at quarterly intervals and a bipartite objective function with one part designed to bestfit market data and the other part to minimize the second derivative measured as quarterly second diﬀerences. Another practical diﬃculty is that the natural objects for analyzing correlation in shifted BGM are forward curves, which must be created from combinations of cash, futures, bond and swap curves by interpolation, diﬀerencing and bestfitting. Apart from smoothness problems, when a day-by-day filmshow of forward curves is run (always a good test of routines producing periodic curves or surfaces), one often observes flapping of the long end of the forward curve, where data is relatively sparse and errors compound. To analyze day-by-day data like yieldcurves with maturities fixed relative to calendar time t, we introduce the concept of relative maturity x = T − t and relative forward K (t, x) = K (t, T )|T =t+x

K (t, T ) = K (t, x)|x=T −t 45

46

Engineering BGM

recycling (despite the possibility of confusion) the notation K (t, ·) to avoid creating a new range of variables. Our convention will be that absolute maturities use capitals and the variables in which they appear, like K (t, T ), are also absolute, while relative maturities use small letters and the variables in which they appear, like K (t, x), are relative. Thus expressions like K (t, t + x)

or

K (t, T − t)

will be avoided as being confusing or meaningless. The th maturity on a yieldcurve changing with calendar time t, will be at the relative maturity x with reference to the moving root of that instantaneous yieldcurve, and we will be interested in the set of relative maturities x = δ ( = 0, .., N − 1). To statistically analyze historical correlation, the underlying models describing the day-to-day movement of yield curves must necessarily be stationary. So we assume the shift is a constant a and the r-factor shifted BGM historical volatility is homogeneous and a function of relative maturity x = (T − t) only, that is ´∗ ³ a (T ) = a, ξ (t, T ) = ξ (T − t) = ξ (x) = ξ (1) (x) , ξ (2) (x) , .., ξ (r) (x) . The correlation function c (·) defined by c (T − t) = c (x) =

ξ (T − t) ξ (x) = kξ (x)k kξ (T − t)k

⇒

kc (·)k = 1

carries the correlation ρj,k between the shifted relative forwards H (t, xj ) and H (t, xk ) because for j, k = 0, .., N − 1 ρj,k =

ξ ∗ (xj ) ξ (xk ) = cT (xj ) c (xk ) . kξ (xj )k kξ (xk )k

(6.1)

The corresponding N × N correlation matrix R for shifted relative forwards at relative maturities x = δ ( = 0, .., N − 1) is then ¢ ¡ R = ρj,k = (c∗ (xj ) c (xk )) .

Note that the relative maturity x0 = 0 corresponds to the present instantaneous forward K (t, t) and so will not figure in option volatility calculations (which are specified as expectations of payoﬀs involving forwards that have not yet matured). For that reason the first row and column of R are often discarded as irrelevant. From the time t independent correlation ρj,k between relative shifted forwards, the instantaneous correlation ρj,k (t) between the absolute forwards H (t, Tj ) and H (t, Tk ) is ρj,k (t) =

ξ (Tj − t)∗ ξ (Tk − t) ξ (t, Tj )∗ ξ (t, Tk ) = . kξ (t, Tj )k kξ (t, Tk )k kξ (Tj − t)k kξ (Tk − t)k

Historical Correlation and Volatility

47

An important property of the vector volatility function ξ (t, T ) is that scaling it by either a deterministic or stochastic scalar quantity ψ (t, T ), leaves the instantaneous correlation ρj,k (t) unchanged because ξ (t, Tj )∗ ξ (t, Tk ) ψ (t, Tj ) ξ (t, Tj )∗ ψ (t, Tk ) ξ (t, Tk ) = = ρj,k (t) . kψ (t, Tj ) ξ (t, Tj )k kψ (t, Tk ) ξ (t, Tk )k kξ (t, Tj )k kξ (t, Tk )k As we will see in Chapter-7, this flexibility to retain instantaneous correlation while scaling by an arbitrary ψ (t, T ) will permit both instantaneous historical correlation and cap and swaption implied volatilities to be jointly fitted. Also, from (3.3), b (t, Tj , Tj+1 ) = ρj,k (t) =

δ j H (t, Tj ) ξ (t, Tj ) [1 + δ j K (t, Tj )]

∗

⇒

∗

b (t, Tj , Tj+1 ) b (t, Tk , Tk+1 ) ξ (t, Tj ) ξ (t, Tk ) = , kξ (t, Tj )k kξ (t, Tk )k kb (t, Tj , Tj+1 )k kb (t, Tk , Tk+1 )k

showing the instantaneous correlation between the forward contracts FTj (t, Tj+1 )

and

FTk (t, Tk+1 )

is the same as that between the corresponding shifted forwards H (t, Tj ) and H (t, Tk ). That raises the possibility that correlation can be estimated in one of the Gaussian, or flat or shifted BGM frameworks, and reasonably used in the others. Various possible ways include: • The Gaussian HJM approach which is straightforward, because within it correlation analysis is on yield-to-maturity curves, which are stable, well defined, and do not have to be particularly smooth. • In contrast, the direct flat BGM approach operates on forward curves, which are essentially diﬀerenced yield-to-maturity curves that must be interpolated, are therefore potentially unstable, and which must be smooth to avoid phantom principle components. • Another flat BGM approach is to first estimate swaprate correlation and then back out forward correlation. The point is that in swap-world the first 5 principal components are stable (sometimes for years), capture most (around 99.99%) of the R2 , and have flatish tails. • An approach more consistent with the data is to estimate an average shift a = a (T ) from option data (see the next Chapter-7), and then find the correlation by adapting the flat BGM techniques.

48

Engineering BGM

6.1

Flat and shifted BGM oﬀ forwards

Diﬀerentiate K (t, x) using the standard SDE dK (t, T ) = K (t, T ) ξ ∗ (t, T ) dWT1 (t) , and setting T = t + x to formally obtain ∂K (t, x) dt = K (t, x) ξ ∗ (x) dWT1 (t) , dK (t, T ) = dK (t, x) − ∂x ½ ¾ ∂K(t,x) 1 dK (t, x) ∂x = dt + ξ ∗ (x) dW0 (t) , K (t, x) K (t, x) +K (t, x) ξ ∗ (x) b (t, T@ , T1 ) = (drift) dt + ξ ∗ (x) dW0 (t) ,

(6.2)

as an SDE for K (t, x) under the spot Libor measure P0 . REMARK 6.1 A rigorous derivation of an equation like (6.2) requires a continuous spectrum of forwards for all maturities as in the framework of the forward BGM construction. But what is really relevant about (6.2) is that the time dependence in the relative maturity x = T −t aﬀects only the drift in the SDE for K (t, x). Moreover, that holds in general; when relative maturities are substituted for absolute maturities in variables like forwards, zero coupon bonds or swaprates, their SDEs change only in their drift. But the actual form of the drift is unimportant because it makes no contribution in the quadratic variation estimators we will use. For similar reasons the measure underlying the SDE, like P0 in (6.2), is also unimportant. A quadratic variation estimator for the covariance matrix q of the N forward volatilities ξ ∗ (xj ) (j = 0, 1, ...N − 1) is then ¶ µ Z t 1 d hK (·, xj ) , K (·, xk )i (s) . q = (qj,k ) = (ξ (xj ) ξ (xk )) = t 0 K (s, xj ) K (s, xk ) ∗

(6.3)

Assembling the first r (by size of eigenvalue) principal components of q into the N × r square root Ξ of q (so q = Ξ ΞT ), the vector volatility function ξ (x) at the discrete points xj can be recovered from q by setting the transpose of ξ (xj ) equal to the appropriate row in Ξ, that is ξ T (xj ) = Ξj,·

j = 0, 1, ...N − 1.

Then, for a continuous ξ (x), simply interpolate between these ξ (xj ).

Historical Correlation and Volatility

49

A similar estimator to (6.3) for the shifted case follows from (3.1)

⇒

dK (t, x) = (drift) dt + ξ ∗ (x) dW0 (t) K (t, x) + a Z 1 t d hK (·, x) , K (·, y)i (s) ∗ , ξ (x) ξ (y) = t 0 [K (s, x) + a] [K (s, y) + a]

(6.4)

where we have made the shift function a (T ) homogeneous, by averaging it into a constant a (the next Chapter-7 shows how that might be done). For a given set of data, the shifted covariance estimator (6.4) will vary with the shifts. But note that its denominator (which is always positive and almost invariant for large shifts) will tend to just scale the covariance matrix and so not alter correlation, in contrast, the numerator has a more dramatic eﬀect. So approximate this estimator by averaging forwards K (t, x) → K (x) and shifts a (T ) → a and using 1 ξ (x) ξ (y) ∼ = t ∗

Z

0

t

d hK (·, x) , K (·, y)i (s) . [K (x) + a] [K (y) + a]

The resulting covariance matrices vary with shift, but the corresponding correlation matrices are the same. That supports the notion of one correlation fits all in shifted BGM. A sensible approach, however, is to use a Gaussian estimator for large shifts, and a lognormal one for small shifts.

6.2

Gaussian HJM oﬀ yield-to-maturity

From (1.4) the SDE for the zero coupon B (t, T ) under the HJM spot measure P0 is

dB (t, T ) = r (t) dt − B (t, T )

Z

T

σ ∗ (t, u) dudW0 (t) = r (t) dt + b∗ (t, T ) dW0 (t) ,

t

in which we now assume

σ (t, T ) = σ (T − t)

⇒

b (t, T ) = −

Z

T t

σ (u − t) du = −

Z

0

T −t

σ (v) dv = b (x) .

50

Engineering BGM

Define the relative maturity zero coupon bond B (t, x) by with SDE

B (t, T ) = B (t, x)|x=T −t

∂B (t, x) dt, dB (t, x) = dB (t, T ) + ∂x ¸ · ∂B (t, x) + B (t, x) r (t) dt + B (t, x) b∗ (x) dW0 (t) , = ∂x dB (t, x) = (drift) dt + b∗ (x) dW0 (t) . B (t, x)

or

Hence an estimator for the bond volatility covariance is Z 1 t d hB (·, x) , B (·, y)i (s) . b∗ (x) b (y) = t 0 B (s, x) B (s, y) An alternative estimator uses yields-to-maturity rather than zero coupons b∗ (x) 1 dW0 (t) Y (t, x) = − ln P (t, x) , with SDE dY (t, x) = (drift) dt − x x Z t 1 b∗ (x) b (y) = d hY (·, x) , Y (·, y)i (s) . ⇒ x y t 0 Principal component analysis readily yields an r × N matrix for xj = jδ ¡ T ¢ ¡ ¢ b (xj ) = bT (jδ) = (bj,· )

containing the principal components in the columns, from which b (x) can be interpolated. The shifted forward volatility ξ (t, T ) can then be recovered by diﬀerencing (recall the remarks of Section-3.1.1) ξ (t, Tj ) = b (t, Tj , Tj+1 ) = b(t, Tj ) − b(t, Tj+1 ) = b(Tj − t) − b(Tj+1 − t), that is ξ (x) = b (x) − b (x + δ) and c (T − t) = c (x) =

6.3

b (x) − b (x + δ) . kb (x) − b (x + δ)k

Flat and shifted BGM oﬀ swaprates

From (2.11), for j = 0, 1, ...N − 1 define the relative swaprate ω (t, j) of a quarterly swap consisting of j + 1 rolls fixing at relative maturities x = δ ( = 0, 1, .., j), to be ω (t, j) =

Pj

B (t, x +1 ) K (t, x ) . Pj =0 B (t, x +1 )

=0

(6.5)

Historical Correlation and Volatility

51

Using the last yield curve in the historical set, in (4.15) set A = w (which is a good enough approximation for correlation work) and diﬀerentiate (6.5) using (6.2). The time dependence in the relative maturity variable x = in the bonds adds terms only to drifts, and so under the spot measure P0 the formal SDE (see Remark-6.1) for the j th swaprate ω (t, j) must have form dω (t, j) = (drift) dt + σ∗ (j) dW0 (t) , ω (t, j) σ (j) =

j X

cj, ξ (x ) ,

j = 0, 1, ..., N − 1 in which

=0

cj, =

where approximately

(6.6) (6.7)

 

B (0, x +1 ) K (0, x ) = 0, .., j Pj =0 B (0, x +1 ) K (0, x )  0 = j + 1, .., N − 1

Collecting the weights cj , into the N × N lower triangular matrix 

  C= 

c0,0 c1,0 .. .

0 c1,1 .. .

··· ··· .. .



0 0 .. .

cN−1,0 cN−1,1 · · · cN −1,N −1

   

(6.8)

the N equations (6.7) for j = 0, 1, ..., N − 1 can be rewritten Ω = C Ξ, in terms of the N × r matrices Ξ and Ω containing discrete values of ξ (·) and σ (·) Ξ = (Ξj,· ) = (ξ (xj )) ,

Ω = (Ωj,· ) = (σ (j))

j = 0, .., N − 1.

Note c0,0 = 1 while the order of magnitude of cj, is roughly 1j , that is, the cj, get smaller by the row. Clearly C is non-singular, so subtracting the th restructured (j − 1) and j th equations in (6.7) for j = 1, .., N − 1 j−1 X =0

j X =0

( j · ) X B (0, x +1 ) K (0, x ) ¸ B (0, x +1 ) K (0, x ) ξ (x ) = σ (j − 1) , −B (0, xj+1 ) K (0, xj ) =0

B (0, x

+1 ) K

(0, x ) ξ (x ) = σ (j)

j X =0

B (0, x

+1 ) K

(0, x )

52

Engineering BGM

gives the forward volatility ξ (·) in terms of the swaprate volatility σ (·) Pj =0 B (0, x +1 ) K (0, x ) + σ (j − 1) , ξ (xj ) = [σ (j) − σ (j − 1)] B (0, xj+1 ) K (0, xj ) µ ¶ 1 1 = 1− σ (j − 1) + σ (j) , cj,j cj,j µ ¶ 1 = σ (j) + [σ (j) − σ (j − 1)] −1 , (6.9) cj,j revealing the inverse of C to be the bi-diagonal matrix  1 0 0 ··· ··· ··· .. . .  .. .. .  1 − c1 c1 0 1,1 1,1   . . . . .. .. .. ..  0 0   .. .. 1 1 C −1 =  . . 0 1 − cj−1,j−1  cj−1,j−1 0  .. ..  1 1 .  0 1 − cj,j . 0 cj,j   . . . . .. .. .. ..  0 0 1 0 ··· ··· 0 0 1 − cN−1,N−1

0 0 .. . .. . .. . 0 1 cN−1,N−1



       .       

(6.10) From (6.6), a quadratic variation estimator for the swaprate covariance matrix Q for the N swaprate volatilities σ (j) is for j, k = 0, .., N − 1, ¶ µ Z τ ¡ T ¢ 1 d hω (·, j) , ω (·, k)i (t) , (6.11) Q = (Qj,k ) = σ (j) σ (k) = τ 0 ω (t, j) ω (t, k) ¡ ¢ from which σ T (j) can be found by principal component analysis Q = ΩΩT

σT (j) = Ωj,·

j = 0, 1, ...N − 1

That yields the corresponding forward volatility function ξ (·) via (ξ (xj )) = (Ξj,· ) = Ξ = C −1 Ω, and the corresponding implied forward covariance matrix ¢T ¡ q = (qj,k ) = ΞΞT = C −1 Q C −1 ,

from which we can compute the correlation function c (T − t) = c (x) = REMARK 6.2

ξ (x) . kξ (x)k

From equation (6.9) the magnitude of ξ (xj ) is roughly ξ (xj ) ∼ = σ (j) + (j − 1) [σ (j) − σ (j − 1)] .

Historical Correlation and Volatility

53

¡ ¢ So if in swap-world a principal component appearing in a column of σ T (j) becomes steep in the³tail (where ´ j approaches N ), then the tail of the corresponding column of ξ T (x ) will be even steeper, leading to gross distortion of the correlation matrix in forward-world.

Using (4.8), (4.15) and (4.16), a similar estimator to (6.11) for the swaprate covariance matrix in the shifted case comes from

⇒

dω (t, j) = (drift) dt + σ ∗ (j) dW0 (t) ω (t, j) + α Z 1 τ d hω (·, j) , ω (·, k)i (t) T , σ (j) σ (k) = τ 0 [ω (t, j) + α] [ω (t, k) + α]

where the maturity dependent swaprate shift has been averaged to a constant α (see Chapter-7 for how that might be done).

Chapter 7 Calibration Techniques

Most banks continuously monitor interest rate market data and after stripping, interpolating, and massaging it, present the processed data as market objects (in the programming sense) ready to be used in pricing algorithms. Typically the object will incorporate yieldcurve and volatility information for a given currency and particular sector (like Treasuries or swaps or municipals in USD). So in this chapter we assume volatility information is available in the form of interpolated swaption and stripped caplet implied volatilities entered in a quarterly implied volatility matrix with exercise times descending row-by-row and tenors moving left to right along the columns. If data is available awayfrom-the-money it is entered into identical but separate volatility matrices referenced either by absolute strike or relative delta strike, the whole making up an implied volatility cube. Example 7.1 In a 40×40 implied volatility matrix position (1, 1) is the top left-hand corner, the volatility of the caplet exercising at 9 34 yrs is in the (39, 1) position, and the volatility of the swaption exercising at 5yrs into a 5yr swap (that is, the swap’s tenor is 5yrs) is in the (20, 20) position. Because (maturity) = (exercise time) + (tenor ) , both are included among the volatilities of the set of 39 coterminal swaptions maturing at 10yrs, which appear along the diagonal {(i, j) : i + j = 40}. Some entries in the cube are direct liquid market quotes which a good calibration would return exactly. Others are interpolated between the liquid quotes in some fashion, which can potentially introduce arbitrage opportunities (the author is unaware of any criteria for ensuring an arbitrage free interpolation, other than it be generated by an arbitrage free interest rate model first fitted to the liquid data). In shifted BGM the shift and volatility are orthogonal in the sense that the shift a (T ) can be found independently, as in Section-7.1 below, leaving the volatility ξ (t, T ) to be fitted in exactly the same way as in flat BGM (though magnitudes will diﬀer with shift). But note that while the volatility ξ (t, T )

55

56

Engineering BGM

with two parameters can potentially fit a whole volatility matrix, the shift a (T ) with only one parameter is more restricted. Many of the volatility functions ξ (t, T ) we use are of the form ξ (t, T ) = χ (t) φ (T ) ψ (T − t) c (T − t) , in which χ (·), φ (·) and ψ (·) are scalars and c (·) is the vector historical correlation function as constructed in Chapter-6. Such functions return the correct instantaneous correlation, can exactly fit caps and a diagonal of coterminal swaptions (allowing vegas with respect to them to be computed), can be made reasonably homogeneous (stationary), and in addition can be made to approximately fit a selection of other swaptions (boosting confidence in allround pricing). Another approach, which essentially depends on the power of optimisers like the NAG [81] or IMSL [64] ones, is to bestfit a selection of liquid caps and swaptions in a generic calibration. Usually the motive is to generate indicative prices to help increase comfort levels with a shortrate model that is actually used for marking-to-market, risk management and hedging. The BGM model incorporates more volatility and correlation information than the shortrate model, whose comparative advantage is being Markov in a few variables and so fast. Among the bestfit genre, Pedersen’s method [85] in which ξ (t, T ) = ξ (t, T )|T =t+x = ξ (t, x) is made piecewise constant over quarterly intervals in both t and x directions, works well and is simple to implement. We also describe the cascade algorithm in which ξ (t, T ) is made piecewise constant over quarterly intervals in both t and T directions, but which then requires smoothing to be made usable (see Brigo and Mercurio [32] for advice on how to do that). Recall that swaption, but not caplet, implied volatilities depend weakly on correlation. Using semi-definite programming and a Pessler type volatility function, we also show how correlation can be freed up to participate in calibration (while staying close to a historical target) and permit an exact fit to the whole matrix. Handling the large covariance matrices involved (120 × 120 or bigger) with presently available semi-definite programming software is, however, problematical although Raise Partner [100] evidently has software that works. The next section shows how to fit the skew with an appropriate term structure of shift a (T ), and at the same time identifies the corresponding zetas ζ (·) and implied volatilities β (·) of caplets and swaptions Z T 2 2 2 |ξ (t, T )| dt, ζ (T ) = β (T ) T = 0

ζ 2 (Tj , TN ) = β 2 (Tj , TN ) Tj =

Z

0

¯

−1 Tj ¯¯N X

¯ ¯ ¯

=j

¯2 ¯ ¯ (j,N ) A ξ (t, T )¯¯ dt, ¯

Calibration Techniques

57

which must then be fitted with a suitable choice of volatility function ξ (t, T ).

7.1

Fitting the skew

Fitting the caplet skew and isolating the volatility is straightforward. Step-1: For fixed Tj suppose we have the implied volatilities β (Tj ) for = 1, 2, ... of a set of caplets with strikes κ (Tj ) all maturing at Tj ; work out their present values Cpl (0, κ (Tj ) , Tj ). Step-2: In the caplet formula (3.3.5), vary the shift a (Tj ) and zeta ζ (Tj ) to bestfit the Cpl (0, κ (Tj ) , Tj ), by minimizing X

° ° ° ° ° ¸ µ· Cpl (0, κ (T¸j ) ,·Tj ) ¶° °, κ L (t, T ) − µ (T ) − µ ω ° j j j j ° −B (t, Tj+1 ) B , , ζ (Tj ) ° ° ° +a (Tj ) +a (Tj )

choosing the weights ω to get a good fit at-the-money. Step-3: Repeating Step-2 for Tj (j = 1, 2, ..) produces the term structure of shift a (Tj ) and also the term structure of zeta ζ (Tj ) (from which the volatilities ξ (t, T ) of the shifted forwards H (t, T ) must now be found). To fit a swaption skew go through the same steps as for caplets using (3.4.16) to get the swaption style shift and zeta

αj =

N−1 X =j

u [a (T ) − µ ] ,

v uZ u ζ j = ζ (Tj , TN ) = t

0

Tj

¯ ¯ NP ¯ −1 A ξ (t, T ¯ ¯ =j

¯2 ¯ ¯ )¯ dt ¯

for a set of swaptions that diﬀer only in strike, that is, have the same exercise time Tj and same maturity TN . Varying j from 1 to N −1 produces the αj (j = 1, .., N − 1) for a set of coterminal swaptions maturing at TN , from which the termstructure of shift a (T ) ( = 1, .., N − 1) can be easily bootstrapped out. Similarly, fixing j and varying N produces a term structure of shift for a set of swaptions with the same exercise, while jointly varying j and N so the diﬀerence (N − j) is fixed produces the term structure of shift for a set of swaptions with the same tenor. Observe that the skew can only be fitted to a selection of data such as the caplets, or a subset of swaptions like a particular as a set of coterminal swaptions. At that point the shift a (T ) is fully determined and cannot be adjusted to fit other data. So to fit larger sets of volatility data, clearly some sort of averaging of shifts over an appropriate set of caplets and swaptions will be required.

58

Engineering BGM

7.2

Maturity only fit

A simple calibration to either caps or a diagonal of swaptions and also correlation, combines c (T − t) and a maturity only dependent function like ξ (t, T ) = φ (T ) c (T − t) .

(7.1)

The simplicity of this calibration makes it eﬀective and robust, and therefore suitable as a default calibration for more sophisticated routines that may fail for some reason. Implied caplet volatilities can be directly inserted into φ (·) , because Z T Z T 2 2 |ξ (t, T )| dt = |φ (T ) c (T − t)| dt = φ2 (T ) T. β 2 (T ) T = 0

0

Fitting the implied volatilities β (Tj , TN ) of N − 1 coterminal swaptions exercising at Tj (j = 1, .., N − 1) and maturing at TN requires ¯2 ¯ ¯ Z Tj ¯NX −1 ¯ ¯ (j,N) ¯ dt, ¯ β 2 (Tj , TN ) Tj = A ξ (t, T ) ¯ ¯ 0 ¯ ¯ =j Z −1 N−1 Tj X NX (j,N) (j,N ) A1 A 2 φ 1φ 2 c∗ (T 1 − t) c (T 2 − t) dt, = 0

1=j 2=j

where the φ = φ (T ) can be bootstrapped as follows: Step-1: For j = 1, .., N − 1 and 1, 2 = j, .., N − 1, precompute Z Tj c∗ (T 1 − t) c (T 2 − t) dt ⇒ X , ,j = Tj and express X 1, 2,j = 0

β 2 (Tj , TN ) Tj =

N −1 N−1 X X

(j,N ) 1

A

(j,N) φ 1 φ 2 X 1, 2,j 2

A

1=j 2=j

j = 1, .., N − 1.

Step-2: Start the routine at j = N − 1 setting φN−1 = β (TN −1 , TN ). Step-3: Knowing φN −1 , .., φj+1 , compute φj as the largest root Γ of h

+

(j,N ) Aj

 −1  NX 

i2

N−1 X

1=j+1 2=j+1

2

Tj Γ + 2 Γ

N −1 X

(j,N ) 1

A

(j,N )

Aj

φ 1X

1,j,j

1=j+1 (j,N ) 1

A

(j,N) φ 1 φ 2 X 1, 2,j 2

A

− β 2 (Tj , TN ) Tj

  

= 0.

The calibration fails when roots are imaginary, for example, when implied volatilities fall steeply, but with normal market data that is unlikely.

Calibration Techniques

59

REMARK 7.1 A frequently heard objection to this calibration is that the volatility function (7.1) is decidedly inhomogeneous. But by recasting φ (T ) = α + φ0 (T ) and then choosing the constant α to minimize N−1 X j=1

2

φ0 (Tk ) =

N −1 X j=1

2

(φ (Tk ) − α)

⇒

N−1 1 X α= φ (Tk ) , N − 1 j=1

it becomes clear that the calibration can, and usually does, have a substantial homogeneous spine ξ (t, T ) = α c (T − t) . Note that here α is eﬀectively the average of the φ (Tk ), and that α c (T − t) is a sensible extension of the calibration to maturities beyond those for which data might be available.

7.3

Homogeneous spines

Volatility functions that are entirely homogeneous ξ (t, T ) = ψ (T − t) c (T − t) , are attractive because of their stationary properties, but exact fits with them are diﬃcult. We describe several that have been used in the past to bestfit caps and which might be used as homogeneous spines ψ (T − t) in the more general volatility function ξ (t, T ) = χ (t) φ (T ) ψ (T − t) c (T − t) , which can then be tweaked using the method of Section-7.5 to fit a selection of caplets and coterminal swaptions. REMARK 7.2 A word of warning is that when bestfitted to caplets only, most of these homogeneous functions, other than perhaps the piecewise linear one, tend to produce functions with high values for small x = T − t and low values for larger x = T − t, and they are therefore relatively useless for imparting information about swaption values relative to cap values.

7.3.1

Piecewise linear

This is the sort of homogeneous volatility function used in the excellent Polypaths [97] mortgage backed securities (MBS) software, where it is bestfitted to a selection of liquid caps and swaptions. Select M nodes T1 , T2 , ..., TM

60

Engineering BGM

(around six or seven is about right, spread out at say 1, 2, 5, 7, 10 and 15 years), linearly interpolate between them and finish with some predetermined exponential decay λ, so that the homogeneous spine takes the form ψ (T − t) = ψ (x)   a0 I [x = 0]  PM {(Ti − x) ai−1 + (x − Ti−1 ) ai }  I [x ∈ (Ti−1 , Ti ]]  =  + i=1  (Ti − Ti−1 ) + {aM+1 + (aM − aM+1 ) exp (−λ [x − TM ])} I [x > TM ] ,

which is determined by the M + 2 constants a0 ,a1 ,...,aM ,aM+1 . It is straightforward but messy, to get quadratic expressions in the a0 ,a1 ,...,aM ,aM+1 for caplet and swaption volatilities, which can then be bestfitted to a selection of liquid caps and swaptions.

7.3.2

Rebonato’s function

Rebonato introduced the following homogeneous spine ψ (T − t) = ψ (x) = (a + bx) exp (−λx) + c, Z T 2 ⇒ β (T ) T = [(a + bx) exp (−λx) + c]2 dx 0

chosen to produce a hump (a defining feature of the forward volatility) at some appropriate point 1 a x= − . λ b

7.3.3

Bi-exponential function

Here the homogeneous spine is ψ (T − t) = ψ (x) = a exp (−λx) + b exp (−µx) + c, Z T ⇒ β 2 (T ) T = [a exp (−λx) + b exp (−µx) + c]2 dx 0

= a2 F (2λ, T ) + b2 F (2µ, T ) + c2 F (0, T ) + 2abF (λ + µ, T ) + 2acF (λ, T ) + 2bcF (µ, T ) where (1 Z T [1 − exp (−λT )] if λ > 0, exp (−λx) dx = λ F (λ, T ) = 0 T if λ = 0.

7.3.4

Sum of exponentials

Given a selection of M (around 5 to 7) caplets maturing at times T1 , T2 , ..., TM (spread out, like 1, 2, 5, 7, 10, 15) with implied volatilities β 1 , β 2 , ..., β M the fol-

Calibration Techniques

61

lowing algorithm will exactly fit a homogeneous spine of form ψ (T − t) = ψ (x) =

M X

aj exp (−λj x)

λj = j

j=1

λ M

λ∼ = 3.

Assuming β 2i+1 Ti+1 ≥ β 2i Ti , evaluate the caplet interval values Vi analytically and also apply the mean value theorem to get for i = 0, 1, ..., M − 1 sZ q Ti+1 Vi = β 2i+1 Ti+1 − β 2i Ti = ψ 2 (x) dx v uM M uX X =t

j=1 k=1

∃ ci ∈ (Ti , Ti+1 )

Ti

¤ aj ak £ −(λj +λk )Ti e − e−(λj +λk )Ti+1 , (λj + λk ) such that Vi =

M X

aj exp (−λj ci ) .

(7.2)

(7.3)

j=1

1 (Ti + Ti+1 ) and solve (7.3) for the corresponding aj . 2 Step-2: Substitute the aj (j = 1, ..., M ) into (7.2) to obtain caplet interval values Vei = Vi + ∆Vi (i = 0, 1, ..., M − 1) corresponding to the aj . Step-3: Adjust the current mean value variables in equations (7.3) ci → ci + ∆ci so that with the current aj Step-1: Set ci =

M X j=1

aj exp (−λj [ci + ∆ci ]) = Vi − ∆Vi

⇒

∆ci ∼ = PM

j=1

∆Vi aj λj exp (−λj ci )

.

Step-4: In the equations (7.3) replace ci by ci +∆ci as computed in Step-3, solve them for a new set of aj , and return to Step-2. The algorithm works quite well in practice, but the author has not established a suﬃcient condition for convergence.

7.4

Separable one-factor fit

Volatility functions which are one-factor and separable ξ (t, T ) = φ (T ) χ (t) , are not homogeneous and carry no correlation information, yet are flexible enough to jointly parametrize caps and a diagonal of coterminal swaptions.

62

Engineering BGM The implied volatilities of a caplet and swaption exercising at Tj , are β 2 (Tj ) Tj = φ2 (Tj ) 

Tj

χ (t)2 dt

0

N −1 X

β 2 (Tj , TN ) Tj = 

Z

(j,N )

A

=j

and

2

φ (T )

Z

Tj

(7.4)

χ (t)2 dt

0

respectively. The task is to fit N −1 caplets exercising at T1 , ..TN −1 , and N −1 coterminal swaptions exercising at T1 , ..TN −1 and settling at TN , altogether 2N − 3 instruments (the last caplet and swaption coincide). Put φ (Tj ) = φj for j = 1, .., N − 1, assume χ (·) is piecewise constant on each interval like χ (t) = χk

for t ∈ (Tk−1 , Tk ] k = 1, .., N and set Z Tj j X χ2 (t) dt = ∆k χ2k , j = 1, .., N − 1. Xj = X (Tj ) = 0

(7.5)

k=1

With this notation, equations (7.4) become

β 2 (Tj ) Tj = φ2j Xj ,



N −1 X

β 2 (Tj , TN ) Tj = 

(j,N)

A

=j

2

φ  Xj ,

(7.6)

and a successful solution to the calibration problem would be to identify the 2N − 2 positive parameters X1 < X2 < ..., < XN−1 ; φ1 , φ2 , .., φN−1 that return the required caplet and swaption volatilities. Work backwards from the final exercise. Step-1: Because the expressions (7.6) involve products of χ (·) and φ (·), instability arising from cross-leakage between the two functions is possible. To prevent this we fit the last caplet/swaption by choosing XN−1 = TN −1 ,

φN−1 = β (TN −1 ) = β (TN−1 , TN ) .

Step-2: Continuing backwards step-by-step, suppose XN −1 , .., Xj+1 and φN −1 , .., φj+1 satisfy (7.6) at TN −1 , .., Tj+1 . To satisfy (7.6) at Tj , divide the two equations to eliminate Xj and make φj the largest root Γ of 

Γ2 β (Tj , TN ) = β (Tj ) Γ 2

2

(j,N ) Aj

+

N−1 X

=j+1

(j,N )

A

2

φ .

Calibration Techniques

63

(j,N )

As long as β (Tj , TN ) > Aj β (Tj ) (an inequality diﬃcult to break with market data) there is one positive root, so PN−1 (j,N ) φ β (Tj ) β 2 (Tj ) =j+1 A = Tj . ⇒ X φj = j (j,N ) φ2j β (Tj , TN ) − Aj β (Tj ) (j,N )

β (Tj ), Step-3: Assuming that at each backward step β (Tj , Tn ) > Aj we will have consecutively computed φj and Xj for k = N − 1, .., 1. Then, so long as the Xj are monotonic increasing, the calibration is completed by retrieving positive s r X1 Xj − Xj−1 , χj = , j = 2, .., N − 1. χ1 = ∆1 ∆j Producing an X (T ) that is not monotonic increasing (so χ (T ) becomes imaginary for some T ) is how the model reacts to a potentially arbitragable situation (like when swaption volatilities are unreasonably greater than caplet volatilities). Otherwise the algorithm is extremely robust; for example, it can parametrize caplet volatilities starting at 40% and converging at 10-years maturity to a diagonal of swaption volatilities each of the order of 20%. REMARK 7.3 Even without correlation, this calibration got satisfactory prices for callable range rate accruals, in which the range rate part requires caplet volatility, and the callability part requires coterminal swaption volatility. First the skew was fitted to return caplet volatilities at the strikes banding the accrual, and then this algorithm fitted caplet volatilities at one of those strikes and also coterminal swaption volatilities at-the-money.

7.5

Separable multi-factor fit

The 1-factor separable algorithm can be articulated to a separable multifactor setting in which correlation, caplets and swaptions have already been best-fitted, either with a homogeneous spine as in Section-7.3 or with a method like Pedersen’s of Section-7.6, but an exact fit to a selection of instruments is now required. Assume therefore that we already have a vector volatility function ψ (t, T ) that incorporates correlation and fits a selection of instruments, and concentrate on finding the scalar functions φ (T ) and χ (t) in   (1) ψ (t, T )   (2) ξ (t, T ) = φ (T ) χ (t) ψ (t, T ) = φ (T ) χ (t)  ψ (t, T )  .. .

64

Engineering BGM

to exactly fit a strip of caplets and diagonal of coterminal swaptions as in Section-7.4. The idea in the following algorithm is to vary dummy swaption volatilities in the 1-factor routine of Section-7.4, so that when the true ones in this multi-factor routine are recomputed they become ever closer to the required ones. So similarly to Section-7.4, our task is to find χ (t) and φ (T ) to fit N − 1 caplet and N − 1 swaption implied volatilities at exercise timesTj (for j = 1, 2, .., N − 1) like 2

Z

2

Tj

2

χ2 (t) |ψ (t, Tj )| dt, (7.7) β (Tj ) Tj = φ (Tj ) 0 ( ) N −1 N −1 (j,N) (j,N ) X X A 2 φ (T 1 ) φ (T 2 ) A 2 R Tj 1 2 β (Tj , TN ) Tj = . × 0 χ (t) ψ ∗ (t, T 1 ) ψ (t, T 2 ) dt 1=j 2=j

Because the expression frequently recurs and can be precomputed (perhaps analytically and exactly) from the given ψ (t, T ), introduce 1 Ψ [ 1, 2, k] = ∆Tk

Z

Tk

ψ ∗ (t, T 1 ) ψ (t, T 2 ) dt

Tk−1

so that, if χ (t) = χk for t ∈ (Tk−1 , Tk ], that is, χ (·) is piecewise constant as in (7.5), then for j = 1, .., N − 1 and 1, 2 ≥ j we can define X

1, 2,j

=

Z

Tj

χ2 (t) ψ ∗ (t, T 1 ) ψ (t, T 2 ) dt =

0

Xj = Xj,j,j =

j X

χ2k Ψ [ 1, 2, k] ∆k

k=1

Z

Tj

0

2

χ2 (t) |ψ (t, Tj )| dt =

j X

χ2k Ψ [j, j, k] ∆k

k=1

in terms of χk and Ψ [·]. Also, knowing the Xj , the corresponding χj j = 1, .., N − 1 can be consecutively recovered, so long as they are real, from X1 = χ21 Ψ [1, 1, 1] ∆T1

and for

Xj = χ2j Ψ [j, j, j] ∆Tj +

j−1 X

j = 2, .., N − 1 from

(7.8)

χ2k Ψ [j, j, k] ∆Tk .

k=1

Setting φ (Tj ) = φj , from (7.7) our task in the multi-dimensional separable case is (as in Section-7.4) to identify φ1 , .., φN−1 and X1 , .., XN −1 that return the required caplet and swaption volatilities for j = 1, .., N − 1 β 2 (Tj ) Tj = φ2j Xj , β 2 (Tj , TN ) Tj =

N −1 N −1 X X 1=j 2=j

(7.9) (j,N ) (j,N ) A 2 φ 1 φT 2 X 1, 2,j . 1

A

Calibration Techniques

65

The following iterative routine to find φ1 , .., φN −1 and X1 , .., XN−1 cycles the 1-factor routine of Section-7.4, and appears quite robust (converging to 8 or 9 decimal places of the target implied volatilities in about 20 loops in the authors R&D program). Step-1: To avoid cross leakage between χ (·) and φ (·), lock φN −1 and th XN−1 to the (N − 1) caplet/swaption volatility via Z TN −1 |ψ (t, TN−1 )|2 dt β 2 (TN −1 ) TN −1 = φ2 (TN −1 ) XN −1 XN−1 = 0

Step-2: Starting with the already specified φN −1 and XN −1 use the 1factor routine of Section-7.4 to obtain the φ1 , .., φN −2 and X1 , .., XN −2 solving the subsidiary problem β 2 (Tj ) Tj = φ2j Xj , e2 (Tj , TN ) Tj = β

−1 N −1 N X X

(j,N) (j,N ) A 2 φ 1 φ 2 Xj , 1

A

1=j 2=j

e (Tj , TN ) are the true ones β e (Tj , TN ) in which the target swaption volatilities β = β (Tj , TN ) during the first passage, but are changed for subsequent passages. Step-3: Using (7.8) recover from X1 , .., XN −1 the corresponding χ1 , ., χN −1 . Step-4: Insert the φ1 , .., φN −1 and χ1 , .., χN−1 computed in Steps-2,3 into the true expression (7.9), and obtain the swaption volatility error εj (note εN −1 = 0 always because of Step-1) satisfying £

2

¤

β (Tj , TN ) − εj Tj =

−1 N −1 N X X

(j,N) (j,N ) A 2 φ 1 φT 2 X 1, 2,j . 1

A

1=j 2=j

Step-5: Return to Step-2, replacing the target swaption volatilities 2 e 2 (Tj , TN ) + εj (when the routine works, the errors εj → 0 e β (Tj , TN ) with β rapidly, forcing the target swaptions into the real ones). REMARK 7.4 In the totally inhomogeneous case when ψ (t, T ) = 1, Step1 gives φ (T ) the flavour of volatility and χ (t) the flavour of perturbation away from 1. In contrast, when ψ (·) is a bestfit, Step-1 lends both φ (T ) and χ (t) the flavour of perturbations away from 1. Also, while the algorithm works quite well in practice, the author has not established suﬃcient conditions for convergence.

7.5.1

Alternatively

An alternative approach is an iterative routine, on each loop first fitting the swaptions by varying the χ (t) and then fitting the caplets by varying the φ (T ). Specifically:

66

Engineering BGM

Step-1: To avoid cross leakage between χ (·) and φ (·), lock φ (TN−1 ) = φN −1 and XN−1 to the (N − 1)th caplet/swaption volatility via XN −1 =

Z

0

TN −1

|ψ (t, TN−1 )|2 dt

β 2 (TN −1 ) TN −1 = φ2 (TN −1 ) XN −1 ,

and then set φ (Tj ) = φj = 1 for j = 1, 2, ..., N − 2. Step-2: With a given φj , use the swaption data β 2 (Tj , TN ) Tj =

N −1 N−1 X X

(j,N ) (j,N ) A 2 φ 1 φT 2 X 1, 2,j , 1

A

1=j 2=j

X

1, 2,j

=

j X

χ2k Ψ [ 1, 2, k] ∆k

k=1

for j = 1, 2, ..to successively back out the χ2j , setting χ2j = 0 if the required number should be negative. Step-3: With the χ2j from the previous step, use the caplet data β 2 (Tj ) Tj = φ2j Xj ,

Xj = Xj,j,j =

j X

χ2k Ψ [j, j, k] ∆k ,

k=1

to successively back out the φj for j = 1, 2, ..., N − 2. Then return to Step-2 and iterate. REMARK 7.5 Rather than bootstrapping to a full spectrum of caplets and coterminal swaptions, this method can clearly be adapted to fitting only liquid ones by using an optimizer at each step to hit the liquid instruments while smoothing the φj and χ2j with respect to j.

7.6

Pedersen’s method

The generic calibration described in this section is based on Pedersen’s paper [85], which the author recommends consulting for its numerous practical implementation details. Given an optimizer like the NAG [81] or IMSL [64] ones (or Powell’s method from Press’s Numerical Recipes [98] as suggested by Pedersen), his algorithm bestfits a selection of liquid caplets and swaptions, incorporates historical correlation and produces a smooth r-factor pseudo-homogeneous volatility function of the form ξ (t, x) = ξ (t, T )|T =t+x .

Calibration Techniques

67

We assume the (N − 1) × (N − 1) matrix from which the target caplet and swaptions (targeti i = 1, 2, ..) are drawn has shifted BGM implied volatilities entered by quarterly exercise in the rows, and quarterly increasing tenors in the columns, after the fashion of Example-7.1. The (N − 1) × (N − 1) matrix X that will be varied by the optimizer in 2 this algorithm, comprises (N − 1) parameters, X = (Xk, )

k, = 1, .., N − 1

with each parameter Xk, approximately the magnitude |ξ (t, x)| of the vector volatility function ξ (t, x) on certain intervals. Specifically we suppose ξ (t, x) is piecewise constant with |ξ (t, x)| ∼ = Xk,

t ∈ (Tk−1 , Tk ] ,

x ∈ (x

−1 , x

]

k, = 1, .., N − 1.

To tolerate this inexactness, bear in mind that in this routine X is simply a vehicle to producing a volatility function ξ (t, x) by the slightly roundabout process clearly specified below, and is not particularly significant in itself. The objective function for the optimizer involves several components and is of form obj (X) = w bestfit (X) + w1 smooth(1) (X) + w2 smooth(2) (X) ,

(7.10)

where the weights w and w1 , w2 are chosen to get a balance between bestfit to the target caplets and swaptions, and smoothness of X in both row and column directions. Both the fit and smoothing functions ¯2 X ¯¯ swpn (X) ¯ i ¯ ¯ bestfit (X) = ¯ target − 1¯ , i

i

smooth(1) (X) = smooth(2) (X) =

N −1 N−1 X X¯ k=1

=2

k=2

=1

¯2 ¯ Xk, ¯ ¯ ¯ , − 1 ¯ Xk, −1 ¯

N −1 N−1 X X¯

¯2 ¯ Xk, ¯ ¯ ¯ , − 1 ¯ Xk−1, ¯

are designed to be independent of scale. From X and the historical correlation function c (·) (see Chapter-6), the swaption prices swpni (X) are computed by first constructing a volatility function ξ (·) and then pricing in the standard fashion. For simplicity, fixed and floating side nodes are assumed to coincide with Tj = δj for j = 0, 1, .., N . For each k corresponding to the kth slice of calendar time t ∈ (Tk−1 , Tk ], given X the (N − 1) × (N − 1) matrix (with 1, 2 = 1, .., N − 1) ³ ´ T Ck = Xk, 1 c (x 1 ) c (x 2 ) Xk, 2

68

Engineering BGM

is a covariance, because the correlation of the relative shifted forwards H (t, x 1 ) and H (t, x 2 ) is cT (x 1 ) c (x 2 ). So if we eigenvalue decompose Ck , let Γ(1) , ..,Γ(r) be the first r eigenvectors ordered by eigenvalue size, multiplied by the square root of the corresponding eigenvalues, and introduce the (N − 1) × r matrix ´ ³ then ΓT Γ ∼ Γ = Γ(1) , ..., Γ(r) = Ck .

The last equation is approximate because we have discarded some eigenvectors, and that is the reason Xk, is only roughly the magnitude of |ξ (t, x)|. Now construct a volatility function ξ (·) for the k th slice of calendar time t ∈ (Tk−1 , Tk ] corresponding to the current value of X by setting the vector volatility function ³ ´ (1) (r) for x ∈ (x −1 , x ] . ξ (t, x) = ξ k. = Γ , ..., Γ

The full vector volatility function ξ (t, x) = ξ k. covering all calendar times t simply requires this step to be repeated for each slice of calendar time t ∈ (Tk−1 , Tk ] for k = 1, 2, .. Note that if t ∈ (Tk−1 , Tk ] the relative maturity x corresponding to Tj is bounded, that is t ∈ (Tk−1 , Tk ]

and

x + t = Tj

⇒

x ∈ [Tj − Tk , Tj − Tk−1 ) .

That means for each quarterly block of calendar time, the volatility ξ (t, Tj ) at absolute maturity Tj is also constant because ξ (t, Tj )|Tj =t+x = ξ (t, xj−k+1 ) = ξ k,

j−k+1

for

t ∈ (Tk−1 , Tk ] .

Caplet and swaption implied volatilities are then easily computed from (4.16), because they are a linear combination of integrals of type Z

T

ξ ∗ (t, Tj1 ) ξ (t, Tj2 ) dt =

0

XZ k

=δ

X

Tk

ξ ∗ (t, Tj1 ) ξ (t, Tj2 ) dt

Tk−1

ξ ∗k,

j1−k+1

ξ k,

j2−k+1 .

k

REMARK 7.6

In general we will not have ³ ´ (1) (r) Γ = Γ , ..., Γ = (X c (x )) ,

that is, the eigenvector decomposition step is not redundant because the components of (c (x )) are usually diﬀerent from the eigenvectors of Ck . Moreover, in practice the eigenvector decomposition applied to each slice of calendar time, appears to induce considerable variability in the vector components of

Calibration Techniques

69

ξ (·). An implementation of the algorithm viewed by the author had the X matrix represented by cells on an Excel spreadsheet with the cell colour changing with size of the corresponding component; the resulting filmshow as X converged to the desired degree of bestfit and smoothness was both entertaining and educational. The author feels Pedersen’s algorithm works well precisely because it can shuﬄe volatility between rows and then between factors within rows in an eﬃcient and coherent way. The overall modus-operandi for Pedersen’s algorithm is thus: Step-1 Start X either with yesterday’s X, or with magnitudes of the historical volatility function ξ (x) by setting Xk, = |ξ (x )| ∀k. Step-2 From X compute the implied volatilities of the target caplets and swaptions using the process described above. Step-3 Insert those values along with X and the target implied volatilities targeti i = 1, 2, .. into the objective function (7.10) and feed into the optimizer to generate a new trial X. Step-4 Return to Step-2 and iterate until the desired fit and degree of smoothness are obtained. Then extract the final ξ (t, x) as the desired bestfit volatility function.

7.7

Cascade fit

This generic calibration attempts to fit all caps, all swaptions and correlation with a piecewise constant volatility function of type ξ (t, T ) = γ (t, T ) c (T − t) ,

γ (t, T ) = γ k,

t ∈ (Tk−1 , Tk ]

k = 1, 2, .., n

for which all coverages are uniform δ j = δ i = δ vis-a-vis the nodes T0 = 0, T1 , ..., Tn , and the parameters γ k, for ≥ k ≥ 1 define the calibration. REMARK 7.7 The algorithm will require solving many quadratic equations which in practice frequently produce imaginary roots. So it frequently fails at a significant number of points where the corresponding γ j, must be made small or zero. The result is an unattractive volatility function with many peaks and troughs, a problem which Brigo and Mercurio [32] showed could be partially tackled by smoothing the cap/swaption volatility surface.

Assume for 1 ≤ j < N ≤ n we are given the 12 n (n − 1) caplet and swaption

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Engineering BGM

volatilities β 2 (Tj , TN ) Tj =

Z

0

=

N −1 N −1 X X

(j,N ) 1

(j,N) 2

A

A

1=j 2=j

Z

Tj

0

¯

−1 Tj ¯¯N X

¯ ¯ ¯

=j

¯2 ¯ ¯ (j,N) A ξ (t, T )¯¯ dt ¯

γ (t, T 1 ) γ (t, T 2 ) c∗ (T 1 − t) c (T 2 − t) dt,

(j,N )

where the weights A (j ≤ < N ) refer to the swaption maturing at Tj with final settlement at TN , and the caplets occur at j = N − 1 when (N −1,N ) AN −1 = 1. The integral term equals j X

γ k,

1

γ k,

2

Z

Tk

Tk−1

k=1

c∗ (T 1 − t) c (T 2 − t) dt = δ

j X

γ k,

1

γ k,

2

X

1, 2,k

k=1

in which we can precompute the correlation terms Z 1 δ ∗ c (T 1 + Tk−1 − t) c (T 2 + Tk−1 − t) dt X 1, 2,k = δ 0

⇒

X

, ,k

= 1.

For 1 ≤ j < N ≤ n the task therefore becomes finding γ k, to satisfy j β 2 (Tj , TN ) =

j N−1 −1 X X NX

(j,N) 1

A

(j,N ) γ k, 1 2

A

γ k,

2

X

1, 2,k .

(7.11)

k=1 1=j 2=j

Given j and γ k, (k < j, k ≤ ≤ n − 1), γ j, (j ≤ ≤ N − 2), the cascade routine satisfies (7.11) by making γ j,N −1 the largest root Γ of

+

  

 +

N −2 ´2 ³ X (j,N ) (j,N) (j,N ) Γ2 + 2 Γ A1 AN−1 γ j, AN −1

PN−2 PN−2 1=j

2=j

1=j

Pj−1 PN −1 PN −1 k=1

1=j

(j,N)

A "1

2=j

1

X

(j,N ) γ j, 1 γ j, 2 X 1, 2,j 2 (j,N) (j,N ) A 2 γ k, 1 γ k, 2 X 1, 2,k A1 −j β 2 (Tj , TN )

A

(7.12)

1,N −1,j

  #  

= 0.

For j = 1, start oﬀ the cascade (7.12) with γ 1,1 = β (T1 , T2 ) and consecutively determine γ 1,2 , γ 1,3 ,... When j > 1 set N = j + 1 in (7.11) and make γ j,j the positive root Γ of (j−1 ) X¡ ¢ 2 γ k,j − j β 2 (Tj , Tj+1 ) = 0, Γ2 + k=1

to set oﬀ the cascade (7.12) with γ j,j and consecutively determine γ j,j+1 , γ j,j+2 ...

Calibration Techniques

7.7.1

71

Extension

A useful extension of the cascade algorithm is to attempt an exact fit after first obtaining a bestfit. For example, suppose we have bestfitted correlation and the swaption matrix by the Pedersen method and produced a vector volatility function ψ (t, T ). We can now attempt an exact fit with the cascade method using a volatility function of type ξ (t, T ) = ψ (t, T ) + γ (t, T ) c (T − t) , γ (t, T ) = γ k, t ∈ (Tk−1 , Tk ] k = 1, 2, .., n where the parameters γ k, for ≥ k ≥ 1 define the new calibration. As above, for 1 ≤ j < N ≤ n and the given 12 n (n − 1) caplet and swaption volatilities, that means satisfying the equations ¯2 ¯ ¯ Z Tj ¯NX −1 ¯ ¯ (j,N) ¯ dt, ¯ β 2 (Tj , TN ) Tj = A [ψ (t, T ) + γ (t, T ) c (T − t)] ¯ ¯ 0 ¯ ¯ =j which generates a system of quadratic equations like (7.12) that can be solved in a similar consecutive fashion.

7.8

Exact fit with semidefinite programming

Starting with the same kind of pseudo-homogenous volatility function ξ (t, x) = ξ (t, T )|T =t+x = ξ k.

when

t ∈ (Tk−1 , Tk ]

x ∈ (x

−1 , x

]

as in Pedersen’s approach (see Section-7.6 above), and also varying correlation (which moves swaption but not caplet volatilities), it is possible to get an exact fit to the swaption volatility matrix under quite adverse market conditions (as during the collapse of Longterm Capital when short-term volatilities were very high) using semidefinite programming (SDP), see Appendix-A.4. Using homogenous by layer type volatility functions like ours, swaption zetas (that is, implied volatility squared multiplied by time to maturity, see 4.16) turn out to be linear combinations of covariance matrices, leading to linear equality constraints in the SDP framework. Moreover, because ξ(·) is pseudohomogenous, instantaneous correlation can be sensibly defined, allowing us to construct an objective function to bestfit historic covariance in a certain sense, and think of the resulting correlation structure as implied correlation. Introduce N − 1 semidefinite matrices X (k) ∈ SN −k for k = 1, 2, . . . , N − 1 (so X (1) ∈ SN −1 , . . . , X (N −1) ∈ S1 ) defined by ³ ´ (k) (k) 1, 2 = 1, 2, . . . , N − k, X 1, 2 = ξ ∗k, 1 ξ k, 2 , X (k) = X 1, 2

72

Engineering BGM

and with them form the 12 N (N − 1) × 12 N (N − 1) block semidefinite matrix ³ ´ X = diag X (1) , .., X (N −1)

Swaption implied volatilities are linear combinations of integrals like Z Tm m Z Tk X ξ ∗ (t, Tj1 ) ξ (t, Tj2 ) dt = ξ ∗ (t, Tj1 ) ξ (t, Tj2 ) dt 0

k=1

=δ

m X

ξ k,

j1−k+1

ξ k,

j2−k+1

Tk−1

=δ

k=1

m X

(k)

Xj1−k+1,

j2−k+1 ,

k=1

which in turn are linear combinations of the elements of X. Hence from (4.16), the implied volatility β (Tm , Tn ) of a swaption exercising at Tm and maturing at Tn can be expressed in the form of a standard SDP equality constraint

=

n−1 X

β 2 (Tm , Tn ) Tm = U (m,n) · X

n−1 X

(m,n)

Aj1

(m,n)

Aj2

j1=m j2=m

δ

m X

(k)

Xj1−k+1,

(7.13) j2−k+1 .

k=1

Our aim now is to satisfy swaption implied volatility constraints like (7.13) using an SDP objective function that produces an X which embodies a satisfactory implied correlation structure. In addressing that task our basic assumptions (which seem to have become fairly accepted wisdom) are: © ª 1. Ordering by eigenvalue size, the principle components e(i) of the historical covariance matrix are stable, but the level of volatility, that is, the corresponding eigenvalues {λi }, may change. 2. Some 3 to 5 factors and no more, are needed to adequately explain movement of the simple forward curves. That suggests a model for covariance like 5 X

λi e(i) e(i)∗ ,

i=1

with the positive eigenvalues λi made variable, but the eigenvectors e(i) held constant. For each i = 1, .., 5, introduce the N − 1 matrices E (i,k) ∈ SN −k for k = 1, 2, . . . , N − 1 defined by ³ ´³ ´∗ (i) (i) E (i,1) = e(i) e(i)∗ , E (i,k) = ej : j = 1, .., N − k ej : j = 1, .., N − k

that is, each E (i,k) is the previous E (i,k−1) less its last row and column. We make a linear combination of the E (i,k) the target for the for the covariance

Calibration Techniques

73

X (k) in the k th layer. Specifically, for each k = 1, .., N − 1 seek semidefinite (k) X (k) and numbers λi ≥ 0 (i = 1, .., 5) to minimize ° ° 5 ° ° ° (k) X (k) (k) ° λi Ei ° . °X − ° ° i=1

2

That can be done by solving the optimization problem find to minimize subject to and

(k)

X (k) , λ°i i = 1, .., 5; k = 1,°.., N − 1 PN−1 ° (k) P5 (k) (k) ° − i=1 λi Ei ° k=1 °X ¢2 ¡ U (m,n) · diag X (1) , .., X (N−1) = β 2 (Tm , Tn ) Tm (one equality constraint per fitted swaption) (k) X (k) º 0, λi ≥ 0 i = 1, .., 5; k = 1, .., N − 1

which may be expressed in the standard form (A.9) of Section-A.4.3.

Chapter 8 Interpolating Between Nodes

The business need for variable coverages δ j , the backward construction technique of Section-3.2, together with a limited calibration set, leads to a situation in which the forwards K (t, T ) and their corresponding volatilities ξ (t, T ) will be known only at a discrete maturity set Tj (j = 1, .., n). To price an instrument depending on an intermediate maturity, for example, a caplet maturing between Tj and Tj+1 , we therefore need to interpolate both volatility and forward. Our approach is to interpolate on deterministic functions like ξ (t, T ), and then use properties of the model to derive interpolations for stochastic variables like K (t, T ) and discount functions B (t, T ). In contrast, direct interpolation on stochastic variables turns out to be inaccurate and unsatisfactory. Note that none of the methods described in this chapter are arbitragefree, though in practice they work fairly accurately. Moreover, they are also inconsistent in that Section-8.3 on interpolating discount factors ought to determine how the forwards in Section-8.1 are interpolated, but they don’t. The author suggests consulting Schlogl [113] for a more exacting analysis.

8.1

Interpolating forwards

For 0 ≤ t ≤ T0 < T < T1 suppose K (t, T0 ) and K (t, T1 ) are known and K (t, T ) is required (for example, to find the intrinsic value of a forward swap whose resets fall between nodes). Start by interpolating the vector volatility function ξ (t, T ) on maturity T , defining 1 θ (T ) {(T1 − T ) ξ (t, T0 ) + (T − T0 ) ξ (t, T1 )} , δ = α (T ) ξ (t, T0 ) + β (T ) ξ (t, T1 ) , θ (T0 ) = 1 = θ (T1 ) .

ξ (t, T ) = with

(8.1)

This interpolation preserves correlation between forwards at nodepoints, θ (·) can be chosen to satisfy some auxiliary condition, and both α (T ) and β (T ) are independent of simulation paths and so can be precomputed.

75

76

Engineering BGM Recalling K (t, T0 ) = K (0, T0 ) E K (t, T0 ) = ln K (0, T0 )

⇒

Z

0

t

½Z

0

t

¾ ξ ∗ (s, T0 ) dWT1 (s)

1 ξ (s, T0 ) dWT1 (s) − 2 ∗

Z

t

0

2

|ξ (s, T0 )| ds

and identifying the forward measures for K (t, T ), K (t, T1 ) with that for K (t, T0 ) ( ) Rt ∗ Z t α (T ) ξ (s, T ) dW (s) 0 T1 R0 t ξ ∗ (s, T ) dWT1 (s) = so +β (T ) 0 ξ ∗ (s, T1 ) dWT1 (s) 0 ln

K (t, T0 ) K (t, T1 ) K (t, T ) ∼ + β (T ) ln (8.2) = α (T ) ln K (0, T ) K (0, T0 ) K (0, T1 ) # " R R 1 α (T ) 0t |ξ (s, T0 )|2 ds + β (T ) 0t |ξ (s, T1 )|2 ds Rt + , 2 2 − 0 |ξ (s, T )| ds

an interpolation for K (t, T ) in terms of K (t, T0 ) and K (t, T1 ) that contains a convexity term which can be precomputed using approximations like Z Z t t T |ξ (s, T )|2 ds = |ξ (s, T )|2 ds. T 0 0

REMARK 8.1 Interpolations that are closer to being arbitrage-free and consistent with the model, can be obtained by distinguishing measures and approximating the corresponding drifts.

8.2

Dead forwards

For T0 < T < T1 if K (T0 , T0 ) and K (T, T1 ) are known and K (T, T ) is required (for example, to price a caplet maturing at T ), the diﬃculty is the dead forward K (T0 , T0 ). To permit sensible approximations while maintaining stochasticity, we extend the life of K (t, T0 ) to T by writing ξ (t, T0 ) = ξ (T0 , T0 )

for T0 ≤ t ≤ T,

and then compute the virtual forward K (T, T0 ), given the three forward values K (T0 , T0 ), K (T0 , T1 ) and K (T, T1 ), from the conditional expectation ) ( ½ ¾ ¯¯Z T K (T, T0 ) ξ ∗ (T0 , T0 ) ¯ ∗ = ET1 E ξ (s, T1 ) dWT2 (s) , ¯ × [WT1 (T ) − WT1 (T0 )] ¯ T0 K (T0 , T0 )

Interpolating Between Nodes

77

where T2 = T1 + δ 1 . Identifying the two measures PT1 and PT2 and setting Z T 2 2 |ξ (s, T1 )| ds, q0 = |ξ (T0 , T0 )| (T − T0 ) , q1 = q12 =

or

T0

T

ξ ∗ (T0 , T0 ) ξ (s, T1 ) ds,

T0

µ

¶ K (T, T1 ) 1 + q1 , Y = K (T0 , T1 ) 2 T0 ¶ µ q q2 12 {ξ ∗ (T0 , T0 ) [WT1 (T ) − WT1 (T0 )] |, Y } ∼ N Y, q0 − 12 , q1 q1 ½ µ ¶ ¾ 2 K (T, T0 ) ∼ q12 1 1 q so q0 − 12 − q0 , Y + = exp K (T0 , T0 ) q1 2 q1 2 ¶ µ ¶ µ K (T, T0 ) ∼ q12 1 q12 K (T, T1 ) ln + q12 1 − . (8.3) ln = K (T0 , T0 ) q1 K (T0 , T1 ) 2 q1 Z

then

Z

T

ξ (s, T1 ) dWT1 (s) ∼ = ln ∗

Combining (8.2) evaluated at t = T and (8.3) gives an interpolation for K (T, T ).

8.3

Interpolation of discount factors

Iterating (1.9) relates the zeros B (t, TN ) and B (t, Tj ) when Tj < TN B (t, TN ) = YN−1 k=j

B (t, Tj ) [1 + δK (t, Tk )]

,

B (Tj , TN ) = YN−1 k=j

1

.

[1 + δK (t, Tk )]

During a simulation at a time t = Tj on the tenor node TN , we therefore know exactly the discount factor B (Tj , TN ). The problem arises when discount factors between nodes are required; for example, to make a decision at t = T ∈ (T0 , T1 ) because B (T, Tn ) = Yn−1 j=1

B (T, T1 )

,

[1 + δK (T, Tj )]

requires an expression for B (T, T1 ). The interpolation scheme (8.3) gives K (T, T ), so assuming a constant infinitesimal HJM forward rate f on [T, T + δ] when t = T , we have B (T, T + δ) =

1 = exp [−f δ] 1 + δK (T, T )

B (T, T1 ) = exp [−f (T1 − T )] =

⇒ 1

(1 + δK (T, T ))

(T1 −T ) δ

.

78

Engineering BGM

Hence the interpolation scheme for discount factors B (T, Tn ) = (1 + δK (T, T ))

8.4

(T1 −T ) δ

1 Yn−1 j=1

. [1 + δK (T, Tj )]

Consistent volatility

The interpolation can be used to define ξ (t, T ) for all T in a fashion consistent with the above interpolations on forwards and zeroes. For example, if we ask that caplet volatilities are linearly interpolated then s   qU T RT 2 0 2  (T1 −T ) 0 |ξ(s,T0 )| ds  |ξ (s, T )| ds 0 δ q U T T0 . = 2 1  + (T −T0 ) T 0 |ξ(s,T1 )| ds  δ

RT

T1

2

Integrating (8.1) and equating 0 |ξ (s, T )| ds in the two equations   2RT 2   Z T (T1 − T ) 0 |ξ (s, T0 )| ds   RT ∗ 2 2 2 δ |ξ (s, T )| ds = θ (T ) +2 (T − T0 ) (T1 − T ) 0 ξ (s, T1 ) ξ (s, T0 ) ds ,   RT 0   + (T − T0 )2 0 |ξ (s, T1 )|2 ds determines θ (T ) and hence the required interpolation for ξ (t, T ).

Chapter 9 Simulation

Two simulation methods are described in this chapter. Glasserman type methods [44], [45] avoid bias from drifts by discretizing the SDEs of positive continuous time martingales, and produce accurate results for time steps of the order of a couple of weeks. Big-step methods [90] use predictor-corrector techniques to approximate drift and volatility, and step in intervals of years between decision times (like Bermudan exercises). The author strongly recommends Glassermans’s book ‘Monte Carlo Methods in Financial Engineering’ [46] as a reference for this chapter.

9.1

Glasserman type simulation

This approach takes zeros B (t, Tj ) discounted by the numeraire N (t) (either M (t) for P0 or B (t, Tn ) for Pn ), and for j = @ (t) , .., n defines martingales Z (t, Tj ) and V (t, Tj ) under the corresponding measure by B (t, Tj ) >0 ⇒ N (t) Z (t, Tj ) − λj Z (t, Tj+1 ) >0 H (t, Tj ) = K (t, Tj ) + a (Tj ) = δ j Z (t, Tj+1 ) Z (t, Tj ) =

(j < n)

in which the λj = 1 − δ j a (Tj ) satisfy 0 ≤ λj ≤ 1; V (t, Tj ) = Z (t, Tj ) − λj Z (t, Tj+1 ) > 0 (j < n) and V (t, Tn ) = Z (t, Tn ) ;

V (t, Tn ) , Z (t, Tj ) = V (t, Tj ) + Πjj V (t, Tj+1 ) ... + Πn−1 j

Πkj = λj λj+1 ...λk .

Discretization of the Z (t, Tj ) and V (t, Tj ) as martingales can be rigorously enforced and decreases bias by avoiding drift (unlike, for example, simulating K (t, Tj ) under Pn ). Both Z (t, Tj ) and V (t, Tj ) should be strictly positive (zeros, numeraires and shifted forwards are strictly positive), which can be ensured by integrating a lognormal SDE with exponential increments. Making diﬀusion coeﬃcients Lipschitz in the discrete case without changing the continuous time dynamics turns out to require a simple adjustment.

79

80

Engineering BGM

Introduce φ {x} = min {1, x+ } and rely on Remarks 3.1.1 that 0 < hj (t) =

V (t, Tj ) δ j H (t, Tj ) = < 1, [1 + δ j K (t, Tj )] Z (t, Tj )

to rewrite bond volatility diﬀerences as b (t, Tj , Tj+1 ) = hj (t) ξ (t, Tj ) = φ {hj (t)} ξ (t, Tj ) = φ

9.1.1

½

V (t, Tj ) Z (t, Tj )

¾

ξ (t, Tj ) .

Under the terminal measure Pn

From (2.4) and (3.3) clearly Z (t, Tn ) = 1 and SDEs for the Z (·) for j = @ (t) , .., n − 1 are dZ (t, Tj ) = b∗ (t, Tj , Tn ) dWn (t) Z (t, Tj )   ¾ n−1 X ½ Z (t, T ) +1 ξ ∗ (t, T ) dWn (t) , = φ 1−λ Z (t, T ) =j

while V (t, Tn ) = 1 and SDEs for the V (·) for j = @ (t) , .., n − 1 are   n−1 X ½ V (t, T ) ¾ ∗ dV (t, Tj )  ∗ = ξ (t, Tj ) + ξ (t, T ) dWn (t) , φ V (t, Tj ) Z (t, T ) =j+1

Z (t, T ) = V (t, T ) + Π V (t, T +1 ) ... + Πn−1 V (t, Tn ) , P in which n−1 =n [·] = 0 when j = n − 1.

9.1.2

Under the spot measure P0

From (5.5) and (3.3) SDEs for the Z (·) are dZ (t, T@ ) = 0 and for j = @ (t) + 1, .., n dZ (t, Tj ) = −b∗ (t, T@ , Tj ) dW0 (t) Z (t, Tj )   ¾ ½ j−1 X V (t, T ) ξ ∗ (t, T ) dW0 (t) . = − φ Z (t, T ) =@(t)

while corresponding SDEs for the V (·) for j = @ (t) , .., n are   ¾ ½ j X dV (t, Tj )  ∗ V (t, T ) = ξ (t, Tj ) − ξ ∗ (t, T ) dW0 (t) , φ V (t, Tj ) Z (t, T ) =@(t)

Z (t, T ) = V (t, T ) + Π V (t, T

+1 ) ...

+ Πn−1 V (t, Tn ) .

Simulation

81

Note there is some redundancy in these (n − @ (t) + 1) SDEs for V (t, Tj ) because © ª dZ (t, T@ ) = 0 = d V (t, T@ ) + Π V (t, T@+1 ) ... + Πn−1 V (t, Tn ) . REMARK 9.1 In simulating in the Glasserman framework, bear in mind: [1] The V (t, Tj ) are diﬀerences of the Z (t, Tj ) and ought to exhibit approximately the same sort of order of magnitude behavior as the forwards K (t, Tj ), hence they simulate more accurately than the Z (t, Tj ). [2] In constructing payoﬀ functions use the property that discounted swaps will be linear combinations of the V (t, Tj ) because from (2.10) Z (t, Tj0 ) − Z (t, TjN ) pSwap (t) ¢ ¡ PiM −1 = PjN −1 + j=j0 δ j µj Z (t, Tj+1 ) − κ i=i0 δ i Z t, T i+1 . N (t)

(9.1)

[3] Integrate the lognormal SDEs in V (·) with exponential increments putting ½ ¾ 1 Vt+∆t = Vt exp (vol-at-t)∗ ∆Wt − |(vol-at-t)|2 ∆t . 2

[4] Usually a quarterly timestep ∆t = 14 is quite accurate (trial it on caps), 1 but always check results with a two-weekly timestep of ∆t = 24 .

9.2

Big-step simulation

From (5.5.1) and (5.5.7) the shifted forward H (t, Tj ) is either i ( R t hP ) n−1 ∗ − 0 h (s) ξ (s, T ) ξ (s, Tj ) ds H (t, Tj ) =j+1 =E Rt H (0, Tj ) + 0 ξ ∗ (s, Tj ) dWn (s) i ( R t hP ) j ∗ H (t, Tj ) h (s) ξ (s, T ) ξ (s, T ) ds j =@(s) 0 =E or Rt H (0, Tj ) + 0 ξ ∗ (s, Tj ) dW0 (s)

(9.2)

(9.3)

under the terminal Pn or the spot P0 measure respectively. In either case, to simulate H (t, Tj ) from H (0, Tj ) in one step (that is, by big-step simulation) we need to separately produce expressions for the drift and volatility terms in (9.2) and (9.3). Simulation of subsequent steps then follows the same pattern, because the model is Markov in the H (t, Tj ).

9.2.1

Volatility approximation

To jointly produce the normally distributed components of a vector µZ t ¶ X = (Xj ) = ξ ∗ (s, Tj ) dW (s) 0

82

Engineering BGM

construct a new high-dimensional model with a volatility function piecewise constant over the simulation interval [0, t] that has the same finite dimensional distributions as the original model. Do that by finding the square root Γ of the covariance matrix µZ t ¶ 1 ∗ T ξ (s, Tj1 ) ξ (s, Tj2 ) ds = ΓΓT , q = EXX = E (Xj1 Xj2 ) = t 0 and defining the volatility function e ξ (·) of the new model to be e ξ (s, Tj ) = Γj,·

s ∈ [0, t] .

√ The simulation step X = tΓε, where ε is a vector of IID standard normal random variables, then produces identical covariance.

9.2.2

Drift approximation

The drift terms in both (9.2) and (9.3) involves integrals like D (t, T , Tj ) =

Z

t

h (s) ξ ∗ (s, T ) ξ (s, Tj ) ds,

0

δ 0 H (s, T ) , h (s) v = 1 + δ 0 H (s, T )

δ0 =

δ , 1−δ a

in which h (s) is a P -martingale lying in (0, 1) that must be approximated knowing either h (t) or H (t, T ) in order to big-step over [0, t]. REMARK 9.2 It turns out that the ‘initial approximation’ h (s) = h (0) performs quite well, so clearly any reasonable approximation incorporating h (t) will perform better. Hence the following approach (similar to that of Pelsser et al [90]) using fairly robust approximations in various places. For 0 < s < t ≤ T < T1 = T + δ, appropriate approximations for a typical shifted forward H (t, T ), are ¯ ½ ¾ δ 0 H (s, T ) ∼ δ 0 H (s, T ) ¯¯ H (t, T ) = ET 1 + δ 0 H (s, T ) 1 + δ 0 H (s, T ) ¯ ¯ ½ ¾ δ 0 H (s, T ) ¯¯ δ 0 ET1 { H (s, T )| H (t, T )} ∼ , = ET1 H (t, T ) = 1 + δ 0 H (s, T ) ¯ 1 + δ 0 ET1 { H (s, T )| H (t, T )} Z s Z s 2 M (s, T ) = ξ ∗ (u, T ) dWT1 (u) , q (s, T ) = |ξ (u, T )| du, 0

0

ª © H (s, T ) = H (0, T ) exp M (s, T ) − 12 q (s, T ) .

Simulation

83

Noting the random variable { M (s, T )| M (t, T )} is normally distributed

o ) M (t, T ) , q(s,T q(t,T ) [q (t, T ) − q (s, T )] ; n o ) 1 q(s,T ) M (t, T ) − q (s, T ) , ET1 { H (s, T ) | H (t, T )} = H (0, T ) exp q(s,T q(t,T ) 2 q(t,T ) { M (s, T ) | M (t, T )} ∼ N

n

q(s,T ) q(t,T )

q(s,T )

q(s,T )

1− ∼ = H (0, T ) q(t,T ) H (t, T ) q(t,T )

and therefore the drift can be approximated by

D (t, T , Tj ) ∼ =

=

Z

δ 0 H (0, T )

t

0

0

q(s,T ) 1− q(t,T )

1 + δ H (0, T

1+δ 0 H(t,T ) ln 1+δ 0 H(0,T ) H(t,T ) ln H(0,T )

q (t, T )

q(s,T ) )

H (t, T ) q(t,T

q(s,T ) 1− ) q(t,T )

if

H (t, T

q(s,T ) ) q(t,T )

ξ (s, Tj ) = ξ (s, T )

ξ ∗ (s, T ) ξ (s, Tj ) ds,

for

s ∈ [0, t] .

So if the following ratio is independent of s ξ ∗ (s, T ) ξ (s, Tj )

⇒

2

|ξ (s, T )|

ξ ∗ (s, T ) ξ (s, Tj ) = fn (T , Tj ) |ξ (s, T )|2

as it is, for example, when the multi-factor volatility vector ξ (t, T ) is either constant or one-factor separable ξ (t, T ) = χ (t) φ (T ), then the drift over the simulation step can be approximated by the computationally eﬃcient D (t, T , Tj ) ∼ =

1+δ 0 H(t,T ) Z ln 1+δ 0 H(0,T ) H(t,T ) ln H(0,T )

t

ξ ∗ (u, T ) ξ (u, Tj ) du,

(9.4)

0

0

=

1+δ H(t,T ) ln 1+δ 0 H(0,T ) H(t,T ) ln H(0,T )

q

,j

t.

REMARK 9.3 An alternative view of (9.4) is that it can be obtained using an exponential bridge with ξ (s, T ) and ξ (s, Tj ) constant on s ∈ [0, t] s

s

H (s, T ) = H (0, T )1− t H (t, T ) t . As a further possibility, that suggests ³ s´ s h (s) = 1 − h (0) + h (t) t t which is a linear bridge.

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Engineering BGM

9.2.3

Big-stepping under the terminal measure Pn

Apply the approximations obtained in Section-9.2.1 and Section-9.2.2 to equation (9.2). Work backwards, and sequentially compute the shifted forwards H (t, Tn−1 ),..., H (t, T@ ); at each step the approximations can be directly applied.

9.2.4

Big-stepping under a tailored spot measure P0

The biggest time steps that can be taken using (9.3) are clearly the coverages δ j , because the summation in the drift in (9.3) is from = @ (s). To big-step between nodes T i (i = 1, .., M ), which (abusing the swap notation of Section 2.2) we assume are decision times for some product (for example, a Bermudan), introduce a new tailored spot measure P0 specific to the product by using as numeraire M (t) the roll-up of zero coupons between the decision nodes T i Y ¢ @−1 ¡ M (t) = B t, T @ =0

1 ¡ B T ,T

+1

¢,

@ (t) = inf

©

∈ Z+ : t ≤ T

ª

.

Similarly from Section-5.3 and using bar notation in the obvious way, if B (t, Tj ) then , t ≤ Tj , M (t) ¢ ¢ ª £ ¡ ¤© ¡ dZ (t, Tj ) = − b∗ t, T@ − b∗ (t, Tj ) dW (t) − b t, T@ dt , Z (t, Tj ) ¡ ¢ where = −b∗ t, T@ , Tj dW 0 (t) , ¢ ¡ which implies dW 0 (t) = dW (t) − b t, T@ dt ¢ ¡ dWj+1 (t) = dW 0 (t) + b t, T@ , Tj+1 dt. Z (t, Tj ) =

Hence under P0 an expression for H (t, Tj ) for j = @ (t) , .., n − 1, is i ( R t hP ) j ∗ h (s) ξ (s, T ) ξ (s, T ) ds H (t, Tj ) j 0 =@(s) =E , Rt H (0, Tj ) + ξ ∗ (s, Tj ) dW 0 (s)

(9.5)

0

and because the summation is now from = @ (s), the diﬃculties that occurred in big-stepping under the spot measure P0 disappear. A further slight diﬃculty is that successive H (t, Tj ) for j = @ (t) , .., n − 1 occur on both sides of (9.5). Tackle by cycling: find a value for H (t, Tj ) by setting hj (s) = hj (0), put that value into the approximations of Section-9.2.1 and Section-9.2.2 to compute a new H (t, Tj ), and then repeat. Finally, payoﬀs are of course present valued using the tailored spot measure P0 . For t ≤ T ∗ ≤ T < Tn the value X (t) of a cashflow X (T ∗ ) determined

Simulation at T ∗ and made at T , is therefore given by ¯ ¾ ¾ ½ ½ X (T ∗ ) ¯¯ B (T ∗ , T ) X (T ∗ ) X (t) . = E = EP0 F t P0 M (t) M (T ) ¯ M (T ∗ )

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Chapter 10 Timeslicers

If the shifted BGM volatility function is separable ξ (t, T ) = χ (t) φ (T ) so Z t Z t ξ ∗ (t, T ) dW (t) = φ∗ (T ) χ (t) dW (t) = φ∗ (T ) M (t) 0

0

then the driver in the model becomes the Gaussian martingale M (t) which (apropos of drifts) permits valuation by PDEs, lattices, trees or timeslicers. But getting a multi-dimensional homogenous correlation function c (T − t) into this framework is problematical (the last section in this chapter shows how it might be approached), so we concentrate on the one-factor case, which usually produces good theta and vega approximations for exotics, and which can also be used to debug code in corresponding simulation routines. Consider a simple Black-Scholes (BS) model (the notation is obvious) ¡ ¢ β t = exp (rt) , St = S0 E (rt + σWt ) = S (t, Xt ) , Xt = σWt ∼ N 0, σ 2 t , driven by the Gaussian martingale Xt . If ft = f (t, Xt ) is the time t price of some derivative, then for 0 < TL < TR (lying on floating-side nodes) ¯ ½ ¾ f (TL , XTL ) f (TR , XTR ) ¯¯ = E0 F ¯ TL . β TL β TR

Timeslicing consists of computing these conditional expectations backwards from maturity to root across consecutive timeslices TR → TL in a semi-analytic fashion (integrating cubics against normal densities) using the fact that XTR conditioned on XTL is also normally distributed (see Section-A.2.1): ¡ ¢ ⇒ ( XTR | XTL ) ∼ N XTL , σ 2 [TR − TL ] Z ∞ ³ ´ p e−r[TR −TL ] f TR , XTL + σ TR − TL x N1 (x) dx. f (TL , XTL ) = −∞

To evaluate integrals across timeslices TR → TL for each XTL -node: Step-1 Represent the random variables XTL and XTR by 50-100 nodes ranging to 5-6 standard deviations of Xt on respective timeslices, Step-2 Values of f (TR , XTR ) will be attached to the XTR -nodes on timesliceTR ; cubic spline them to produce a continuous representation, Step-3 Analytically evaluate the integrals, which are cubics against Gaussian densities, to find f (TL , XTL ) at each.XTL -node on timeslice-TL .

87

88

Engineering BGM

REMARK 10.1 To calibrate judgement about accuracy, node spacing and speed, value a BS call, whose price is known, back through about 20 timeslices. Depending on node density 4 to 6 figures of accuracy are possible.

REMARK 10.2 The timeslice approach is flexible. For example, if the timeslices are the resets of a Bermudan swaption, node values at timesliceT will be the continuation value for some XT and the intrinsic value for other XT .

10.1

Terminal measure timeslicer

With a one-factor separable volatility function ξ (t, T ) = χ (t) φ (T ), from (9.2) the shifted forwards under the terminal measure Pn take the form ( ) φ (Tj ) Mn (t) − 12 φ2 (Tj ) q (t) H (t, Tj ) Rt Pn−1 = exp , −φ (Tj ) H (0, Tj ) =j+1 φ (T ) 0 h (s) dq (s) Z t Z t χ (s) dWn (s) , q (t) = hMn i (t) = χ2 (s) ds, j = L, .., n − 1 Mn (t) = 0

0

and are essentially driven by the Gaussian martingale Mn (t), whose quadratic variation is q (t). Once nodes on say timeslice-TL are distributed to cover 5 to 6 standard deviations of Mn (TL ), values of H (TL , Tj ) can be assigned to those nodes using the drift approximation (9.4) Z

TL

0

hl (s) dq (s) ∼ =

ln

1+δ 0 H(TL ,Tj ) 1+δ 0 H(0,Tj ) H(TL ,Tj ) ln H(0,T j)

q (TL ) ,

δ0 =

δ 1−δ a

(10.1)

by starting with H (TL , Tn−1 ) and working backwards to specify H (TL , Tj ) for j = n − 1, .., L. With the forwards H (TL , Tj ) specified the values of other quantities that depend on them, like swaps, swaprates, and the numeraire bond B (TL , Tn ), can also be assigned to nodes on this timeslice-TL , for example, B (TL , Tn ) = FTL (TL , Tn ) =

n−1 Y j=j

1 . 1 + δ j [H (TL , Tj ) − a (Tj )]

Because (apropos of drift) the Pn -numeraire B (t, Tn ) is not path dependent (in contrast to the spot numeraires) one can load onto the timeslicer either

Timeslicers

89

actual or discounted values and then work consistently with that choice; that is, the discounted by B (t, Tn ) value f (t) of an asset f (t) is a Pn -martingale f (t) =

f (t) B (t, Tn )

⇒

¯ ¯ © ª © ª f (TL ) = En f (TR )¯ FTL ∼ = En f (TR )¯ Mn (TL ) ,

from our assumption that the model is approximately Markov in Mn (TL ). Because values of f (t) and B (t, Tn ) can be assigned to nodes, there is no problem loading the discounted value f (·) itself, which can then be timesliced back to the root, where the PV can be recovered as f (0) = B (0, Tn )f (0). To step down timeslices TR → TL the procedure is: Step-1 Discounted values f (TR , Mn (TR )) will already be attached to the nodes representing Mn (TR ) on timeslice-TR , so spline them to get a continuous cubic representation. Step-2 Assign nodes to timeslice-TL to represent the Gaussian random variable Mn (TL ), and then work out via H (TL , Tj ) the discounted intrinsic value i[Mn (TL )] of relevant instruments on each node Mn (TL ), for example, the underlying swap in the case of a Bermudan (values would also be initially attached to the furthest, that is, the very first timeslice in this same fashion). Step-3 With the aid of the formulae in Section-10.4 and the fact that { Mn (TR )| Mn (TL )} ∼ N (Mn (TL ) , q (TR ) − q (TL )) , compute discounted continuation values c[Mn (TL )] at each node Mn (TL ) ¯ © ª c [Mn (TL )] = En f (TR , Mn (TR ))¯ Mn (TL ) .

Step-4 From the intrinsic and continuation values, compute timeslice-TL values at each node Mn (TL ) on timeslice-TL ; for example, for Bermudans f (TL , Mn (TL )) = max {c [Mn (TL )] , i [Mn (TL )]} . Step-5 Go to Step 1, iterate and get f (0) = B (0, Tn ) f (0) at the root.

10.2

Intermediate measure timeslicer

Here the modus operandi is to attach nodes to the timeslicers under their respective forward measure, that is, nodes ML (TL ) to timeslice-TL under PTL and nodes MR (TR ) to timeslice-TR under PTR , and then step down timeslices TR → TL computing actual values (that is, not discounted values) under the PTR -forward measures while making some necessary drift adjustments. Assuming actual (that is, not discounted) values f (TR , MR (TR )) are hung on timeslice-TR , then actual values on timeslice-TL will be given by f (TL , ML (TL )) = B (TL , TR ) ETR { f (TR , MR (TR ))| FTL } .

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Engineering BGM

With separable volatility ξ (t, T ) = χ (t) φ (T ), from (3.3) some relevant Brownian motions, forwards and driving martingales under PL are: dWTj (t) = dWTL (t) +

j−1 X

h (t) χ (t) φ (T ) dt,

j>L

(10.2)

=L

¾ ½ H (t, Tj ) φ (Tj ) ML (t) − 12 φ2 (Tj ) q (t) R P , = exp t 2 H (0, Tj ) +φ (Tj ) j−1 =L φ (T ) 0 h (s) χ (s) ds Z t Z t ML (t) = χ (s) dWL (s) , q (t) = hML i (t) = χ2 (s) ds, 0

0

allowing, after a drift approximation like (10.1), the time-TL intrinsic values of relevant instruments including B (TL , TR ) = FTL (TL , TR ) to be hung on the nodes ML (TL ) of timeslice-TL . All that remains is to make a drift adjustment in the conditional expectation, that is, find the distribution under PTR of MR (TR ) conditional on ML (TL ), which from (10.2) are connected by MR (TL ) = ML (TL ) +

R−1 X

φ (T )

=L

Z

TL

h (s) χ2 (s) ds = ML (TL ) + D (TL ) ,

0

MR (TR ) = {ML (TL ) + D (TL )} E ⇒

(Z

TR

)

χ (s) dWR (s) , TL

{ MR (TR )| ML (TL )} ∼ N (ML (TL ) + D (TL ) , q (TR ) − q (TL )) ,

because from (10.1), D (TL ) is determined solely by ML (TL ). The procedure for stepping down nodes TR → TL is then similar to the terminal case Section10.1, except that at Step-2 the drift term D (TL ) must also be attached to the nodes ML (TL ) ready to use in the conditional expectation of Step-3.

10.3

A spot measure timeslicer is problematical

Unfortunately constructing a timeslicer under either the spot measure P0 or the tailored spot measure P0 is problematical due to the path dependent nature of the spot numeraires, which from (5.4) and Section-9.2.4 are respectively M (t) = B (t, T@ )

@−1 Y j=0

1 B (Tj , Tj+1 )

and

Y ¢ @−1 ¡ M (t) = B t, T @ j=0

1 ¢. ¡ B T j , T j+1

In big-step simulation the problem is circumvented by using the tailored spot measure P0 and conditioning on information at the beginning of each simula-

Timeslicers

91

tion step. But here, from (9.5), the shifted forwards will take the form ( ) φ (Tj ) M0 (t) − 12 φ2 (Tj ) q (t) H (t, Tj ) R t Pj = exp , +φ (Tj ) 0 φ (T ) h (s) χ2 (s) ds H (0, Tj ) =@(s) Z t Z t χ (s) dW 0 (s) , q (t) = hM0 i (t) = χ2 (s) ds, j ≥ @ (t) , M0 (t) = 0

0

which involves integrating the drift from time zero, rather than from the previous timestep. Hence in general, while we can approximate each hk (s) as a function of s and its initial and final values, the drift as a whole φ (Tj )

Z

0

t

j X

φ (T ) h (s) χ2 (s) ds

(10.3)

=@(s)

cannot be reasonably approximated in terms of a driver M0 (T ) at some timeslice-T due to the limit @ (s) in the summation. In particular, this path dependency prevents nodes being loaded with either initial or intrinsic values. Approximating hk (s) by its initial value hk (0) works, but leads to inaccuracies at distant timeslices and in out-of-the-money options. REMARK 10.3 The spot measure jumps through successive forward measures, see Section-5.3, so if an intermediate measure timeslicer works, why not a spot one? The respective conditional distributions of the driving martingales { MR (TR )| ML (TL )} ∼ N (ML (TL ) + D (TL ) , q (TR ) − q (TL )) and

{ M (TR )| M (TL )} ∼ N (M (TL ) , q (TR ) − q (TL ))

under PTR , under P0

show there is a diﬀerence. The point is that if we want the jump from PTR back to PTL to be regarded as under P0 then appropriate drifts must be used. For P0 that involves the path dependent term (10.3).

10.4 10.4.1

Some technical points Node placement

Density and spacing of nodes will determine speed and accuracy, and two alternatives present themselves: we could distribute nodes non-uniformly so that they are denser where the driver is most likely to be or where sensitive decisions have to be made (for example, close to a barrier); alternatively, we could assume all nodes on all timeslices are equally spaced, and then speed calculation by optimizing code with lookup tables.

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Engineering BGM

Equal spacing is easier to program and optimize for speed, but creates nodes where the driver has low probability of going. Distributing nodes where unconditional densities are highest (for example, as in the Gauss-Hermite integration routine) leads to inaccuracies in splining due to big gaps where the density is low. Whichever method is used an accuracy of five to six decimal places is an appropriate target to aim for. Whether or not we can expect to use just one lookup table for the whole valuation process, depends largely on whether or not the conditional variances change with timeslice. Because in shifted BGM that is the case for any parameterization other than ξ (t, T ) = φ (T )

or

χ (t) = 1,

we accept finding a new table for each timeslice, and place nodes at each timeslice in any way that makes sense.

10.4.2

Cubics against Gaussian density

Integrals of cubics (or polynomials in general) against the normal density function can be found by expressing them as the sum of integrals of the form Ã ! Z δ 1 [u − a]2 uν ∂Iν−1 p exp − du ⇒ Iν = a Iν−1 + β 2 Iν (δ, β, a) = 2 2 2 ∂a β 0 2πβ

by diﬀerentiating Iν (δ, β, a) with respect to a under the integral sign. The first few partial derivatives of I0 (δ, β, a) with respect to a are known ¶ µ ¶ ¶ ¶ µ µ µ a a δ−a δ−a 0 0 0 −N − , βI0 = −N +N − , I0 = N β β β β ¶ ¶ ¶ ¶ µ µ µ µ δ−a a δ−a a 2 00 3 000 00 00 000 000 β I0 = N −N − , β I0 = −N +N , − β β β β while from the recurrence relationship ¢ ¡ I1 = a I0 + β 2 I00 , I2 = a2 + β 2 I0 + 2aβ 2 I00 + β 4 I000 , ¢ ¡ ¢ ¡ I3 = a a2 + 3β 2 I0 + 3β 2 a2 + β 2 I00 + 3aβ 4 I000 + β 6 I0000 .

10.4.3

Splining the integrand

Given a set of m equally spaced points {(xj , zj ) : zj = z (xj ) ,

(xj+1 − xj ) = δ

j = 1, 2, .., m}

(with spacing δ) a cubic spline interpolation for z = z (x) on the interval x ∈ [xj , xj+1 ] only, can be expressed, setting u = x − xj , in the form £ 00 ¤ ¸ · − zj00 3 zj00 2 zj+1 δ 00 zj+1 − zj δ 00 (j) − zj+1 − zj u+ u + u . z (x) = z (u) = zj + δ 6 3 2 6δ

Timeslicers

93

¢ ¡ Hence if X ∼ N µ, σ 2 and we know z (X) for discreet values xj of X, then ( ) 1 [x − µ]2 z (x) √ E (z (X)) = exp − dx 2 σ2 2πσ 2 −∞ ( ) 2 X Z δ z (j) (u) 1 )] [u − (µ − x j ∼ √ exp − du, = 2 σ2 2πσ 2 0 j Z

∞

which can be evaluated using the results of Section-10.4.2.

10.4.4

Alternative spline

Instead of splining the integrand as in Section-10.4.3 above, one could spline the product of integrand and density and integrate that. A cubic spline interpolation through m points {(xj , zj ) : zj = z (xj )

j = 1, 2, .., m}

can be expressed, see Press et al [98], in the form 00 z = Azj + Bzj+1 + Czj00 + Dzj+1 , 1 1 (xj+1 − x) , B = 1 − A = (x − xj ) , A= δj δj ¢ ¢ 1¡ 3 1¡ 3 A − A δ 2j , D = B − B δ 2j , δ j = (xj+1 − xj ) . C= 6 6

The integral of the splined function is then x Zm x1

10.5

z (x) dx =

m ½ X 1 j=1

2

δ j (zj + zj+1 ) −

¾ ¢ 1 3 ¡ 00 00 . δ j zj + zj+1 24

Two-dimensional timeslicer

The problem is that the homogenous correlation part of a general volatility function is basically incompatible with the separable volatility structure needed to construct a timeslicer. Only exponential functions, which are both homogenous and separable, because exp (T − t) = exp (T ) exp (−t), seem to oﬀer a solution Start with the homogeneous part of a general volatility function ξ (t, T ) = χ (t) φ (T ) ψ (T − t) c (T − t) ,

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Engineering BGM

set x = T − t and bestfit its first two components with exponential functions µ ¶ µ ¶ a1 b1 exp (−λx) a + exp (−µx) b, a = b = . ψ (x) c (x) ∼ = a2 b2 With six parameters λ, µ, a1 , a2 , b1 , b2 a not unreasonable fit is possible. Now recalibrate using the separable multi-factor method of Section-7.5 to obtain ξ (t, T ) = ψ (t) φ (T ) {exp (−λ [T − t]) a + exp (−µ [T − t]) b} as the new volatility function for our two-dimensional timeslicer. Apropos of drifts, under say the terminal measure Pn the driving martingale will be Z t ∗ ψ (t) φ (T ) [exp (−λ [T − t]) a + exp (−µ [T − t]) b] dWn (t) , 0 Z t a∗ dWn (t) = |a| φ (T ) exp (−λT ) ψ (t) exp (λt) |a| 0 Z t b∗ dWn (t) , + |b| φ (T ) exp (−µT ) ψ (t) exp (µt) |b| 0 = |a| φ (T ) exp (−λT ) Mna (t) + |b| φ (T ) exp (−µT ) Mnb (t) ,

a linear combination of two jointly normally distributed random variables Z t Z t a b ψ (t) exp (λt) dWa (t) , Mn (t) = ψ (t) exp (µt) dWb (t) , Mn (t) = 0

0

that have zero mean, and variances, covariance and correlation given by Z t Z t 2 a b ψ (s) exp (2λs) ds, var Mn (t) = ψ 2 (s) exp (2µs) ds, var Mn (t) = 0 0 Z t ¡ a ¢ a · b cov Mn (t) , Mnb (t) = ψ 2 (s) exp ([λ + µ] s) ds, |a| |b| 0 ¡ ¢ cov Mna (t) , Mnb (t) p ρ (t) = p . var Mna (t) var Mnb (t)

A correlated two-dimensional timeslicer carrying both Mna (t) and Mnb (t) can now be constructed in a similar way to the one-dimensional equivalent.

Chapter 11 Pathwise Deltas

The most important risk measures are deltas (sensitivity to movements of the underlying yield curve) and vegas (sensitivity to changes in the implied volatilities of the instruments to which the model is calibrated); in this chapter we show how to compute pathwise deltas along the lines of Glasserman and Zhao [44]. Denote by P (numeraire Nt expectation E) whichever measure, spot P0 or terminal Pn , we are using. Let C0 be the present value of the discounted ∂ payoﬀ stream ‘payoﬀ C (·)’, and D = ∂K(0,T ) denote partial diﬀerentiation with respect to the initial value K (0, T ) of the forward K (t, T ). Then µ ¶ C0 giving C0 = N0 E payoﬀ C ⇒ D C0 = D N0 N0 µ ¶ C0 D C0 = N0 D + C0 D ln N0 = N0 D E payoﬀ C + C0 D ln N0 , N0 δ for Pn . with D ln N0 = 0 for P0 and D ln N0 = − 1 + δ K (0, T ) Knowing ³all the ´ option deltas D C0 , or equivalently the discounted option C0 deltas D N , permits sensitivities with respect to any other yieldcurve de0 pendent instrument like swaps to found with a little extra work (see Remark11.1). One reasonably accurate method of computing D C0 is to bump and grind: price the option, bump the forward, reprice and diﬀerence, that is, set µ ¶ C0 ∼ 1 D = {E payoﬀ C (K (0, T ) + ε) − E payoﬀ C (K (0, T ))} , N0 ε 1 = E {payoﬀ C (K (0, T ) + ε) − payoﬀ C (K (0, T ))} , ε → ED payoﬀ C (K (0, T )) as ε → 0, which allows the delta to be computed as the average of trajectory by trajectory diﬀerences. More accurate is the pathwise method : diﬀerentiate through the expectation and analytically compute D payoﬀ C (K (0, T )) in µ ¶ C0 = D E payoﬀ C (.., K (0, T ) , ..) = ED payoﬀ C (.., K (0, T ) , ..) . D N0 95

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Engineering BGM

That is, compute the partial derivative of the payoﬀ and then average that derivative over all trajectories. We now develop a suit of methods for finding D payoﬀ C (·) for forward, zero, swap and option payoﬀs.

11.1

Partial derivatives of forwards

Because the shift is a function of maturity only, for j, = 0, .., n − 1, clearly

∆j,

∂ ∂ = and ∂K (0, T ) ∂H (0, T ) ∂H (t, Tj ) ∂K (t, Tj ) = . (t) = D K (t, Tj ) = ∂K (0, T ) ∂H (0, T )

Recall from (5.1) and (5.7) that the SDE for H (t, Tj ) has form

d∆j,

dH (t, Tj ) = µj (t) dt + ξ ∗ (t, Tj ) dW (t) ⇒ (11.1) H (t, Tj ) n X ∂µj (t) dH (t, Tj ) + H (t, Tj ) ∆j1, (t) dt, (t) = ∆j, (t) H (t, Tj ) ∂K (0, Tj1 ) j1=1 ∆i, (0) = I {i = } .

with initial condition

Solving (11.1) numerically along with the Glasserman SDEs of Section-9.1 is too numerically intensive, and ignoring the drift leading to the solution ∆j, (t) =

K (t, Tj ) I [j = ] , K (0, T )

is too biased. So use the low variance property of hj (t), see (3.4), and set hj (t) = hj (0) =

δ j H (0, Tj ) [1 + δ j K (0, Tj )]

leading to the following approximate solution for H (t, Tj ) under P0 H (t, Tj ) =E H (0, Tj ) where

½Z

t 0

(0) µj

(0) µj

(t) dt +

Z

0

∗

(t) = ξ (t, Tj )

t

¾ ξ (t, Tj ) dW0 (t) , ∗

j X

j1=@

hj1 (0) ξ (t, Tj1 ) ,

(11.2)

Pathwise Deltas

97

and an alternative approximation under the terminal measure Pn H (t, Tj ) =E H (0, Tj ) where

½Z

t 0

(n) µj

(t) dt +

Z

0

(n)

µj (t) = −ξ ∗ (t, Tj )

t

¾ ξ (t, Tj ) dWn (t) , ∗

n−1 X

(11.3)

hj1 (0) ξ (t, Tj1 ) .

j1=j+1

Partially diﬀerentiating H (t, Tj ) with respect to H (0, T ), or equivalently K (0, T ), then yields ∆j, (t) =

H (t, Tj ) I {j = } + H (t, Tj ) H (0, T )

(0)

Z

0

t

(·)

D µj (s) ds,

(11.4)

D µj (t) = ξ (t, Tj ) ξ ∗ (t, T ) D h (0) I [@ (t) ≤ ≤ j] , (n)

D µj (t) = −ξ (t, Tj ) ξ ∗ (t, T ) D h (0) I [j < < n] , D h (0) =

11.2

1 − δ a (T )

[1 + δ K (0, T )]2

with

.

Partial derivatives of zeros and swaps

From (9.1) the discounted swap Sj0 at node t = Tj0 is S (Tj0 ) =

= Z (Tj0 , Tj0 )−Z (Tj0 , TjN )+

jN −1 X

pSwap (Tj0 ) N (Tj0 )

δ j µj Z (Tj0 , Tj+1 )−κ

j=j0

iM −1 X i=i0

¢ ¡ δ i Z Tj0 , T i+1 ,

so if we can find D Z (Tj0 , Tk ) (k ≥ j0), then we have D S (Tj0 ). Under P0 B (Tj0 , Tk ) = Z (Tj0 , Tk ) = M (Tj0 )

( Qj0−1

1 [1+δj K(Tj ,Tj )] Qj=0 k−1 1 × j=j0 [1+δj K(T j0 ,Tj )]

D Z (Tj0 , Tk ) = −Z (Tj0 , Tk )

( Pj0−1

)

δj D K(Tj ,Tj ) j=0 j K(Tj ,Tj )] Pk−1 [1+δ δj D K(Tj0 ,Tj ) + j=j0 [1+δ j K(Tj0 ,Tj )]

⇒ )

,

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Engineering BGM

while under Pn Z (Tj0 , Tk ) =

n−1 Y B (Tj0 , Tk ) = [1 + δ j K (Tj0 , Tj )] ⇒ B (Tj0 , Tn ) j=k

D Z (Tj0 , Tk ) = Z (Tj0 , Tk )

n−1 X j=k

REMARK 11.1

Letting the present values of the first N swaps be

pSwap (j) = 1 − B (0, Tj ) − κ the Jacobian J = (Jj, ) = the swap deltas

11.3

δ j D K (Tj0 , Tj ) . [1 + δ j K (Tj0 , Tj )]

∂C0 ∂ pSwap(j)

³

j−1 X

δ B (0, T

+1 ) ,

j = 1, .., N,

=0

∂ pSwap(j) ∂K(0,Tl )

´

can be found and inverted, allowing

to be expressed in terms of the forward deltas.

Diﬀerentiating option payoﬀs

Following are some not very rigorously derived results that are useful in diﬀerentiating option payoﬀs, see Section-A.5 for further information. Option Greeks usually involve the Heaviside (or characteristic) function I (·) and the positive value function (·)+ defined respectively by   1 when x > 0 I (x) = 12 when x = 0 , (x)+ = max (0, x) = x I (x) ,  0 when x < 0

and also the Dirac delta function δ (·), a probability measure such that ½ Z b 1 when 0 ∈ (a, b) δ (y) dy = . 0 when 0 ∈ / (a, b) a Integrating these functions yields expressions for their derivatives Z x d (x)+ = I (x) , I (u) du = (x)+ ⇒ dx −∞ Z x d I (x) = δ (x) , δ (u) du = I (x) ⇒ dx −∞

(11.5)

moreover, writing (x)+ = x I (x) and diﬀerentiating gives

d d (x)+ = x I (x) = I (x) + x δ (x) = I (x) dx dx ⇒ xδ (x) = 0.

(11.6)

Pathwise Deltas

99

REMARK 11.2 I (0) = 12 is consistent with δ (·) being an even function, and follows Fourier transform practice, see Section-A.5. The next lemma, which follows from (11.6), deals with a recurring kind of expression: LEMMA 11.1 If X = X ε , Y = Y ε , Z = Z ε are random variables dependent on a parameter ε, and A = A {X ≥ Y } is the event that X ≥ Y , then ¸¾ ½ · ∂ ∂ c I (A) + Y I (A ) = E {Z (X − Y ) δ (X − Y )} = 0. E Z X ∂ε ∂ε

REMARK 11.3 In practice, when the characteristic function represents the payoﬀ of a digital option, it is usually approximated by a call straddle o 1n + (x) − (x − ε)+ I (x) ∼ = ε

⇒

1 δ (x) ∼ = I [0 ≤ x ≤ ε] , ε

which expression is useful for simulating the Dirac delta function according to 1 E Z δ (X) = E Z I [0 ≤ X ≤ ε] ε for random variables X and Z.

11.4

Vanilla caplets and swaptions

The deltas of vanilla caplets and swaptions can of course be found by analytic methods, and in the case of a caplet, exactly. That provides a way of checking, see [44], the accuracy of the pathwise delta method. For a T -maturing caplet, the discounted payoﬀ at T1 is δ (K (T, T ) − κ)+ + = [Z (T, T ) − (1 + δκ) Z (T, T1 )] N (T1 ) ¾ ½ I [Z (T, T ) − (1 + δκ) Z (T, T1 )] , D payoﬀ C = × [D Z (T, T ) − (1 + δκ) D Z (T, T1 )]

payoﬀ C =

⇒

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Engineering BGM

while for a Tj0 -maturing swaption, the discounted payoﬀ at Tj0 is +

payoﬀ C = [S (Tj0 )] =

·

Z (Tj0 , Tj0 ) − Z (Tj0 , TjN ) ¢ ¡ PiM −1 −κ i=i0 δ i Z Tj0 , T i+1

¸+

⇒

¸ D Z (Tj0 , Tj0 ) − D Z (Tj0 , TjN ) ¢ ¡ P . D payoﬀ C = I [S (Tj0 )] iM −1 −κ i=i0 δ i D Z Tj0 , T i+1 ·

Finish by substituting for the D Z (·) and D K (·).

11.5

Barrier caps and floors

A barrier cap or floor is a vanilla cap knocked out or in by the current forward crossing a barrier. Because a knock-in together with a knock-out is the same as the underlying option, barrier options will be generally cheaper than their vanilla equivalents. Hence their attraction to customers who are willing to take a view in exchange for lower premiums. An illustration is the following: Example 11.1 With Libor at 5%, a customer takes a loan paying floating and for protection buys for about $10k a cap struck at 7.4%. But he doesn’t think rates will go above 6% and in any case can tolerate 7.4%, so he sells for about $100k an up-and-in floor struck at 7.4% with barrier at 6%. If his view is right, he has reduced the cost of his loan by $90k, if wrong he’ll pay 7.4% when floating is between 6% and 7.4%. As with all other barrier options, for example, FX barriers, there are sixteen flavours corresponding to combinations of cap or floor, up or down, in or out, and barrier less or greater than strike. Value a barrier floor struck at κ with barrier β (< κ), by valuing the component floorlets Ck (k = 1, .., N ) fixed at Tk and paying at Tk+1 . Let Aj = {K (Tj , Tj ) ≥ β} be the event (complement Acj ) that the barrier is crossed at Tj . Then the floorlet payoﬀ is ¾ ½ I [A1 ] + I [Ac1 ] I [A2 ] + I [Ac1£] I [Ac2¤] I [A3 ] δ(κ−K(Tk ,Tk ))+ , payoﬀ Ck = E N (Tk+1 ) +... + I [Ac1 ] I [Ac2 ] I [Ac3 ] ..I Ack−1 I [Ak ] ½ ¾ I [A1 ] + I [Ac1£] I [A2 ]¤ + ... + =E [(1 + δκ) Z (Tk , Tk+1 − Z (Tk , Tk ))] , I [Ac1 ] I [Ac2 ] ..I Ack−1 I [Ak ]

Pathwise Deltas

101

and its delta is a rather complex expression into which the D Z (·) and D K (·) from the previous sections must be substituted  ¶  µ 1 − I [A2 ] − I [A£c2 ] I [A3¤] − ...     D K (T , T ) δ [A ]   1 1 1 c c c   I [A ] I [A ] ..I A ] −I [A   k 2 3 k−1   ¶ µ c c I [A ] − I [A ] I [A ] − ... 3 1 1£ D payoﬀ Ck = E +D K (T2 , T2 ) δ [A2 ] ¤  −I [Ac1 ] I [Ac3 ] ..I Ack−1 I [Ak ]          ..   . × [(1 + δκ) Z (Tk , Tk+1 − Z (Tk , Tk ))]+ ¾ ½ I [A1 ] + I [Ac1 ] I [A2 ] + I [Ac1£] I [Ac2¤] I [A3 ] + +... + I [Ac1 ] I [Ac2 ] I [Ac3 ] ..I Ack−1 I [Ak ] ¾ ½ I [(1 + δκ) Z (Tk , Tk+1 − Z (Tk , Tk ))] . × × [D Z (T, T ) − (1 + δκ) D Z (T, T1 )]

In simulating this expression, the Dirac delta functions are of course handled along the lines of Remark-11.3.

Chapter 12 Bermudans

The owner of a payer (receiver) Bermudan swaption has the right to exercise into an underlying payer (receiver) swap at some subset of its fixed side reset dates. A good source of revenue for banks is an arrangement popular with investors, in which they receive higher than usual fixed interest in exchange for giving the bank the right to cancel the deal if it does not suit. At the centre of the structure is a callable swap, which is a payer swap that can be cancelled at any of its reset dates. A callable swap in which the bank pays fixed, is thus equivalent to the bank owning a vanilla payer swap plus a receiver Bermudan, because if the bank exercises the Bermudan, it gets a receiver swap that oﬀsets (cancels) the remainder of the payer swap. Roughly speaking, these sorts of deals are set up as follows. The investor deposits funds with the bank which are safely invested and generate floating Libor back to the bank. The bank then sets up (with its exotic derivatives desk) a callable swap in which the exotics desk receives that floating Libor and pays a fixed coupon. But because the swap is cancellable, that is equivalent to the exotics desk getting a free receiver Bermudan, which is worth money. That free money is then in part used to increase the fixed coupon in the underlying payer swap, which goes back to the investor as enhanced yield. A moment’s thought reveals that such deals will tend to get cancelled if Libor rates fall, because the payer swap is then getting more expensive for the exotics desk. So the investor takes a view and bets that interest rates will remain steady; if he is right, he receives enhanced coupon for the full duration of the deposit, if he is wrong, he gets his deposit back early and must find an alternative investment at probably lower rates. Diﬀerent views for the consideration of the investor can be created by substituting exotic coupons for the fixed coupon. For example, a callable constant maturity spread swap might, instead of the fixed coupon, pay a constant coupon plus a positive multiple of the 2-year constant maturity swaprate less the 30-year constant maturity swaprate. The investor receives enhanced yield betting that long-term rates don’t fall relative to short-term rates (because in that case the exotics desk must pay increased coupon perhaps causing them to cancel). Receiver Bermudans are also often used to partially hedge fixed coupon mortgaged backed securities (MBS); if rates drop and mortgagees refinance at a lower coupon, banks may still want to receive the original higher coupon.

103

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Engineering BGM

The huge size of the US MBS market thus accounts for the massive Bermudan books carried by many US banks. Bermudans are now so common as to be almost vanilla products; some traders calibrate to them! Nevertheless, because they are the archetypical callable product, their pricing exhibits many of the techniques needed to price more esoteric callable exotics; hence this chapter. They are also very stable instruments, relatively easy to price and hedge, and not overly sensitive to correlation or the swaption volatility smile. We now show how to combine Glasserman style simulation with LongstaﬀSchwartz’s regression technique [73] for conditional expectations, to value a payer Bermudan which can be exercised into a notional underlying fixed maturity payer swap at any one of its (M − 1) reset times T i (i = 0, 1, ..., M − 1). The author would also suggest Chapter-8 of Glasserman’s book [46] for parallel reading with this chapter.

12.1

Backward recursion

Denote by P (numeraire N (t) expectation E) whichever measure (spot P0 or terminal Pn ) we are using, let S (t) be the time-t discounted to t = 0 intrinsic value of the underlying swap with next reset T i ¢ ¡ pSwap t, T i , T M , S (t) = N (t) so that at a reset when t = T i we can use the notation ¡ ¢ ¡ ¢ pSwap T i , T i , T M ¡ ¢ Si = S T i = , N Ti

(12.1)

and write expectations conditioned on Fi = FT i as E(i) X = E { X| Fi } .

REMARK 12.1 Here, and generally¡ throughout this chapter, we work ¢ with variables like the discounted Si = S T i in equation (12.1) rather than ¡ ¢ the undiscounted pSwap t, T i , T M . That is, values of all relevant instruments will be discounted by the numeraire N (t) to time t = 0 making them P-martingales. Note also that any exercise decisions determined by inequalities can be based on either non-discounted or discounted values; because numeraires are positive the results must be the same. The present value Bm (0) of a Bermudan taken © is therefore the supremum ª over all discrete stopping times τ ∈ T = T i : i = 0, 1, ..., M − 1 of potential

Bermudans

105

discounted payoﬀs, namely Bm (0) = sup ES (τ ) = ES (τ ∗ ) ,

(12.2)

τ ∈T

where τ ∗ ∈ T is the optimal stopping time. If Bm (t) is the discounted to ¡t =¢ 0 Bermudan value at time t, then at consecutive resets T i clearly Bm T i is the maximum of the intrinsic value ¡ ¢ © ¡ ¢¯ ª S T i and the continuation value E Bm T i+1 ¯ FT i £ ¡ ¢ © ¡ ¢¯ ª¤ ¡ ¢ Bm T i = max S T i , E Bm T i+1 ¯ FT i , ¡ ¢ which, using the above notation and setting Bm T i = Bmi , can be written more compactly as (12.3) Bmi = Si ∨ E(i) Bmi+1 . Hence Bm (t) is a supermartingale that dominates S (t) because both © ¡ ¢¯ ª ¡ ¢ ¡ ¢ ¡ ¢ and Bm T i ≥ S T i . Bm T i ≥ E Bm T i+1 ¯ FT i

Repeated application of (12.3) for i = M − 1, .., 0 then generates the following backward recursion for Bm (0) = Bm0 + BmM−1 = SM−1 ∨ 0 = SM−1 , (i)

Bmi = Si ∨ E

Bmi+1 ,

(12.4)

(i = M − 2, .., 0) .

¡ ¢ Letting Ai be the event that at T i the intrinsic value Si = S T i of the underlying swap is greater than the continuation value E(i) Bmi+1 , that is n o Ai = Si ≥ E(i) Bmi+1 , i = 0, 1, .., M − 2 and

AM−1 = {SM−1 ≥ 0} ,

the recursion (12.4) combines to give iiii h h h h Bm0 = S0 ∨ E(0) S1 ∨ E(1) S2 ∨ E(2) ... SM−2 ∨ E(M−2) [SM−1 ∨ 0]

= E(0) E(1) E(2) ...E(M−2) S0 ∨ S1 ∨ S2 ∨ S3 ∨ S4 ∨ ....SM−2 ∨ SM−1 ∨ 0, + = E S0 ∨ S1 ∨ S2 ∨ S3 ∨ ....SM−2 ∨ SM−1 , ¾ ½ I [A0 ] S0 + I [Ac0 ] I [A1 ] S£1 + I [A¤c0 ] I [Ac1 ] I [A2 ] S2 + .. . =E ... + I [Ac0 ] I [Ac1 ] ..I AcM−2 I [AM−1 ] SM−1

Now our problem is to decide which of Si or E(i) Bmi+1 is greater at reset time T i , in the context of a simulation algorithm with a large number of simulated trajectories, and that is where the Longstaﬀ-Schwartz technique is appropriate. For simplicity, we will illustrate his method via an example in Section-12.2 below.

106

12.1.1

Engineering BGM

Alternative backward recursion

For a more numerically eﬃcient backward recursion introduce ¡ ¢ Hi = H T i = E(i) Bmi+1 −Si , HM−1 = −SM−1 .

Using the recurrence (12.4) for Bmi , the recurrence between Hi−1 and Hi for i = M − 1, .., 1 is therefore n o + Hi−1 = E(i−1) {Si ∨ [Hi + Si ]} − Si−1 = E(i−1) Si + (Hi ) − Si−1 , +

= E(i−1) (Hi ) + E(i−1) {Si } − Si−1 .

But from (9.1) and (12.1) M−1 X ¢ ¢ ¡ ¡ Si = 1 − Z T i , T M − κ δ i1 Z T i , T i1+1 , i1=i

⇒

(i−1)

E

¢ ¡ {Si } − Si−1 = κδ i−1 Z T i−1 , T i .

Hence a recursion involving only one coupon

Hi−1

HM−1 = −SM −1 , B0 = H0 + S0 , n o ¢ ¡ i = M − 1, .., 1 = E(i−1) (Hi )+ + κδ i−1 Z T i−1 , T i

as opposed to the many time consuming coupon calculations needed in (12.4). REMARK 12.2 If the coupon payment κ in the underlying¡swap ¢ S (t) is exotic it may be diﬃcult to compute the intrinsic values Si = S T i , a problem that this algorithm solves because to compute Hi−1 only the immediate method of computing, coupon determined at T i−1 is required. An alternative ¡ ¢ or rather estimating, intrinsic values Si = S T i for exotic coupon swaps is to apply the Longstaﬀ-Schwartz regression technique of the next Section12.2 and regress realized coupon values along each trajectory against values of relevant variables at time T i .

12.2

The Longstaﬀ-Schwartz lower bound technique

We illustrate Longstaﬀ-Schwartz’s method of computing lower bound values of callable options like Bermudans, through an example. Consider a 10-year Bermudan struck at κ which can be exercised annually into a 10-year swap with annual rolls; that is, M = 10 and T i = i.

Bermudans

12.2.1

107

When to exercise

On each of K simulated trajectories ω k (k = 1, .., K), the Bermudan could be exercised at just one of the 9 reset times T 1 , .., T 9 , say T i , into one of the following 9 swaps with discounted intrinsic value ¡ ¢ ¢ ¡ ¢ pSwap T i , T i , T 10 ¡ ¡ ¢ i = 1, .., 9. I ωk , T i = Si = S T i = N Ti

ª © For trajectory ω k suppose τ (ω k ) ∈ T 1 , .., T 9 is the exercise time. Each value I (ω k , τ (ω k )) is discounted to the root, so their average 1 X I (ω k , τ (ω k )) ≤ Bm (0) K k

will be a lower bound estimate of the Bermudan’s price. Proceed as follows ¡ ¢ to carry back along ω k the up-to T i best discounted intrinsic value Yk T i so far. Step-1 Start with reset T 9 = 9 on trajectory ¢ ω k . Because exercise is deter¡ mined solely by the intrinsic value I ω k , T 9 at reset T 9 , clearly the corresponding best discounted value at T 9 is ¢ ¡ ¢ ½ ¡ ¡ ¢ ¡ ¢ I ωk , T 9 if¡ I ω¢k , T i > 0 . Yk T 9 = Yk ω k , T 9 = 0 if I ω k , T i ≤ 0

Step-2 Move ¡ ¢ to reset T 8 . On trajectory ¢ ω k the choice for best discounted ¡ value Yk T 8 at T 8 is either I ω k , T 8 if it is best to exercise at T 8 or ¡ ¢ Yk T 9 if it is best to exercise later. In general to step from T i+1 to T i use ¢¯ ¢ ¡ ¡ the continuation value E Bm T i+1 ¯ FT i to make the exercise decision, so that the best discounted value at T i is ¢  ¡  I ωk , T i ≤ ·  ¡ 0 or¢ ¡ ¢ ¸    I ω k ,¢¯T i >¢ 0 and¡ Y if T  k i+1 ¡ ¡ ¢ ¡ ¢ ,  E Bm¡ T i+1 ¢¯ FT i (ω k ) > I ω k , T i Yk T i = ¸ ·  ¡  ¢  and¡  ¡ ¡I ω k ,¢¯T i >¢ 0 ¢ .  I ωk , T i if E Bm T i+1 ¯ FT i (ω k ) ≤ I ωk , T i

Step-3 Repeat Step-2 through the resets T 7 , T 6 , ..., T 1 . On reaching the first reset T 1 we have found a lower bound on Bm (0) ¡ ¢ I (ω k , τ (ω k )) = Yk T 1

⇒

1 X ¡ ¢ Yk T 1 ≤ Bm (0) . K k

It remains to identify or approximate the continuation values.

108

12.2.2

Engineering BGM

Regression technique

Because our example is Markov in the 40 (quarterly for 10 years is 4 × 10) Glasserman V (·) variables, the information set FT i will be determined by their time T i values or any linearly independent combination thereof. So the conditional expectation that is the continuation value will be a function of those variables evaluated at T i . ¡ ¡ ¢¯ ¢ ¡ ¢ ¡ ¡ ¢ ¡ ¢ ¡ ¢¢ E B T i+1 ¯ FT i = fi T i = fi V1 T i , V2 T i , .., V40 T i ¡ ¢ and if we find or approximate fi T i , we have the continuation values. Using linear L2 regression, ¡ ¢ we estimate on timeslice T i the corresponding continuation function fi T i from the information embodied in the simulated trajectories. Some robust heuristic approximation will be needed, because it is not practical to regress on a large number of basis functions. The art is to choose for basis functions those linear combinations of the Glasserman V (·) variables that account for most of the value of the Bermudan. Reasonable results can be obtained by regressing on just the intrinsic values. For example, in stepping down from reset T 9 → T 8 let ¢¤+ £ ¡ U = U (ω k ) = I ω k , T 8 , which is the positive part of the intrinsic value at time T 8 and regress on 1, U and U 2 . The constant 1 is included to add useful stability. Including a in the regression X¡ ¢2 Yk − a − bUk − cUk2 , min a,b,c..

k

makes the means of target and estimate equal on minimizing with respect to a X X ∂ =0 ⇒ Yk = a + bUk + cUk2 . ∂a k

k

The regression runs as follows: Step-1 For negative intrinsic values I (ω k , R8 ) at reset T 8 no decision is required, so first identify the set containing K (8) trajectories ¢ ª © ¡ Ω+ 8 = ω k (k = 1, .., K (8)) : I ω k , T 8 > 0 ¢ ¡ for which that intrinsic value I ω k , T 8 is strictly positive. We will regress on the corresponding Uk = U (ω k ), where the ω k ∈ Ω+ for k = 1, 2, .., K (8). ¢T 8 ¡ Step-2 Look for the best solution C = c1 c2 c3 in the L2 sense to the over-specified system of equations ¡ ¢     U12 1 U1 Y1 T 9 ..    .. ..  .. . .  .  . ¢      ¡   1 Uk  Uk2   C =  Yk T 9  .    .. ..  ..  .. . .  .  .¡ ¢  2 1 UK(8) UK(8) YK(8) T 9

Bermudans

109

Step-3 Having found C, compute the estimates ¡ ¢ ¡ ¡ ¢¯ ¢ f8 T 8 = E B T 9 ¯ F8     ¡ ¡ ¢¯¯ ¢ U12 1 U1 E B T 9 F8 (ω 1 ) ..    .. ..  ..  . . . .      ¡ ¡ ¢¯ ¢ ∼    Uk2  =  E B T 9 ¯ F8 (ω k )  =  1 Uk C   .. .. . . ..   . .   .. ¡ ¡ ¢¯ ¢ ¡ ¢ 2 ¯ 1 UK(8) UK(8) E B T 9 F8 ω K(8) ¡ ¡ ¢¯ ¢ of the conditional expectations E B T 9 ¯ F8 (ω k ) for each of the trajectories ¢ ¡ ω k ∈ Ω+ 8 with positive intrinsic values I ω k , T 8 > 0. Step-4 Repeat Step-3 backwards through the resets T 7 , T 6 , ..., T 1 . A good technique to reduce bias (see [46] for further comment), is to use one set of trajectories (about 1k is enough) to compute the regression coeﬃcients (and thus eﬀectively the stopping times), and another set (4k - 10k) to compute prices.

12.2.3

Comments on the Longstaﬀ-Schwartz technique

The Longstaﬀ-Schwartz method clearly gives a lower bound on the price of a payer Bermudan, because optimizing with a restricted set of regression variables is an approximation that might be improved with a larger and better choice. The author’s general experience with the technique, however, is that it is robust and with proper choice of regression variables the lower bound obtained is an accurate measure of the Bermudans value. The choice of regression variables is partly an art and partly a science, and generally they should be tailored to the callable instrument being valued and involve variables that in a linear combination can mimic its potential behavior. Thus for vanilla Bermudans useful regression variables might include: swap values, because they are linear combinations of the finite number of Markov variables (the Glasserman V (·) variables) underlying the model, and so make good variables for regression; zero coupons to reflect the level of the yieldcurve; diﬀerences of zero coupons to reflect tilt and flex in the yieldcurve; European swaptions maturing at the reset dates, because they form lower bounds etc. The author has tended to rely on a combination of judicious use of the timeslicer (recall that it does not incorporate correlation but is otherwise accurate) to check simulation results, and experiment with diﬀerent sets of regression variables (trying to push up prices and get them stable) to increase his comfort levels with the lower bound Longstaﬀ-Schwartz prices. He has yet to implement the time-consuming upper bound method described in Section12.3 below, and thus get a genuine confidence interval for prices. Moreover, at the present moment he can’t help feeling that perhaps the proper place for the upper bound method is more as a risk management tool to check models, rather than an everyday front oﬃce pricer.

110

12.3

Engineering BGM

Upper bounds

The lower bound on the Bermudan price as obtained by Longstaﬀ-Schwartz’s technique corresponds to the buyer’s bid. He owns the option, has perhaps paid less than the optimal price for it, and can decide to exercise it according to the optimal routine he has used to compute his bid, or indeed at any other time of his choosing. An upper bound on the Bermudan price corresponds naturally to the seller’s oﬀer, he must hedge the sold Bermudan and cater for the possibility that the buyer might exercise at any time, including perhaps accidentally at a time that is more advantageous to him than the optimal time. A series of papers have tackled the upper bound problem starting with essentially equivalent independent approaches by Rogers [109] and Haugh and Kogan [51], and an alternative approach by Jamshidian [67]. Subsequently Andersen and Broadie [8] added operational depth, which was refined by Joshi [69], [70], [71]. From (12.2), for any stopping time τ ∈ T ES (τ ) ≤ Bm (0) = sup ES (τ ) = ES (τ ∗ ) , τ ∈T

showing that any stopping time can be used to compute a lower bound on the Bermudan price Bm0 . To get an upper bound, let H be the set of all adapted martingales h (t) for which supt∈T |h (t)| < ∞. Then for any h ∈ H, we have Bm (0) = sup E [S (τ ) + h (t) − h (t)] = Eh (τ ∗ ) + sup E [S (τ ) − h (t)] , τ ∈T

τ ∈T

= h (0) + sup E [S (τ ) − h (t)] , τ ∈T

from the optional sampling theorem applied to the martingale h (t). Hence Bm (0) ≤ h (0) + E max [S (t) − h (t)] ,

(12.5)

t∈T

where the maximum is now taken over the exercise times in © ª T = T i : i = 0, 1, ..., M − 1 ,

and not the stopping times. Because h ∈ H was arbitrary, that yields a problem dual to (12.2), namely ½ ¾ Bm (0) ≤ inf h (0) + E max [S (t) − h (t)] . h∈H

t∈T

(12.6)

Bermudans

111

In fact, as Rogers showed, equality holds because Bm (t), being a supermartingale that dominates S (t) according to the Doob-Meyer decomposition can be written Bm (t) = M (t) − A (t) , where M (t) is a martingale and A (t) is an increasing process with A (0) = 0. Substituting M (t) for h (t) in (12.5) gives Bm (0) ≤ Bm (0) + E max [S (t) − Bm (t) − A (t)] ≤ Bm (0) , t∈T

where the second inequality holds because Bm (t) ≥ S (t) and A (t) ≥ 0. Upper bounds can now be computed by judicious choices of martingales h (t) to insert in (12.5), after which the maximums on each trajectory are averaged to get the bound. Andersen and Broadie used a martingale defined as follows. Let L (t) be the discounted to time t = 0 of the lower bound estimate of Bm (t) given by the Longstaﬀ-Schwartz method, and for t ∈ T set ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ © ¡ ¢ ¡ ¢ª h T i = h T i−1 + L T i − L T i−1 − Ii−1 E(i−1) L T i − L T i−1 ,

where Ii−1 = 0 if continuation is indicated at T i−1 , and Ii−1 = 1 if exercise is indicated at T i−1 . Joshi’s h (t) martingale (essentially the same as the Andersen and Broadie one) is the seller’s self-financing hedge consisting of one unit of Bermudan option bought at the buyer’s price. The seller follows the buyer’s optimal strategy giving rise to four possibilities at each exercise date: in the two cases where buyer and seller agree there is a perfect hedge; if the buyer exercises and the seller does not then the price from the optimal strategy is greater than the exercise value and the seller makes money; if the buyer does not exercise and the seller does, then the seller can re-buy the option for less than the exercise and again makes money. Spare money can be invested in the numeraire resulting in a self-financing strategy that must be a martingale after discounting. Note that each of these techniques requires on each ¢ ¡ ¢running ¡sub-simulations trajectory at each reset to find either L T i or Bm T i , a total of say N1 outer simulations, followed by say N2 inner simulations at each reset making N1 × N2 × M simulations altogether, which can evidently take up to 20 times longer than the lower bound simulation. On the other hand, variances tend to be lower cutting the number of paths needed.

12.4

Bermudan deltas

∂ Setting D = ∂K(0,T ) as in Chapter-11, and diﬀerentiating the recursion (12.4) under the expectation with respect to the initial value K (0, T ) of

112 the

Engineering BGM th

forward ·

¸ I [A0 ] S0 + I [Ac0 ] I [A1 ] S£1 + I [A¤c0 ] I [Ac1 ] I [A2 ] S2 + .. , D Bm0 = ED ... + I [Ac0 ] I [Ac1 ] ..I AcM−2 I [AM−1 ] SM−1   I [A0 ] D S0 + I£[Ac0 ] I [A 1 ] D S1 ¤  +.. + I [Ac0 ] ..I AcM−2 I [AM −1 ] D SM−1  ª  ©   + D I [A0 ] S0 + D I [Ac0 ] E(0) B1    ¤ª ©£  +I [Ac1 ] D I [A1 ] S1 + D I [Ac1 ] E(1) B2    = E.   ..    ½ ¾   ¤ £ D I [A ] S M−2 M−2  +I [Ac ] I [Ac ] ..I Ac  ¤ £ 1 2 c (M−2) M−3   E +D I A B M−1 M −2 ¤ £ +I [Ac1 ] I [Ac2 ] ..I AcM−2 {D I [AM−1 ] SM−1 }

and then repeatedly applying Lemma 11.1 gives ¸ · I [A0 ] D S0 £+ I [Ac0¤] I [A1 ] D S1 + ..+ , D B0 = E .. + I [Ac0 ] ..I AcM−2 I [AM−1 ] D SM−1

(12.7)

in which the D Si can be found from (9.1) and the results of Chapter-11 by partially diﬀerentiating ¡ ¢ ¡ ¢ pSwap T i , T i , T M ¡ ¢ , Si = S T i = N Ti M−1 X ¢ ¡ ¢ ¢ ¡ ¡ = Z T i, T i − Z T i, T M − κ δ i1 Z T i , T i1+1 , i1=i

and substituting for the D Z (·) and D K (·). The critical simulation equations are (12.4) and (12.7). After simulating a trajectory the Ai (that is, the optimal stopping times) will be known from the the Longstaﬀ-Schwartz regression technique, while the Si and D Si being vanilla entities are easy to find on any trajectory. Computing B0 and D B0 is then simply a matter of averaging individual contributions over all trajectories.

Chapter 13 Vega and Shift Hedging

The vegas of an option, that is, its sensitivity to changes in the implied volatilities of the instruments to which the model is calibrated, are as important a risk measure as the option’s deltas. But because in the shifted version of BGM, the shift a (T ) and volatility ξ (t, T ) functions are jointly fitted to swaption values during the volatility part of the calibration, the vega hedge must comprise both volatility and shift components. In this chapter we show, along the lines of Pelsser et al [89], how to compute vegas (including shift hedges) by perturbing the underlying BGM shift a (T ) and volatility ξ (t, T ) functions in such a way that only swaptions pSwpn (t, κ, Tj , TN ) of a particular maturity Tj (but diﬀerent strikes) are aﬀected. The corresponding changes in value of the exotic option that we wish to hedge, then yield the required hedge parameters. Denote by P (numeraire Nt expectation E) whichever of the spot P0 or terminal Pn measures we are using, and let C0 be the present value of the discounted payoﬀ stream ‘payoﬀ C (·)’ comprising our exotic option. From Chapter-7, two swaptions say pSwpn (t, κ1 , Tj , TN ) and pSwpn (t, κ2 , Tj , TN ) at diﬀerent strikes κ1 and κ2 suﬃce to fix first the shift and then the zeta at a particular exercise time Tj , so we assume two such swaptions will figure in the hedge. Note that these swaptions have exactly the same implied volatility and shift, diﬀerences in their values arise only from the strikes. Perturbing by a small amount ∆θ the shift α (Tj , TN ) of just the j th swaption pSwpn (t, κ, Tj , TN ) changes its value by ∆θ pSwpn (0, κ, Tj , TN ) where α (Tj , TN )

→

∆θ pSwpn (0, κ, Tj , TN ) = =

½

(1 + ∆θ) α (Tj , TN )

⇒

pSwpn (0, κ, (1 + ∆θ) α (Tj , TN )) − pSwpn (0, κ, α (Tj , TN ))

∂ pSwpn (0, κ, α (Tj , TN )) × α (Tj , TN ) ∆θ. ∂α

¾

Similarly, perturbing by a small amount ∆ε the instantaneous swaption volatility σ (t, Tj , TN ) of just the j th swaption changes its implied volatility β (Tj , TN ), which in turn changes its value by ∆ε pSwpn (0, κ, Tj , TN ) where ⇒ σ (t, Tj , TN ) → (1 + ∆ε) σ (t, Tj , TN ) Z Tj β 2 (Tj , TN ) Tj = |σ (t, Tj , TN )|2 dt → (1 + ∆ε)2 β 2 (Tj , TN ) Tj

and

0

113

114

Engineering BGM ¾ ½ pSwpn (0, κ, (1 + ∆ε) β (Tj , TN )) ∆ε pSwpn (0, κ, Tj , TN ) = − pSwpn (0, κ, β (Tj , TN )) ∂ pSwpn (0, κ, β (Tj , TN )) × β (Tj , TN ) ∆ε. = ∂β

The partial derivatives are easy to compute; from (4.16) and (A.2.3) n p o pSwpn (0, κ, α, β) = level ×B ω (0) + α, κ + α, β Tj ⇒ ³ ´ p p ∂ pSwpn (0, κ, α, β) = level × (κ + α) N0 h − β Tj Tj , ∂β n ³ ´o p ∂ pSwpn (0, κ, α, β) = level × N (h) − N h − β Tj , ∂α 1 2 ln ω(0)+α κ+α + 2 β Tj p where h= . β Tj Crucially, we will construct perturbations in the BGM shift a (T ) and volatility ξ (t, T ) functions, that respectively change the shift α (Tj , TN ) and volatility σ (t, Tj , TN ) of just the j th swaptions pSwpn (0, κ, Tj , TN ) and no others. That enables us to find the changes ∆θ C0 and ∆ε C0 in value of the exotic option corresponding to those perturbations in a (T ) and ξ (t, T ) respectively (on a trajectory-by-trajectory basis if pricing is by simulation), equate those changes to the corresponding swaption changes like ∆θ C0 = a1 ∆θ pSwpn (0, κ1 , Tj , TN ) + a2 ∆θ pSwpn (0, κ2 , Tj , TN ) , ∆ε C0 = a1 ∆ε pSwpn (0, κ1 , Tj , TN ) + a2 ∆ε pSwpn (0, κ2 , Tj , TN ) , and then solve these two equations for the vega and shift hedge pair (a1 , a2 ) of the exotic option into the pair of swaptions pSwpn (0, κ1 , Tj , TN ) and pSwpn (0, κ2 , Tj , TN ). For clarity of exposition, we first derive the required BGM shift and volatility perturbations for coterminal swaptions and then consider the case when calibration is to a smaller miscellaneous set of liquid instruments.

13.1

When calibrated to coterminal swaptions

Assume fixed and floating nodes coincide, and there is a full complement of coterminal quarterly swaptions (with the last ones caplets) exercising at Tj for j = 1, .., N − 1 and all maturing at TN .

Vega and Shift Hedging

13.1.1

115

The shift part

Ignoring the spread µj , for j = 1, 2, .., N − 1 α (Tj , TN ) =

N −1 X

(N )

(N )

uj, a (T ) ,

uj,

=j

that is

α (N −1)×1

=

u

a

(N −1)×(N −1)

(N −1)×1

  (N) α (T1 , TN ) u1,1    ..   = 0 . α (TN−1 , TN ) 0 

δ B (0, T +1 ) = PN −1 k=j δ k B (0, Tk+1 )

(13.1)

or

  (N) a (T1 ) u1,N −1   ..  .. .. . .  . . (N ) ) a (T N −1 0 uN −1,N −1

..

.

Being a non-singular upper triangular (N − 1) × (N − 1) matrix, u has both a right and left inverse u−1 which is also (N − 1)×(N − 1) and upper triangular. Specifically, subtracting (13.1) at j and (j + 1) and simplifying a (Tj ) = α (Tj+1 , TN ) + a

=

u−1



u−1 α 1

1 (N ) uj, j

[α (Tj , TN ) − α (Tj+1 , TN )]

where ..

1

. 1 − (N ) 0  u(N) u1,1  1,1  .. .. ..  0 . . .  1 1 = 0 . . . 1 − (N) (N)  uj, j uj, j   ..  0 . ... ...  0 ... ... 0

0 .. . 0 .. . 1 (N ) uN −1,N−1

⇒



     .     

Now proportionally perturb just the j th swaption shift by an amount ∆θ α (Tj , TN ) that is

α

→

→

(1 + ∆θ) α (Tj , TN ) ¡ ¢T α + ∆θ 0, . . . , α (Tj , TN ) , . . . , 0 .

The corresponding perturbation in the BGM shift function that changes the shift α (Tj , TN ) in just the j th swaption and no others must be n ¢T o ¡ a → u−1 α + ∆θ 0, . . . , α (Tj , TN ) , . . . , 0 , ´T ³ P −1 (N) . = a + ∆θ u−1 0, . . . , N=j uj, a (Tj ) , . . . , 0

116

Engineering BGM

Note that this perturbation aﬀects just a (Tj−1 ) and a (Tj ) with all other a (Tj ) remaining unchanged Ã ! N−1 X (N) 1 uj, a (T ) , (13.2) a (Tj−1 ) → a (Tj−1 ) + ∆θ 1 − (N ) uj−1, j−1 =j a (Tj )

→

a (t, Tj ) + ∆θ

1 (N) uj, j

N −1 X

(N )

uj, a (T ) .

=j

(N)

REMARK 13.1 Given the orders of magnitude of uj, (roughly N1−j ) 1 clearly the numbers (N ) can be quite large, but will be kept under control uj,

j

so long as swaption shifts α (Tj , TN ) are not changing rapidly from tenor Tj (N ) to tenor Tj+1 . Similarly for the Aj, appearing in the volatility part below, where the swaption volatility σ (t, Tj , TN ) must be stable from tenor to tenor.

13.1.2

The volatility part

Construction of the volatility perturbation must allow for ξ (t, T ) being vector valued and time dependent (unlike the shift a (T )); hence the following die-at-exercise convention: Condition 1 Forward and swaprate volatilities satisfy ξ (t, Tj ) = 0

and

σ (t, Tj , TN ) = 0

for

t > Tj .

(13.3)

(N)

From (4.16) with weights Aj, that depend only on the initial yieldcurve and assuming k factors, the volatilities of a set of coterminal swaptions maturing at TN can be written (the range of σ and ξ must be k-dimensional) for j = 1, 2, .., N − 1 as σ (t, Tj , TN ) =

N−1 X

(N )

Aj, ξ (t, Tj ) ,

(13.4)

=j

that is

σ (t) (N −1)×k

=

A

ξ (t)

(N −1)×(N −1)

(N −1)×k

or

   (N) .  (N) ξ (t, T1 ) σ (t, T1 , TN ) A1,1 . . A1,N−1      .. .. ..  ..  = . . .  0  . . (N ) σ (t, TN −1 , TN ) ξ (t, TN−1 ) 0 0 AN −1,N −1 

Being a non-singular upper triangular (N − 1)×(N − 1) matrix, A has both a right and left inverse A−1 which is also (N − 1)×(N − 1) and upper triangular,

Vega and Shift Hedging

117

and therefore ξ (t)

A−1 σ (t) .

=

REMARK 13.2 The equations σ (t) = Aξ (t) and ξ (t) = A−1 σ (t) must hold for all times t ∈ [0, TN ], during which time the component swaptions are consecutively exercising. The time convention Condition-13.3 ensures there is no discrepancy. For example, deleting the 1st row from A and 1st column from A−1 is equivalent to eliminating the first calibration instrument: A−1 → A−1 (1 : N , 2 : N ) ,

A → A (2 : N , 1 : N )

⇒ A (2 : N , 1 : N ) ∗ A−1 (1 : N , 2 : N ) = IN −2 ,

because of the upper triangular nature of A. Then time can be moved forward a quarter by deleting a column in A and row in A−1 : A (2 : N , 1 : N ) → A (2 : N , 2 : N ) −1

A

⇒

−1

(1 : N , 2 : N ) → A −1

A (2 : N , 2 : N ) ∗ A

and

(2 : N , 2 : N ) ,

(2 : N , 2 : N ) = IN−2 .

The point is that it does not matter whether we work with the full (N − 1) × (N − 1) matrix A or its submatrices; due to the triangular nature of A there is no inconsistency in ignoring the first instrument according to Condition-13.3 once it has matured. (N )

In the case of the usual simple approximation to the Aj, (N)

Aj,

=

 

0 for <j δ B (0, T +1 ) H (0, T )

 PN −1 k=j

(13.5)

δ k B (0, Tk+1 ) H (0, Tk )

the inverse A−1 can be computed like u−1 for the shift. Subtracting (13.4) at j and (j + 1) and simplifying ξ (t, Tj ) = σ (t, Tj+1 , TN ) + ξ

A−1

=



A−1 σ 1

1 (N ) Aj, j

[σ (t, Tj , TN ) − σ (t, Tj+1 , TN )]

where 1

..

. 1 − (N) 0  A(N) A1,1  1,1  .. .. ..  0 . . .  1 1 = 0 . . . 1 − (N) (N)  Aj, j Aj, j   ..  0 . ... ...  0 ... ... 0

0 .. . 0 .. . 1 (N) AN−1,N−1



     .     

⇒

118

Engineering BGM

Now proportionally perturb just the j th swaprate volatility by an amount ∆ε σ (t, Tj , TN ) that is

σ

→

→

σ + ∆ε

¡

(1 + ∆ε) σ (t, Tj , TN ) 0, . . . , σ (t, Tj , TN ) , . . . , 0

¢T

in a scalar fashion that importantly leaves swaprate correlation unaltered. The perturbation in the BGM volatility function corresponding to this change in just the j th swaprate volatility σ (t, Tj , TN ) (and no others) is given by n ¢T o ¡ ξ → A−1 σ + ∆ε 0, . . . , σ (t, Tj , TN ) , . . . , 0 , ´T ³ P −1 (N ) . = ξ + ∆ε A−1 0, . . . , N=j Aj, ξ (t, Tj ) , . . . , 0

In the case of the usual simple approximation (13.5), the perturbation will aﬀect just ξ (t, Tj−1 ) and ξ (t, Tj ) with all other ξ (t, Tj ) remaining unchanged Ã ! N −1 X (N ) 1 Aj, ξ (t, T ) , ξ (t, Tj−1 ) → ξ (t, Tj−1 ) + ∆ε 1 − (N) Aj−1, j−1 =j (13.6) ξ (t, Tj )

→

ξ (t, Tj ) + ∆ε

1 (N) Aj, j

N −1 X

(N )

Aj, ξ (t, T ) .

=j

REMARK 13.3 Problems may also arise in practice if N and hence the (N) reciprocals of Aj, j are large and concurrently the relative perturbation ∆ε is substantial to get distinct changes in value under simulation. REMARK 13.4 Together (13.2) and (13.6) give some idea of the kinds of perturbation of the BGM shift a (t) and volatility ξ (t, T ) functions needed to produce changes in one particular swaption, namely joint changes in just two adjacent (by maturity) values of ξ (t, T ). Bump and grind methods for computing vegas involve a recalibration that must try to mimic these perturbations in some way; that is not easy, and leads to inaccuracies when the functions are overly distorted.

13.2

When calibrated to liquid swaptions

Assume calibration is to a liquid set of m swaptions and caplets exercising at diﬀerent times Tj and with mixed maturities Tn where n ≤ N . The matrices u and A in (13.1) and (13.4) actually specify the calibration instruments, so if we

Vega and Shift Hedging (N)

119

(N)

make the convention that the uj, and Aj, for our m calibration instruments are entered in rows by exercise, then we can talk without ambiguity about the calibration hu, Ai. For N nodes, u and A will therefore be of order m×(N − 1) with m ≤ N − 1, and also block triangular Xj,1 = Xj,2 = ... = Xj, −1 = 0 and ⇒ Xj1, 1 = 0 for j1 > j and

Xj, 6= 0 1< .

Deleting a row removes a calibration instrument, while deleting columns of zeroes at the front of the matrix corresponds to moving forward in time. Practically speaking, with quarterly nodes N will be over 100 and m much less. Moreover, we can expect rank u = rank A = m (otherwise the linear dependence indicates an unsatisfactory calibration set) so right inverses u−1 and A−1 will exist (see Section-A.4.2) such that a = u−1 α solves and

−1

ξ=A

σ

solves

α = u a with σ=Aξ

kak2

with each

minimal, ° ° ° (k) ° minimal. °ξ ° 2

Similarly to Section-13.1, perturbations in the shift and volatility of the j th instrument ¡ ¢T α → α + ∆θ 0, . . . , α (Tj , TN ) , . . . , 0 , ¡ ¢T σ → σ + ∆ε 0, . . . , σ (t, Tj , TN ) , . . . , 0 , are induced by the corresponding perturbations n ¢T o ¡ a → u−1 α + ∆θ 0, . . . , α (Tj , TN ) , . . . , 0 , n o ¢ ¡ T , ξ → A−1 σ + ∆ε 0, . . . , σ (t, Tj , TN ) , . . . , 0

in the BGM shift and volatility functions.

REMARK 13.5 As in Section-13.1, the volatility perturbation is consistent with the time convention (13.3), because if we strike calibration instruments as they exercise along with columns corresponding to the past, the resulting submatrices will still be mutually inverse though no longer yielding an ξ with minimal 2-norm. Alternatively, to ensure an ξ with minimal 2-norm, we could recompute ξ for each internodal period working with consecutive matrices of decreasing size as instruments exercise and time moves forward. REMARK 13.6 A diﬀerent approach to finding the required perturbations when calibration is to liquid instruments, is to use linear programming with extra constraints designed to produce a stable and accurate result.

Chapter 14 Cross-Economy BGM

The cross-economy version of shifted BGM links foreign and domestic shifted BGM models via the forward FX exchange rate. In the Gaussian HJM framework the link is possible with deterministic volatilities for domestic and foreign instantaneous forwards, and also the FX rate; that is, deterministic volatilities are totally compatible with lognormal models for the prices of domestic and foreign bonds and the spot and forward FX rates. But, as shown by Schlogl [112], in cross-economy BGM some among the domestic and foreign interest rate volatilities, and FX forward volatilities must be stochastic. Nevertheless, as with swaption volatilities in domestic BGM, with appropriate choices it is possible to obtain approximations for stochastic volatilities in cross-economy BGM that are good enough to return by simulation fairly accurate values for the implied volatilities to which the model is calibrated. To set the scene, in the following Section-14.1 we first work through relevant ideas in a cross-economy HJM framework, before considering the BGM equivalent. Our notation for the foreign economy will be to superfix f to domestic variables to denote the equivalent foreign variable; for example, if PT and B (t, T ) are respectively the T -forward measure and zero coupon in the domestic economy, then PfT and B f (t, T ) are the equivalent in the foreign economy.

14.1

Cross-economy HJM

The extra ingredients needed in HJM to cope with a foreign economy are: • a spot FX rate S (t), which is the price of one foreign zlotty in domestic dollars Z1 = $S (t), with volatility function ν (t), drift αS (t) and SDE £ ¤ dS (t) = S (t) αS (t) dt + ν ∗ (t) dW0 (t) ,

• an instantaneous foreign forward rate f f (t, T ) at time t for maturity T with volatility function σ f (t, T ), drift αf (t, T ) and SDE df f (t, T ) = αf (t, T ) dt + σ f ∗ (t, T ) dW0 (t)

(14.1)

121

122

Engineering BGM • a foreign spot rate rf (t) and foreign bank account β f (t) defined by β f (t) = exp

rf (t) = f f (t, t) ,

µZ

0

t

¶ rf (s) ds ,

• foreign zero coupon bonds B f (t, T ) maturing at T defined by Ã Z B (t, T ) = exp −

!

T

f

f

f (t, u) du .

t

(14.2)

The FX drift αS (t) is fixed by the fact that Z1 invested in the foreign bank account and converted to domestic dollars constitutes a domestic asset, and so its discounted value must be a P0 -martingale; that is, the drift in ³ f ´ S(t) d β (t) £ ¤ β(t) = αS (t) dt + ν ∗ (t) dW0 (t) + rf (t) dt − r (t) dt, β f (t) S(t) β(t)

¤ £ = αS (t) + rf (t) − r (t) dt + ν ∗ (t) dW0 (t)

must be zero, giving αS (t) = r (t) − rf (t). Hence the SDE for S (t) is £ ¤ dS (t) = r (t) − rf (t) dt + ν ∗ (t) dW0 (t) . S (t)

(14.3)

Similarly the forward drifts αf (t, T ) are fixed by the fact that all foreign zero coupon bonds converted to dollars are domestic assets, and so their discounted values must be P0 -martingales. Because  ³R ´  Z T  −rf (t) dt + T αf (t, u) du dt  t ³R ´ , d f f (t, u) du =  + T σ f ∗ (t, u) du dW0 (t)  t t

from (14.1), (14.2), (14.3) and using Ito d

³

(

B f (t,T ) S(t) β(t)

B f (t,T ) S(t) β(t) f

+ r (t) dt −

´

=

ÃZ

t

T

©£ ¤ ª r (t) − rf (t) dt + ν ∗ (t) dW0 (t) − r (t) dt f

!

α (t, u) du dt −

ÃZ

t

T

σ

f∗

!

(t, u) du dW0 (t)

)

¯Z ¯2 ÃZ ! ¯ T 1 ¯¯ T f ¯ ∗ + ¯ σ (t, u) du¯ dt − ν (t) σ (t, u) du dt, ¯ 2¯ t t

Cross-Economy BGM

123

and for the drift in this SDE to be zero ¯Z ¯2 Z T Z T ¯ 1 ¯¯ T f ¯ f α (t, u) du = ¯ σ (t, u) du¯ − ν ∗ (t) σ f (t, u) du ¯ 2¯ t t t (Z ) T ⇒ αf (t, T ) = σ f ∗ (t, T ) σf (t, u) du − ν (t) . t

Hence SDEs for the foreign forwards and zero coupons are Z T df f (t, T ) = σf (t, T ) σ f (t, u) du dt + σ f ∗ (t, T ) (dW0 (t) − ν (t) dt) , t ÃZ ! T dB f (t, T ) = rf (t) dt − σ f ∗ (t, u) du (dW0 (t) − ν (t) dt) , (14.4) B f (t, T ) t and letting W0f (t) be Brownian motion under Pf0 , these two expressions show the foreign arbitrage free measure Pf0 is related to P0 by µZ t ¶ dW0f (t) = dW0 (t) − ν (t) dt ⇒ Pf0 = E ν ∗ (s) dW0 (s) P0 . (14.5) 0

14.2

Forward FX contracts

The value in the domestic economy of a forward contract GT (t, T1 ) maturing at T , on a foreign zero B f (t, T1 ) maturing at T1 = T + δ, must satisfy ©£ ¤¯ ª ET GT (t, T1 ) − B f (T, T1 ) S (T ) ¯ Ft = 0,

which implies GT (t, T1 ) is a PT -martingale with time t value ¯ ¾ ½ f ¯ B (T, T1 ) B f (t, T1 ) GT (t, T1 ) = ET S (T )¯¯ Ft = S (t) . B (T, T ) B (t, T )

When T1 = T the payoﬀ from B f (t, T ) at T is one zlotty Z1 = $ S (T ) and GT (t, T ) becomes the FX forward contract ST (t) =

B f (t, T ) S (t) B (t, T )

(14.6)

This is the well known interest rate parity relationship; for example, if AUS short interest rates rise (B f (t, T ) gets smaller) and US rates and forward FX are unchanged (B (t, T ) and ST (t) remain fixed), then S (t) must increase to compensate (the AUS strengthens against the USD).

124

Engineering BGM

Because ST (t) is a strictly positive PT -martingale it will have an SDE of form dST (t) = ν ∗T (t) dWT (t) , ST (t) where ν T (t) is the instantaneous forward FX volatility. REMARK 14.1 It is important to appreciate that so far the relationships derived in this section are completely model free.

14.2.1

In the HJM framework

The interest rate parity relationship (14.6) holds for all t, so on diﬀerentiating it using the HJM version of the component SDEs (1.4), (14.3), (14.4) and (14.6) ÃZ ! T dB (t, T ) ∗ = r (t) dt − σ (t, u) du dW0 (t) , B (t, T ) t £ ¤ dST (t) dS (t) = r (t) − rf (t) dt + ν ∗ (t) dW0 (t) , = ν ∗T (t) dWT (t) , S (t) ST (t) ÃZ ! T dB f (t, T ) f f∗ = r (t) dt − σ (t, u) du (dW0 (t) − ν (t) dt) , B f (t, T ) t

the stochastic components on both sides must tally, giving the HJM volatility parity relationship ν T (t) = ν (t) +

Z

T

σ (t, u) du −

t

Z

T

σf (t, u) du.

(14.7)

t

Note that at maturity T , the spot and forward FX volatilities converge, that is ν (T ) = ν T (T ) . Evaluating the spot volatility parity (14.7) relationship at two maturities T and T1 (> T ) and subtracting gives the forward volatility parity relationship ν T1 (t) = ν T (t) +

Z

T1

T

σ (t, u) du −

Z

T1

σ f (t, u) du.

T

REMARK 14.2 In the cross-economy version of Ho & Lee, which can be surprisingly useful for work on portfolios of underlying instruments like bonds (that is, not primarily volatility dependent instruments like options) both domestic and foreign interest rate volatilities are constant at say σ and σ f respectively. Then, various possibilities for the parity relationship include:

Cross-Economy BGM

125

the spot volatility ν (t) is constant and the forward volatilities ν T (t) are time dependent ¢ ¡ ν (t) = ν ⇒ ν T (t) = ν + σ − σ f (T − t) ;

the forward FX volatilities ν T (t) are dependent only on maturity T but the spot volatility is time t dependent ¢ ¡ ¢ ¡ ν T (t) = ν + σ − σ f T ⇒ ν (t) = ν + σ − σ f t.

14.2.2

In the BGM framework

We now develop the BGM equivalents of the cross-economy HJM measure change (14.5), and the HJM volatility parity relationship (14.7). Dividing the (model free) spot parity relationship (14.6) at the two maturities T and T1 (> T ) gives a forward version of the parity relationship FTf (t, T1 ) =

ST1 (t) FT (t, T1 ) . ST (t)

(14.8)

Note that in this equation FTf (t, T1 ) is a T -maturing forward contract on the zero B f (t, T ) all within the foreign economy, and is quite diﬀerent to GT (t, T1 ). This equation holds for all t, so matching SDEs on each side must yield the required measure changes and volatility parity relationships between the two economies. Under the T -forward measures PT and PfT , SDEs for the components are dFTf (t, T1 ) FTf (t, T1 )

= −bf ∗ (t, T, T1 ) dWTf (t) ,

(14.9)

dFT (t, T1 ) dST (t) = −b∗ (t, T, T1 ) dWT (t) , = ν ∗T (t) dWT (t) , FT (t, T1 ) ST (t) dST1 (t) = ν ∗T1 (t) [b (t, T, T1 ) dt + dWT (t)] , ST1 (t) from which an SDE for the right-hand side of (14.8) is ³ ´ S (t) d STT1(t) FT (t, T1 ) ´ = − [b (t, T, T1 ) − ν T1 (t) + ν T (t)]∗ [dWT (t) − ν T (t) dt] . ³ ST1 (t) ST (t) FT (t, T1 )

Matching this SDE to that of the foreign forward contract FTf (t, T1 ) gives ∗

[b (t, T, T1 ) − ν T1 (t) + ν T (t)] [dWT (t) − ν T (t) dt] = bf ∗ (t, T, T1 ) dWTf (t) .

126

Engineering BGM

That means that the measure change between the domestic T -forward measure PT and foreign T -forward measure PfT must be determined by dWTf (t) = dWT (t) − ν T (t) dt ⇒ PfT = E

µZ

t 0

¶ ν ∗T (s) dWT (s) PT , (14.10)

and the BGM volatility parity relationship connecting FX forward volatilities and domestic and foreign bond volatility diﬀerences is ν T1 (t) − ν T (t) = b (t, T, T1 ) − bf (t, T, T1 ) ,

(14.11)

f

=

δH (t, T ) δH (t, T ) ξ (t, Tj ) − ξ f (t, Tj ) . [1 + δK (t, T )] [1 + δK f (t, T )]

The change of measure relationship (14.10) with maturity T set at the end of the current interval T = T@ , also relates the spot measures in the domestic and foreign economies dW0f (t) = dW0 (t) − ν @ (t) dt

(14.12)

where ν @ (t) is the instantaneous volatility of the immediately maturing FX forward contract ST@ (t) = S@ (t) which has SDE dS@ (t) = ν ∗@ (t) dW0 (t) . S@ (t)

(14.13)

REMARK 14.3 A version of shifted cross-economy BGM is possible with the FX also having a term structure of shift f (T ). Suppose T1 = T + δ, relevant SDEs are then the SDEs for FTf (t, T1 ) and FT (t, T1 ) as in (14.9), with the two new SDEs for ST (t) and ST1 (t) ST (t) + f (T ) ∗ dST (t) = ν T (t) dWT (t) , ST (t) ST (t) ST (t) + f (T1 ) ∗ dST1 (t) = 1 ν T1 (t) [b (t, T, T1 ) dt + dWT (t)] . ST1 (t) ST1 (t) Similarly to (14.11), the volatility parity relationship becomes ST (t) + f (T ) ST1 (t) + f (T1 ) ν T1 (t) − ν T (t) ST1 (t) ST (t) ¤ £ δ K f (t, T ) + af (T ) f δ [K (t, T ) + a (T )] ξ (t, T ) − ξ (t, T ) , = [1 + δK (t, T )] [1 + δK f (t, T )] creating a problem of how to handle the stochastic parts of the FX volatility components!

Cross-Economy BGM

14.3

127

Cross-economy models

Assume for simplicity that fixed and floating nodes Tj (j = 0, 1, 2, ...) coincide in the two economies. The volatility parity equation (14.11) for successive nodes Tj and Tj+1 is ν Tj+1 (t) = ν Tj (t) + b (t, Tj , Tj+1 ) − bf (t, Tj , Tj+1 ) . Then, substituting for the bond volatilities from (3.3) b (t, Tj , Tj+1 ) = bf (t, Tj , Tj+1 ) =

δ j H (t, Tj ) ξ (t, Tj ) = hj (t) ξ (t, Tj ) , [1 + δ j K (t, Tj )] δ j H f (t, Tj ) ξ f (t, Tj ) = hfj (t) ξ f (t, Tj ) , [1 + δ j K f (t, Tj )]

we obtain a vector system of forward volatility parity equations for j = 1, 2, ... ν Tj+1 (t) = ν Tj (t) + hj (t) ξ (t, Tj ) − hfj (t) ξ f (t, Tj ) .

(14.14)

Within this system of equations the hj (·) and hfj (·) are stochastic, and so unlike HJM, it is not possible to concurrently have domestic rates modelled by shifted BGM (that is ξ (t, T ) deterministic), foreign rates also modelled by shifted BGM (that is ξ f (t, T ) deterministic), and FX forward volatilities lognormal (that is ν T (t) deterministic). Practically useful possibilities reduce to one of: 1. Domestic and foreign rates are shifted BGM and just one of the FX forward volatilities, say ν Tk (t), is deterministic. In that case, the parity equations (14.14) say all other forward volatilities ν Tj (t) (j 6= k) must be stochastic. Note that this includes the possibility, see Section-14.4 below, of the chosen deterministic forward volatility being at one maturity on some time intervals and other maturities at other times. 2. Domestic rates are shifted BGM and all FX forward volatilities are deterministic, which means foreign rates cannot be shifted BGM and foreign shifted forward volatilities ξ f (t, Tj ) must be stochastic. 3. Foreign rates are shifted BGM and all FX forward volatilities are deterministic, so domestic rates are not shifted BGM and domestic shifted forward volatilities ξ (t, Tj ) are stochastic. 4. Recall that hfj (t) and hj (t) (j = 1, 2, ..) are low variance martingales, and use the deterministic approximation f ν Tj+1 (t) ∼ = ν Tj (t) + hj (0) ξ (t, Tj ) − hj (0) ξ f (t, Tj )

(14.15)

in a setup which assumes domestic and foreign rates are shifted BGM and all FX forward volatilities are deterministic.

128

Engineering BGM

Often the choice will be suggested by the instrument to be valued, with the change of measure formula (14.10) allowing stochastic variables in both domestic and foreign economies to be expressed in terms of measures convenient for computation. In the next example, for instance, the best choice is clearly foreign rates and FX deterministic. Example 14.1 The T -maturity roll Q (t, T ) of a quanto paying foreign Libor in domestic dollars at T1 will have time-t value ¯ ª © Q (t, T ) = B (t, T1 ) ET1 δK f (T, T )¯ Ft , ¯ ª © = δB (t, T1 ) ET1 H f (T, T ) − af (T )¯ Ft , ¯ ) ÃZ ! ( ¯ T ¯ f f∗ f f = δB (t, T1 ) ET1 H (t, T ) E ξ (s, T ) dWT1 (s) − a (T )¯ Ft , ¯ t ³R ´¯ ) ( T ¯ H f (t, T ) E t ξ f ∗ (s, T ) [dWT1 (s) − ν T1 (s) ds] ¯ = δB (t, T1 ) ET1 ¯ Ft , ¯ −af (T ) (£ ³ ´ ) ¤ RT K f (t, T ) + a (T ) exp − t ξ f ∗ (s, T ) ν T1 (s) ds = δB (t, T1 ) , −af (T ) if ξ f (s, T ) and ν T1 (s) are taken to be deterministic.

14.4

Model with the spot volatility deterministic

In order to both approximately fit the implied volatilities of forward FX options and be able to simulate accurately, we will assume that: 1. Both domestic and foreign forward rates are shifted lognormal, that is both ξ (t, Tj ) and ξ f (t, Tj ) are deterministic. 2. On consecutive intervals the instantaneous forward volatility ν T@ (t) = ν @ (t) of the next maturing forward FX contract ST@ (t) = S@ (t) is deterministic. From (14.14), these two assumptions mean FX forward volatilities away from the current interval ought to be stochastic, but we will use (14.15) to approximate them deterministically. The implied volatility α (Tj ) of the Tj exercising forward FX option is α2 (Tj ) Tj =

Z

0

Tj

j X ¯ ¯ ¯ν Tj (t)¯2 dt = k=1

Z

Tk Tk−1

¯ ¯ ¯ν Tj (t)¯2 dt,

(14.16)

Cross-Economy BGM

129

in which the forward volatilities ν Tj (t) on t ∈ (Tk−1 , Tk ] are computed from the following table k 1 2 3 .. .

t ∈ (Tk−1 , Tk ] (T0 , T1 ] (T1 , T2 ] (T2 , T3 ] .. .

ν Tj (t) = ν T1 (t) + b (t, T1 , Tj ) − bf (t, T1 , Tj ) ν T2 (t) + b (t, T2 , Tj ) − bf (t, T2 , Tj ) ν T3 (t) + b (t, T3 , Tj ) − bf (t, T3 , Tj ) .. .

j−1 j

(Tj−2 , Tj−1 ] (Tj−1 , Tj ]

ν Tj−1 (t) + b (t, Tj−1 , Tj ) − bf (t, Tj−1 , Tj ) ν Tj (t)

with the domestic b (t, Tj1 , Tj2 ) and foreign bf (t, Tj1 , Tj2 ) bond volatility differences deterministically approximated by repeatedly applying (3.3) and setting stochastic parts to their initial value b (t, Tk , Tj ) ∼ =

j−1 X =k

bf (t, Tk , Tj ) ∼ =

j−1 X =k

j−1

X δ H (0, T ) ξ (t, T ) = h (0) ξ (t, T ) , (14.17) [1 + δ K (0, T )] =k

j−1

X f δ H f (0, T ) ξ f (t, T ) = h (0) ξ f (t, T ) . f [1 + δ K (0, T )] =k

Putting it all together 2

α (Tj ) Tj =

¯ P ¯ ν Tk (t) + j−1 h (0) ξ (t, T =k ¯ P ¯ − j−1 hf (0) ξ f (t, T ) Tk−1 =k

j Z X

k=1

Tk

¯2 ) ¯¯ ¯ dt

(14.18)

REMARK 14.4 In the Pedersen cross-economy calibration scheme below equation (14.18) will connect the varying trial values for ξ (t, Tj ), ξ f (t, Tj ) and ν @ (t) with the target implied volatilities α (Tj ). Simulation can now be done under the spot measure using virtually any scheme. For example, with the notation of Section-9.1, a Glasserman type simulation under P0 would in the domestic economy, have SDEs for the V (t, Tj ) for j = @ (t) , .., n like   ¾ ½ j X dV (t, Tj )  ∗ V (t, Tk ) = ξ (t, Tj ) − ξ ∗ (t, Tk ) dW0 (t) , φ V (t, Tj ) Z (t, Tk ) k=@(t)

B (t, Tk ) = V (t, Tk ) + Πkk V (t, Tk+1 ) ... + Πn−1 Z (t, Tk ) = V (t, Tn ) , k M (t)

130

Engineering BGM

and in the foreign economy, would have SDEs for the V f (t, Tj ) for j = @ (t) , .., n like f



j X

½



¾

f

V (t, Tk ) dV (t, Tj )  f ∗ = ξ (t, Tj ) − ξ f ∗ (t, Tk ) dW0f (t) , φ V f (t, Tj ) Z f (t, Tk ) k=@(t) ¾ ½ f f f B (t, Tk ) V (t, Tk ) + Πf,k k V (t, Tk+1 ) + , = Z f (t, Tk ) = M f (t) ... + Πf,n−1 V f (t, Tn ) k with the foreign BM W0f (t) connected to the driving BM W0 (t) under P0 by (14.12) dW0f (t) = dW0 (t) − v@ (t) dt. That permits simulation of domestic and foreign rates because the deterministic volatility v@ (t) will be specified for all t through the calibration. To simulate the FX, proceed as follows. On the current interval t ∈ (T@−1 , T@ ] FX forwards STj (t) maturing at Tj > T@ are connected to ST@ (t) through the parity relationship (14.6) STj (t) =

FTf@ (t, Tj ) ST (t) FT@ (t, Tj ) @

t ∈ (T@−1 , T@ ] ,

and the domestic and foreign interest rate forward contracts FT@ (t, Tj ) =

Z (t, Tj ) , Z (t, T@ )

FTf@ (t, Tj ) =

Z f (t, Tj ) . Z f (t, T@ )

The SDE for the immediately maturing FX forward contract ST@ (t) = S@ (t) is dS@ (t) = ν ∗@ (t) dW0 (t) , t ∈ (T@−1 , T@ ] , S@ (t) with initial condition at the start of the interval t = T@−1 , given by the finishing value of the previously maturing contract S@ (t) according to ST@ (T@−1 ) =

FTf@−1 (T@−1 , T@ ) FT@−1 (T@−1 , T@ )

ST@−1 (T@−1 ) .

And, of course, each contract STj (t) dies at its maturity Tj . The overall modus-operandi is therefore to use the same Brownian motion trajectory W0 to first simulate domestic rates, then the foreign rates, and finally the FX forward contracts, the latter according to the scheme

Cross-Economy BGM S@ (t) ST1 (t)

dST1 (t) ST1 (t)

T@ = T1 = ν ∗T1 (t) dW0 (t)

ST2 (t) ST2 (t) = .. . STj (t)

STj (t) =

... .. . .. . live live

14.5

FTf (t,T2 ) 1 FT1 (t,T2 ) ST1

T@ = T2 dead

(t)

.. .

FTf (t,Tj ) 1 FT1 (t,Tj ) ST1

131

(t)

dST2 (t) ST2 (t)

T@ = T3 dead

= ν ∗T2 (t) dW0 (t) .. .

STj (t) =

FTf (t,Tj ) 2 FT2 (t,Tj ) ST2

(t)

dead live live

T@ = Tj−1 .. .

T@ = Tj .. .

S@ (t) .. .

dead

dead

STj−2 (t)

dead

STj−1 (t)

dSTj−1 (t) STj−1 (t)

= ν ∗Tj−1 (t) dW0 (t)

STj (t) =

FTj−1 (t,Tj ) STj−1

FTf (t,Tj ) j−1

(t)

dSTj (t) STj (t)

= ν ∗Tj (t) dW0 (t)

STj (t)

Cross-economy correlation

We now extend to two economies, and also include FX, the method of getting forward correlation from swaprate correlation that was developed in Section-6.3. Assume uniform coverage δ with both foreign and domestic notes coinciding on the floating nodes Tj = jδ (j = 0, 1, ..., N − 1), and as usual add an f superfix to domestic variables to denote the equivalent foreign variable. Using the relationship (14.12) between the domestic P0 and foreign Pf0 spot measures, under the domestic spot measure P0 relevant SDEs for the domestic and foreign swaprates will have form dω (t, j) = (drift) dt + σ ∗ (j) dW0 (t) , ω (t, j) + α dω f (t, j) = (drift)f dt + σ f ∗ (j) dW0 (t) , ω f (t, j) + αf

(14.19)

where α and αf are constant average estimates of the domestic and foreign swaprate shifts, and the volatilities σ (j) and σ f (j) are σ (j) =

j X

k=0

cj,k ξ (xk ) ,

σ f (j) =

j X

k=0

cfj,k ξ f (xk ) ,

j = 0, 1, ..., N − 1

132

Engineering BGM

in which, on the domestic side, ξ (t, T ) = ξ (T − t) = ξ (x) , and  B (0, x ) K (0, xk ) k+1   for  Pj k=0 B (0, xk+1 ) K (0, xk ) cj,k =    0 for k = j + 1, .., N − 1,

k = 0, .., j,

and, on the foreign side,

and ξ f (t, T ) = ξ f (T − t) = ξ f (x) ,  f f B (0, x ) K (0, xk ) k+1    Pj f f k=0 B (0, xk+1 ) K (0, xk ) cfj,k =    0 for k = j + 1, .., N − 1.

for

k = 0, .., j,

To condense equations to matrix form below, it will be convenient to introduce the (2N + 1) × (2N + 1) upper triangular and non-singular matrix C constructed from the cj,k and cfj,k according to 

  C= 



  C =   f

c0,0 c1,0 .. .

0 c1,1 .. .

··· ··· .. .

0 0 .. .

cN−1,0 cN−1,1 · · · cN −1,N −1 cf0,0 cf1,0 .. .

0 cf1,1 .. .

··· ··· .. .

0 0 .. .

cfN−1,0 cfN −1,1 · · · cfN −1,N −1   C 0 0 C =  0 Cf 0  . 0 0 1

    



  ,  

−1

as given From (6.8), the inverses of C and C f are respectively C −1 and C f by (6.10). Hence the inverse of C is the (2N + 1) × (2N + 1) matrix  −1  C 0 0 −1 C−1 =  0 C f 0. 0 0 1

Our assumption that ν T@ (t) = ν 0 (t) is deterministic means the relevant data for correlation analysis must be the timeseries of values of the next maturing forward FX contract ST@ (t) = S0 (t). That can be easily constructed from the spot and domestic and foreign three month discount rate timeseries

Cross-Economy BGM

133

using (14.6), or one can simply confuse the spot data with the forwards because successive diﬀerences as used in our quadratic variation estimator below will be about the same. Setting T@ = t+δ and assuming the forward volatility is a constant so as to be homogeneous, that is ν T@ (t) = ν 0 (t) = ν 0 , a formal SDE for the relative FX forward contract Sδ (t) = ST (t)|T =t+δ can then be obtained in the usual way dSδ (t) = (drift) dt + ν ∗0 dW0 (t) . Sδ (t)

(14.20)

Using (14.19) and (14.20) quadratic variation estimators for the covariances of domestic rates, foreign rates, domestic against foreign rates, FX against domestic rates, and FX against foreign rates are then respectively: ¶ µ Z τ d hω (·, j) , ω (·, k)i (t) 1 , Q = (Qj,k ) = τ 0 [ω (t, j) + α] [ω (t, k) + α] Ã Z ! ® ³ ´ d ω f (·, j) , ω f (·, k) (t) 1 τ f f Q = Qj,k = , τ 0 [ω f (t, j) + αf ] [ω f (t, k) + αf ] Ã Z ! ® 1 τ d ω (·, j) , ω f (·, k) (t) X = (Xj,k ) = , τ 0 [ω (t, j) + α] [ωf (t, k) + αf ] ¶ µ Z τ d hω (·, j) , Sδ (·)i (t) 1 , Y = (Yj ) = τ [ω (t, j) + α] Sδ (t) Ã Z0 f ® ! τ ³ ´ d ω (·, j) , S (·) (t) 1 δ Y f = Yjf = , τ 0 [ω f (t, k) + αf ] Sδ (t) Z 1 τ d hSδ (·)i (t) . Z= τ 0 Sδ2 (t) These can be assembled into the (2N ance matrix  Q X Q =  X Qf Y Yf

+ 1) × (2N + 1) cross-economy covari Y Y f  = Ω ΩT Z

which by principal component analysis can be decomposed in terms of its rank r square-root Ω (a (2N + 1) × r matrix). Generally one seems to need r∼ = 7 to have enough flexibility to jointly cover action in domestic and foreign rates and also the FX. The entries in the first N rows of Ω are the domestic swaprate volatilities σ (·), the second N entries the foreign swaprate volatilities

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Engineering BGM

σ f (·), and the final entry the constant FX volatility ν 0 , and are connected to the domestic and foreign forward volatilities via 

σ (0) .. .





ξ (x0 ) .. .



              ξ (xN −1 )   σ (N − 1)  C 0 0     f f     f Ω=  σ (0)  = 0 C 0  ξ (x0 )  = CΞ.     . . 0 0 1 .. ..           ξ f (x  σ f (N − 1)  N −1 ) ν0 ν0 In turn, that yields: 1. The corresponding domestic rate ξ (x) and foreign rate ξ f (x) volatility functions at xj = δj, and the constant FX volatility ν 0 via the (2N + 1) × r matrix 

ξ (x0 ) .. .



       −1  ξ (xN−1 )  C 0   f  ξ (x0 )  = Ξ = C−1 Ω =  0 C f −1     .. 0 0   .   f  ξ (x ) N−1

ν0



σ (0) .. .



      0  − 1)    σ (N f  0  σ (0)  ,   . 1  ..     σ f (N − 1)  ν0

from which the vector volatility functions ξ (x) and ξ f (x) can be interpolated for all x. 2. The corresponding cross-economy (2N + 1) × (2N + 1) implied forward covariance matrix ¡ ¢T ¢T ¡ q = ΞΞT = C−1 ΩΩT C−1 = C−1 Q C−1 .

3. The cross-economy (2N + 1)×(2N + 1) implied forward correlation matrix R as 1 1 R = (diag q)− 2 q (diag q)− 2 . (14.21) 4. The domestic c (·) and foreign cf (·) correlation functions c (T − t) = c (x) =

ξ (x) , kξ (x)k

ξ f (x) °. cf (T − t) = cf (x) = ° ° ° f °ξ (x)°

Cross-Economy BGM

14.6

135

Pedersen type cross-economy calibration

This section is based on a lecture note by Dun [41] who extended the Pedersen calibration of Section-7.6 to the cross-economy model. At least three factors (one for domestic rates, one for foreign rates and one for FX) and more usually seven to nine factors are needed in a crosseconomy model to properly capture correlations between the forwards in both economies and the FX rates, all of which must be calibrated to interest rate and FX implied volatilities subject to the volatility parity relationship. This is a complex structure with some data constraints: 1. Reliable implied volatilities for FX forward options out to maturities of 5 to 10 years usually exist, and together with domestic and foreign interest rate implied volatilities constitute a full set of data to which to calibrate. 2. Beyond 5 to 10 year maturities, where only interest rate volatilities are dependable, parity is used to define the FX forward volatilities. But to do that, and produce a successful and believable calibration, we need some notion of how FX forward implied volatilities might behave at more distant maturities (say beyond 10 years). Using (14.16), the behavior with respect to T of the FX implied volatility α (T ) can be gauged with an illustrative example in which we suppose that domestic and foreign initial rates are flat, the shift is flat, all coverages are equal, and domestic, foreign and FX volatilities are flat with each determined by one of three independent factors. That is, for all t and Tj suppose δ j = δ fj = δ, K (0, Tj ) = K K f (0, T ) = K f

a (Tj ) = a, ⇒ ⇒

af (Tj ) = af ,

ν @ (t) = ν (1, 0, 0) ,

δ (K + a) =h ξ (t, Tj ) = ξ (0, 1, 0) , [1 + δK] ¡ ¢ δ K f + af f = hf hj (0) = ξ f (t, T ) = ξ f (0, 0, 1) . [1 + δK f ] hj (0) =

In that case for @ (t) = T1 Section-14.4 yields

⇒

ν Tj (t) ∼ = ν (1, 0, 0) + (j − 1) hξ (0, 1, 0) − (j − 1) hf ξ f (0, 0, 1) , · ³ ´2 ¸ ¯ ¯ ¯ν Tj (t)¯2 = ν 2 + (j − 1)2 (hξ)2 + hf ξ f , ⇒ 1 α (Tj ) = jδ 2

Z

0

Tj

· ³ ´2 ¸ ¯ ¯ ¯ν Tj (t)¯2 dt = ν 2 + (j − 1)2 (hξ)2 + hf ξ f .

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Engineering BGM

The graph of the implied volatility α (Tj ) against (j − 1) is a hyperbola, with r ³ ´ lim α (Tj ) = (j − 1)

j→∞

2

(hξ) + hf ξ f

2

.

(14.22)

Hence forward FX implied volatilities will tend to be concave up and asymptotically linear with maturity, suggesting that for more distant maturities where no market data exists, it is reasonable to require that for Tj > 10 say, A α (Tj ) ∼ = (j − 1) where A is some constant that will be determined during calibration. That is, we will let the interest rate volatility data and a linear constraint determine the long term FX implied volatilities. We now generalize the Pedersen method of Section-7.6 to work in a crosseconomy framework. The ingredients are: 1. Pseudo-homogeneous volatility functions for domestic and foreign rates like ¯ ¯ and ξ f (t, x) = ξ f (t, T )¯ ξ (t, x) = ξ (t, T )|T =t+x T =t+x

and a deterministic instantaneous forward volatility ν T@ (t) = ν @ (t) for the next maturing forward FX contract.

2. The historical correlation matrix R given by (14.21) for correlations between domestic and foreign rates and FX with the irrelevant first row and column stripped so its size is (2N − 1) × (2N − 1). 3. Term structures of shift a (T ) and af (T ) chosen to fit skews in the domestic and foreign economies as outlined in Section-7.1. 4. An (N − 1)×(N − 1) matrix from which the target domestic caplet and swaption volatilities (targetDi ) are selected, a similar sized (N − 1) × (N − 1) matrix for the target foreign caplet and swaption volatilities (targetFi ), and an (N − 1) vector from which are selected target FX forward implied volatilities (targetFXi ). We then suppose these three matrices are assembled side-by-side (domestic, foreign, FX) into a (N − 1)× (2N − 1) matrix from which the combined targets are selected. The optimizer in this algorithm will vary the (N − 1) × (2N − 1) matrix X comprising the (N − 1) (2N − 1) parameters, X = (Xk, )

k = 1, .., N − 1;

= 1, .., 2N − 1

and also a further parameter A for the linear constraint on distant FX volatilities. Each parameter Xk, in the first (respectively second) (N − 1)×(N − 1)

Cross-Economy BGM

137

block of the magnitude |ξ (t, x)| of the domestic (respec¯ X is approximately ¯ ¯ ¯ tively ¯ξ f (t, x)¯ of the foreign) vector volatility function on certain intervals, while the last column of X is approximately the magnitude |ν @ (t)| of the FX vector volatility function. Specifically we suppose ξ (t, x), ξ f (t, x) and ν @ (t) are piecewise constant with |ξ (t, x)| ∼ = Xk, for

¯ ¯ ¯ ¯ = Xk, ¯ξ (t, x)¯ ∼

t ∈ (Tk−1 , Tk ] ,

x ∈ (x

|ν @ (t)| ∼ = Xk,2N −1

+N −1

−1 , x

]

k, = 1, .., N − 1.

The objective function obj (X, A) for the optimizer involves varying the matrix X and the scalar A and has a number of components of form obj (X, A) = (1) (2) wD bestfitD (X) + wD1 smoothD (X) + wD2 smoothD (X) (1) (2) +wF bestfitF (X) + wF 1 smoothF (X) + wF 2 smoothF (X) (2) +wF X bestfitF X (X) + wF X2 smoothF X (X) +wL distant (A)

(14.23)

where the weights wD , wD1 , wD2 , wF , wF 1 , wF 2 , wF X , wF X2 and wL are chosen to get a balance between bestfit to the target caplets and swaptions, smoothness of X in both row and column directions, and linearity of distant FX implied volatilities. The domestic rate components are bestfitD (X) =

¯2 X ¯¯ swpnD (X) ¯ i ¯ ¯ , − 1 ¯ targetD ¯ i

i

¯ ¯2 N −1 N−1 X X ¯ Xk, ¯ (1) ¯ ¯ smoothD (X) = ¯ Xk, −1 − 1¯ , (2) smoothD

(X) =

k=1

=2

k=2

=1

N −1 N−1 X X¯

the foreign rate components are

¯2 ¯ Xk, ¯ ¯ ¯ ¯ Xk−1, − 1¯ ,

¯2 X ¯¯ swpnF (X) ¯ i ¯ ¯ bestfitF (X) = ¯ targetF − 1¯ , i

i

¯2 N−1 X 2N−2 X ¯¯ Xk, ¯ (1) ¯ ¯ smoothF (X) = ¯ Xk, −1 − 1¯ , k=1 =N +1

(2) smoothF (X)

=

N−1 X 2N−2 X

k=2 =N +1

¯ ¯2 ¯ Xk, ¯ ¯ ¯ ¯ Xk−1, − 1¯ ,

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Engineering BGM

the FX components are bestfitF X (X) =

¯ ¯2 X ¯ ivolFXi (X) ¯ ¯ ¯ , − 1 ¯ targetFX ¯ i

i

(2) smoothF X

(X) =

N −1 ¯ X k=2

and the distant FX component is distant (A) =

¯2 ¯ Xk,2N −1 ¯ ¯ ¯ ¯ Xk−1,2N −1 − 1¯ ,

¯2 X ¯¯ α (Tj ) ¯ ¯ . ¯A − 1 ¯ ¯ (j − 1)

Tj >10

Note that all components of the objective function are designed to be independent of scale, making the weights wD , wD1 , wD2 , wF , wF 1 , wF 2 , wF X , wF X2 and wL comparable and thus easier to adjust to emphasize diﬀerent aspects of the calibration (like making wD , wF and wF X larger to get a better fit at the cost of smoothness of the volatility function). From X and the historical correlation matrix R, the domestic swpnDi (X) and foreign swpnFi (X) swaption implied volatilities, and also the FX forward implied volatilities ivolFXi (X) are computed by first constructing volatility functions ξ (·), ξ f (·) and ν @ (·), and then pricing in the standard fashion. For simplicity, fixed and floating side nodes will be assumed to coincide with Tj = δj for j = 0, 1, .., N in both foreign and domestic economies. For each k corresponding to the k th calendar time slice t ∈ (Tk−1 , Tk ], for the current X the (2N − 1) × (2N − 1) matrix Ck = diag (Xk, ) R diag (Xk, )

( = 1, .., 2N − 1)

(where for an n-vector X, diag X is an n × n matrix with X along the diagonal and zeros elsewhere) is a covariance. So we can eigenvalue decompose Ck (1) (r) letting Γ· , ..., Γ· be the first r eigenvectors (ordered by eigenvalue size) multiplied by the square root of the corresponding eigenvalue, so that if ´ ³ (1) (r) , for = 1, .., 2N − 1 Γ = Γ , ..., Γ then

ΓT Γ ∼ = Ck

(the equation is approximate because we have discarded some eigenvectors). For the k th slice of calendar time t ∈ (Tk−1 , Tk ], now construct the volatility functions ξ (·), ξ f (·) and ν @ (·) from Γ corresponding to the current value of X by setting for = 1, .., N − 1 and x ∈ (x −1 , x ] ³ ´ (1) (r) ξ (t, x) = ξ k. = Γ , ..., Γ ³ ´ (1) (r) ξ f (t, x) = ξ fk. = ΓN −1+ , ..., ΓN −1+ ³ ´ (1) (r) ν @ (t) = ν k = Γ2N−1 , ..., Γ2N−1

Cross-Economy BGM

139

The full volatility function ξ (t, x) = ξ k. covering all calendar times t simply requires this step to be repeated for each slice of calendar time t ∈ (Tk−1 , Tk ] for k = 1, 2, .. Note that if t ∈ (Tk−1 , Tk ] the relative maturity x corresponding to Tj is bounded, that is

t ∈ (Tk−1 , Tk ]

and

x + t = Tj

⇒

x ∈ [Tj − Tk , Tj − Tk−1 ) .

That means for each quarterly block of calendar time, the domestic volatilities ξ (t, Tj ) and foreign volatilities ξ f (t, Tj ) at absolute maturity Tj are also constant because for t ∈ (Tk−1 , Tk ]

ξ (t, Tj )|Tj =t+x = ξ (t, xj−k+1 ) = ξ k, j−k+1 , ¯ ¯ = ξ f (t, xj−k+1 ) = ξ fk, j−k+1 . ξ f (t, Tj )¯ Tj =t+x

Caplet and swaption implied volatilities (or their zetas) in both domestic and foreign economies are then easily computed from (4.16), because for T ≤ Tj1 , Tj2 they are a linear combination of integrals like

Z

T

∗

ξ (t, Tj1 ) ξ (t, Tj2 ) dt = 0

=δ Z

X

k

∗

ξ f (t, Tj1 ) ξ f (t, Tj2 ) dt = 0

=δ

X

k

∗

ξ (t, Tj1 ) ξ (t, Tj2 ) dt ξ k,

XZ

ξ fk, ∗j1−k+1

Tk

Tk−1

ξ ∗k, j1−k+1

k

T

XZ

j2−k+1 ,

Tk

∗

ξ f (t, Tj1 ) ξ f (t, Tj2 ) dt

Tk−1

ξ fk,

j2−k+1 .

k

Similarly from (14.18) the zeta of the FX forward option maturing at Tj

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Engineering BGM

will be a linear combination of expressions (for , l1, l2 ≥ k) like

Z

Z

Z

Tk Tk−1 Tk

Tk−1

Z

Tk

Tk Tk−1

|ν Tk (t)|2 dt = δ |ν k |2 ,

ν ∗Tk (t) ξ (t, T ) dt = δν ∗k ξ k, ν ∗Tk (t) ξ f (t, T ) dt = δν ∗k ξ fk,

−k+1 , −k+1 ,

h 1 (0) h 2 (0) ξ ∗ (t, T 1 ) ξ (t, T 2 ) dt

Tk−1

Z

= δh 1 (0) h 2 (0) ξ ∗k,

Tk Tk−1

Z

hf1 (0) hf2 (0) ξ f

∗

1−k+1 ξ k, 2−k+1 ,

(t, T 1 ) ξ f (t, T 2 ) dt

= δhf1 (0) hf2 (0) ξ fk, ∗ 1−k+1 ξ fk, Tk

h 1 (0) h 2 (0) ξ f

∗

2−k+1 ,

(t, T 1 ) ξ (t, T 2 ) dt

Tk−1

= δhf1 (0) h 2 (0) ξ fk, ∗ 1−k+1 ξ k,

2−k+1 .

The overall modus-operandi for Pedersen’s algorithm is thus: Step-1 Start X either with yesterday’s values, or with the magnitudes of ¯ ¯ ¯ ¯ the domestic and foreign historical volatility functions |ξ (xf )| and ¯ξ f (xf )¯, and magnitude of the historical spot FX volatility ν. Start A either with yesterday’s value or a number estimated from the asymptote (14.22) in the simple example above. Step-2 From X compute the implied volatilities swpnDi (X) and swpnFi (X) of the target domestic and foreign caplets and swaptions, and implied volatilities ivolFXi (X) of the target FX forward options using the methods described above. Step-3 Insert those values along with X and A, and the target implied volatilities targetDi , targetFi and targetFXi into the objective function (14.23) and feed into the optimizer to generate new trial values of X and A. Step-4 Return to Step-2 and iterate until the desired fit and degree of smoothness are obtained. Then extract the final domestic and foreign rate volatilities ξ (t, x) and ξ f (t, x), and the FX volatility ν @ (t) as the desired bestfit volatility functions.

Chapter 15 Inflation

A major application of the cross-economy BGM model of Chapter-14 is to inflation (see Jarrow et al [68] for the original paper articulating cross-economy HJM to inflation), where the consumer price index (CPI) takes the place of the foreign currency exchange rate. Thus in this chapter the variable S (t) denotes the CPI, which is the price in inflatable dollars in our nominal-world of one inflation proof zlotty in real-world, that is Z1 = $S (t). As in the crosseconomy model, superfix f to nominal-world variables to denote real-world variables. Calibration involves stripping nominal and real curves, and identifying forward CPI volatility functions (with correlations) along with a satisfactory forward inflation curve. Once that is done, the pricing and hedging of products is usually straightforward.

15.1

TIPS and the CPI

In the US, basic ingredients available to build an inflation model are: 1. Standard Treasury bonds (nominal variables), which can be stripped using standard techniques. 2. The lagged consumer price index CPI. 3. Inflation indexed bonds called TIPS (Treasury Inflation Protected Securities). 4. Inflation information in the form of futures, inflation forecasts or traders’ views that the inflation rate is mean-reverting. The essential idea behind TIPS bonds is that their coupons and final redemption be defined in terms of the CPI index at payoﬀ time. Specifically, if a TIPS bond BT IP S is issued at time T0 when the CPI was S (T0 ), then • A coupon payment of real-world Zc occurring at time Tj pays oﬀ in S(T ) nominal-world dollars as $ S(T0j ) c dollars. 141

142

Engineering BGM • Final redemption of real-world Z1 h at timei TN is floored, paying oﬀ in N) nominal-world dollars as $ max 1, S(T S(T0 ) .

Hence the time t nominal-value of a TIPS bond issued at T0 (≤ t) is ¯ o n   PN S(T ) ¯ E0 β(T1 j ) S(T0j ) c¯ Ft j=1  ½ ¯ o n h i+ ¯¯ ¾  BT IP S (t; T0 ) =  , S(TN ) S(TN ) ¯ 1 1 +E0 β(TN ) S(T0 ) ¯ Ft + E0 β(TN ) 1 − S(T0 ) ¯¯ Ft ¯ o n   PN S(Tj ) ¯ cB (t, T ) E F ¯ j T t j j=1 S(T0 ) ½  ¯ o h n i+ ¯¯ ¾  = , S(TN ) ¯ N) ¯ Ft 1 − S(T +B (t, TN ) ETN S(T0 ) ¯ Ft + B (t, TN ) ETN S(T0 ) ¯ o nP  N 1 cB (t, T ) S (t) + B (t, T ) S (t) j T N T j N j=1 S(T ) 0   ½h i+ ¯¯ ¾ = , S(TN ) ¯ 1 − S(T0 ) ¯ Ft +B (t, TN ) ETN  o S (t) nPN j=1 cBf (t, Tj ) + Bf (t, TN )   S (T0 ) ½h = i+ ¯¯ ¾   . S(TN ) +B (t, TN ) ETN 1 − S(T0 ) ¯¯ Ft

So stripping the TIPS bond is complicated by the floor on the CPI index for which we need the forward CPI volatility ν TN (t) at time TN . In practice, the option is often ignored because several years after issue usually the CPI S (t) has increased well beyond its initial value S (T0 ) putting the floor well out-of-the-money. Nevertheless, for recently issued TIPS it can be a problem. The CPI is often expressed in terms of the zero coupon CPI swap rate (eﬀectively a quarterly average) defined as that value of κ satisfying (1 + δκ)

N

=

S (T ) S (T0 )

where

T − T0 = N δ.

As an accumulation index the CPI usually increases and so must have a positive drift µ (t) under the reference measure P. From the modelling point of view that is not unreasonable: recall that Black-Scholes is perfectly compatible with a steeply drifting stock! In the inflation case, however, the CPI cannot climb in isolation from the bond drifts. We have, using (2.1), dS (t) ST (t) B (t, T ) , = µ (t) dt + ν ∗ (t) dW (t) , S (t) S (t) dST (t) = −ν T (t) b (t, T ) dt + ν ∗T (t) dW (t) , ST (t) dB (t, T ) = f (t) dt + b∗ (t, T ) dW (t) , B (t, T ) dB f (t, T ) = f f (t) dt + bf ∗ (t, T ) dW f (t) , B f (t, T )

B f (t, T ) =

Inflation

143

where, following the reasoning of Section-2.1, the model for B f (t, T ) is similar to that of B (t, T ) under P in nominal-world, but is now under a diﬀerent reference measure Pf in real-world. Having diﬀerent reference measures in real- and nominal-worlds makes sense, because real-world zlottys constitute a diﬀerent set of numbers with a diﬀerent set of statistics to zlottys converted to dollars. Diﬀerentiating both sides of the parity relationship and equating volatility and drift terms in the only way that avoids maturity dependence in the drifts, defines the measure change between P and Pf , and gives connections dW f (t) = dW (t) − ν (t) dt,

bf (t, T ) = −ν T (t) + b (t, T ) − ν (t) ,

µ (t) = f (t) − f f (t) .

Because the CPI drift µ (t) is the diﬀerence of the bond drifts f (t) and f f (t) (clearly an analogue of the Fisher equation), it cannot climb too quickly.

15.2

Dynamics of the forward inflation curve

Products of interest usually revolve around the inflation rate; for example, inflation forwards, inflation futures, inflation to Libor swaps, and options on inflation. So understanding the dynamics of inflation contracts will be important. Let I (t; δ) be the δ-period inflation rate and IT (t; δ) the T -maturing forward inflation contract on inflation, defined respectively by S (t) −1 S (t − δ) ½ IT (t; δ) = ET { I (T ; δ)| Ft } = ET I (t; δ) =

and ¯ ¾ S (T ) ¯¯ Ft − 1. S (T − δ) ¯

The forward version of interest rate parity applied to inflation FTf −δ (t, T ) =

ST (t) FT −δ (t, T ) ST −δ (t)

says that real rates, nominal rates and the forward CPI are closely connected. The relationship is usually written as the so-called Fisher equation £ ¤ 1 + δK (t, T − δ) = [1 + HT (t, δ)] 1 + δK f (t, T − δ) , where

HT (t, δ) =

ST (t) −1 ST −δ (t)

(15.1)

144

Engineering BGM

is called the pseudo-forward because its forward CPI components are forward contracts, and at time T − δ it does equal true forward inflation ¯ ½ ¾ ST (T ) ¯¯ IT (T − δ; δ) = ET FT −δ − 1 ST −δ (T − δ) ¯ ST (T − δ) − 1 = HT (T − δ, δ) . = ST −δ (T − δ) In words, the Fisher equation (15.1) says (1 + nominal rate) = (1 + inflation compensator) × (1 + real rate) . Though HT (t; δ) and IT (t; δ) are similar, connecting the two at an arbitrary t < T − δ requires a convexity correction. We have ¯ ¾ ¯ ¾ ½ ½ ST (T ) ¯¯ ST (T − δ) ¯¯ 1 + IT (t; δ) = ET Ft = ET Ft , ST −δ (T − δ) ¯ ST −δ (T − δ) ¯ nR o ¯   ¯   E tT −δ ν ∗T (s) dWT (s) ¯ ST (t) ST (t) nR o ¯¯ Ft = ET exp α (t, T ) , = T −δ   ST −δ (t) ST −δ (t) E ν ∗ (s) dWT −δ (s) ¯ t

T −δ

where

nR o¯   T −δ ∗ ¯   exp ν (s) b (s, T − δ, T ) ds  ¯  t n T −δ  o   ¯  R T −δ ∗ ¯ E ν (s) dW (s) exp α (t, T ) = ET T ¯ Ft , T t  ¯  o × nR T −δ    ¯   ∗ ¯  E t ν T −δ (s) dWT (s)  nR o ¯   exp T −δ ν ∗ (s) bf (s, T − δ, T ) ds ¯¯  T −δ nR t o ¯ Ft , = ET T −δ ¯  ∗  ×E [ν T (s) − ν T −δ (s)] dWT (s) ¯ t

yields the Ft -measurable stochastic convexity variable α (t, T ). If forward CPI volatilities ν T (t) are all assumed deterministic, and the bond volatility diﬀerence b (s, T − δ, T ) is approximated deterministically by setting its stochastic components to their time t values, an approximation to α (·) will be α (t, T ) ∼ =

Z

T −δ

∗

[b (s, T − δ, T ) − ν T (s) + ν T −δ (s)] ν T −δ (s) ds, ht i 2 δH(t,T −δ) ∼ = 1+δK(t,T −δ) ξ ∗ (T − δ) ν T −δ − ν ∗T ν T −δ + |ν T −δ | (T − δ − t) .

The Fisher equation in terms of the true forward and convexity adjustment is £ ¤ 1 + δK (t, T − δ) = exp {−α (t, T )} [1 + IT (t, δ)] 1 + δK f (t, T − δ) .

Inflation

145

REMARK 15.1 In HJM all volatilities are deterministic, so Z T −δ α (0, T ) = [b (s, T − δ, T ) − ν T (s) + ν T −δ (s)]∗ ν T −δ (s) ds 0

exactly. To roughly estimate the magnitude of α (·) assume the forward CPI volatilities ν T (t) are 2%, cash forwards K (t, T ) are 6%, the BGM volatility ξ (t, T ) is flat at 10%, and δ = .25. Then Z T δK (t, T ) ξ (t, T ) ∼ σ (t, u) du = b (t, T − δ, T ) = = .25 × .06 × .1, 1 + δK (t, T ) T −δ ∼ (T − δ) × .25 × .06 × .1 × .02 = .00003 × (T − δ) , α (0, T ) = giving corrections like α (0, 10) ∼ = 15 bpts. = 3 bpts, α (0, 50) ∼ The dynamics of forward inflation IT (t; δ) will be determined by IT (t; δ) =

ST (t) exp {α (t, T − δ, T )} − 1, ST −δ (t)

and is a PT -martingale. So if we diﬀerentiate it and express the resultant SDE in terms of WT (t), all drift terms must cancel. Hence dIT (t; δ) = [1 + IT (0; δ)] [ν T (t) − ν T −δ (t) + ν α (t)]∗ dWT (t) , where stochastic ν α (t) comes from α (·) and is assumed small. Thus IT (t; δ) is approximately a shifted lognormal process with a shift of 1, making it almost normal with approximate small volatility [ν T (t) − ν T −δ (t)]. REMARK 15.2 That IT (t; δ) is normally distributed with low variance, may account for traders’ observations that inflation is mean reverting. Moreover, because the forward volatility expression [ν T (t) − ν T −δ (t)] will aﬀect the value of inflation options, we will need to be careful in our calibration.

15.2.1

Futures contracts

Similarly to the definition of the forward contract, the futures contract JT (t, δ) maturing at T on the inflation rate I (t, δ) is ¯ ¾ ½ S (T ) ¯¯ JT (t, δ) = E0 { I (T, δ)| Ft } = E0 Ft − 1, S (T − δ) ¯ so that if, as with the forwards, the CPI volatilities ν T (t) are deterministic ¯ ¾ ½ ST (T ) ¯¯ 1 + JT (t, δ) = E0 Ft , ST −δ (T − δ) ¯ nR o ¯   T ¯   E t ν ∗T (s) dWT (s) ¯ ST (t) ST (t) nR o ¯¯ Ft = E0 exp β (t, T ) = T −δ ST −δ (t)  E ν ∗ (s) dW (s) ¯  ST −δ (t) t

T −δ

T −δ

146

Engineering BGM

where nR o ¯  T ¯  E t ν ∗T (s) [dW0 (s) + b (s, s, T ) ds] ¯ n o¯ F , exp β (t, T ) = E0  E R T −δ ν ∗ (s) [dW (s) + b (s, s, T − δ) ds] ¯¯ t  0 T −δ t " #¯   RT ∗ ¯   ν (s) b (s, s, T ) ds  ¯  T  R T −δt ∗ exp   ¯   − t ν T −δ (s) b (s, s, T − δ) ds ¯    o nR ¯ T −δ ∗ , = E0 F ¯ ν T (s) dW0 (s) E t  ¯ t     ¯ o × nR T −δ    ¯   ¯  E t ν ∗T −δ (s) dW0 (s)   ¯  RT ∗ ¯   ν (s) b (s, s, T ) ds  ¯  T  R T −δt ∗    ¯   exp  − t ν T −δ (s) b (s, s, T − δ) ds  ¯  ¯ Ft , R T −δ = E0 ∗ −n t [ν T (s) − ν T −δ (s)] ν T −δ (s)ods ¯¯      R T −δ   ¯   ×E t [ν T (s) − ν T −δ (s)]∗ dW0 (s) ¯   

yields the F0 -measurable stochastic convexity variable β (t, T ). Approximating the drift deterministically, yields a convexity adjustment like β (t, T ) # "R R T −δ ∗ T ∗ ν T (s) b (s, s, T ) ds − t ν T −δ (s) b (s, s, T − δ) ds t R T −δ , = − t [ν T (s) − ν T −δ (s)]∗ ν T −δ (s) ds Z T −δ Z T ν ∗T (s) b (s, s, T ) ds − ν ∗T −δ (s) b (s, s, T ) ds. = α (t, T ) + t

t

Thus the Fisher equation expressed in terms of JT (t, δ) is £ ¤ 1 + δK (t, T − δ) = exp {−β (t, T )} [1 + JT (t, δ)] 1 + δK f (t, T − δ) .

15.2.2

The CME futures contract

With an inflation period of 3 months (δ = .25) the time T settlement futures price of this contract is specified as CPI (T ) − CPI (T − 3 months) %, CME (T ) = 100 − 4 × 100 × CPI (T − 3 months) ¸¶ µ · S (T ) −1 = 100 (1 − 4JT (T, δ)) %, = 100 1 − 4 S (T − δ) ⇒ CME (t) = E0 { 100 (1 − 4JT (T, δ))| Ft } = 100 (1 − 4JT (t, δ)) %. ] (t) is an estimate of the futures contract using the pseudo-forward If CME ] (t) = 100 (1 − 4HT (t, δ)) %, CME

Inflation

147

then the convexity correction required to get a more accurate price is ] (t) = −400 [JT (t, δ) − HT (t, δ)] %, CME (t) − CME = −400 [exp {β (t, T − δ, T )} − 1]

ST (t) %. ST −δ (t)

Chapter 16 Stochastic Volatility BGM

Our aim in this chapter is not to try to fit a volatility smile exactly, but to add a measure of convexity to an existing skew. We develop a shifted-stochastic volatility model with zero correlation between the forwards and their stochastic volatility, in which the shift (rather than the absent correlation) is used to fit the skew. Independence of the yieldcurve and stochastic volatility drivers simplifies the mathematics, and yields a model that when compared to Shifted BGM: 1. Has only a few more parameters and is only marginally more complex to calibrate using techniques already developed. 2. Has a similar term structure of skew, but no term structure of convexity. 3. Makes forward measure changes in an equally straightforward way. 4. Also permits caplets and swaptions to be priced by a single technique. 5. Uses almost identical methods to compute deltas and vegas. An added measure of flexibility is that the stochastic volatility or variance process can be any reasonable positive process such as the square-root process considered here (other possibilities include, for example, the exponential Ornstein Uhlenbeck processes).

16.1

Construction

Return to (2.1) and suppose that under the reference measure P, all bonds have an additional stochastic component in their volatility depending on a positive variance process V (t) like p dB (t, T ) = f (t) dt + V (t)b∗ (t, T ) dW (t) t ≤ T, (16.1) B (t, T ) where the multi-dimensional BM W (t) appears only in the bond SDEs, and is independent of the extra one-dimensional BM U (t) driving V (t) dV (t) = µ (t, V (t)) dt + γ (t, V (t)) dU (t) ,

V (t) > 0 V (0) = 1.

149

150

Engineering BGM

Note that V (0) can always be set at 1, otherwise simply re-scale V (·) and b (·). For any zero coupon bond B (t, S) with an earlier maturity t ≤ S < T ´. ³ ´ ³ B(t,S) B(t,S) d B(t,T ) B(t,T ) i h p p ∗ = V (t) [b (t, S) − b (t, T )] dW (t) − V (t)b (t, T ) dt ,

so we can define a change of measure from P to the forward measure PT induced by B (t, T ) as numeraire by dWT (t) = dW (t) −

p V (t)b (t, T ) dt

dUT (t) = dU (t) .

Note that if W (t) and U (t) were correlated, there would be a non-zero component of the bond volatility b (·) corresponding to U (t), and consequently the BM U (t) driving V (t) would not be the same under both reference and forward measures. Thus bonds discounted by B (t, T ) as numeraire are martingales under PT ³ ´. ³ ´ p B(t,S) B(t,S) d B(t,T V (t)b (t, S, T )∗ dWT (t) . ) B(t,T ) = For T < S = T1 the measure change between PT and PT1 is given by dWT1 (t) = dWT (t) +

p V (t)b (t, T, T1 ) dt,

and the ratio of the bonds is the forward contract or its reciprocal FT (t, T1 ) =

B (t, T1 ) , B (t, T )

1 B (t, T ) = , FT (t, T1 ) B (t, T1 )

which for t ≤ T have respective SDEs

p ∗ = − V (t)b (t, T, T1 ) dWT (t) , ³ ´. ³ ´ p 1 1 V (t)b (t, T, T1 )∗ dWT1 (t) . d FT (t,T FT (t,T1 ) = 1) dFT (t,T1 ) FT (t,T1 )

We are now in a position to describe a backward construction for our model similar to that of Section 3.2. Temporarily simplify notation by indexing variables by maturity and dropping time t, for example BT = B (t, T )

WT = WT (t)

bT = b (t, T )

a (T ) = aT

etc.

For 0 ≤ t ≤ R < S < T < U we will illustrate the method by moving back through the intervals (T, U ], (S, T ] and (R, S] successively defining the bond volatility diﬀerences [bT − bU ], [bS − bT ] and [bR − bS ] in the process. Our basic assumption and starting point is that the deterministic volatility

Stochastic Volatility BGM

151

functions ξ R , ξ S and ξ T are specified exogenously and are bounded, and that KT is a martingale under PU . Thus on the three intervals respectively BT δ T KT = − 1 HT = KT + aT B √ U ∗ dHT = V HT ξ T dWU , BS − 1 HS = KS + aS δ S = |(S, T ]| δ S KS = BT h i √ dHS = HS µS dt + V ξ ∗S dWU ,

δ T = |(T, U ]|

BR δ R KR = − 1 HR = KR + aR BS h i √ dHR = HR µR dt + V ξ ∗R dWU .

δ R = |(R, S]|

On the first interval (T, U ], comparison of volatility terms in √ δ T dHT = δ T V HT ξ ∗T dWU = δ T dKT ¶ µ ¶ µ BT √ BT = =d V [bT − bU ]∗ dWU BU BU identifies the diﬀerence in the bond volatilities δ T HT ξ , [bT − bU ] = 1 + δ T KT T which must be bounded (assuming δ T aT < 1) because the SDE for HT has a positive solution, and that defines a change of measure from PU to PT by √ δ T HT √ V ξ T dt. dWT = dWU − V [bT − bU ] dt = dWU − 1 + δ T KT On the second interval (S, T ], in the equation h i √ δ S dHS = δ S HS µS dt + V ξ ∗S dWU ¶ µ Á ¶ µ BS BT BS =d = δ S dKS = d BT BU BU ¶ µ o n √ BS √ V [bS − bT ]∗ dWU − V [bT − bU ] dt , = BT

comparison of volatility and drift terms yields ¶ µ BS δ S HS ξ S = [bS − bT ] , and BT ¶ µ BS δ S HS µS = − V [bS − bT ]∗ [bT − bU ] ⇒ BT ⇒ µS = −V ξ ∗S [bT − bU ] i √ ∗h √ √ dHS = HS V ξ S dWU − V [bT − bU ] dt = HS V ξ ∗S dWT .

152

Engineering BGM

The SDE for HS has a solution that is strictly positive, which in turn defines the bond volatility diﬀerence [bS − bT ] =

δ S HS ξ , 1 + δ S KS S

and a change of measure from PU to PS by h√ i √ √ dWS = dWU − V [bS − bU ] dt = dWU − V (bS − bT ) + V (bT − bU ) dt ¾ ½ δ S HS √ δ T HT √ = dWU − V ξS + V ξ T dt. 1 + δ S KS 1 + δ T KT On the third interval (R, S] proceed as on the second. h i √ δ R dHR = δ R HR µR dt + V ξ ∗R dWU ¶ µ o n √ BR √ V [bR − bS ]∗ dWU − V [bS − bU ] dt = BS i √ h √ √ ⇒ dHR = HR V ξ ∗R dWU − V [bS − bU ] dt = V HR ξ ∗R dWS ⇒

[bR − bS ] =

δ R HR ξ . 1 + δ R KR R

As before, the SDE for HR has a strictly positive solution which defines [bR − bS ] and the next measure change from PU to PR by √ dWR = dWU − V [bR − bU ] dt, ¾ ½ δ S HS √ δ T HT √ δ R HR √ V ξS + V ξT + V ξ R dt. = dWU − 1 + δ S KS 1 + δ T KT 1 + δ R KR The measures PR , PS and PT thus defined are clearly the forward ³ measures ´ B at R, S and T . For example, for any bond BQ (Q < T ) the ratio BQ must T be a PT -martingale because ¶ µ Á ¶ µ ¶ µ BQ BT BQ √ BQ ∗ =d = V [bQ − bT ] dWT . d BT BU BU BT In general, starting with the terminal measure Pn at the terminal node Tn and working backwards in the above fashion constructs a system of equations with each forward H (t, Tj ) an exponential martingale under the corresponding forward measure Pj+1 . The general structure of the model is now becoming clear;√ whenever a volatility function occurs in shifted BGM simply multiply it by V to get the stochastic BGM equivalent. The next section shows the same result holds for our approximations to swaprate volatilities; all the results of√Chapter-4 hold in stochastic volatility BGM if volatilities are multiplied by Vt .

Stochastic Volatility BGM

16.2

153

Swaprate dynamics

Recall the swaprate ω (t) in Chapter-4 was split into stochastic ω1 (t) and shift ω2 (t) parts:

ω (t) = ω1 (t) − ω2 (t) where PN−1 N −1 N−1 X X j=0 δ j FT (t, Tj+1 ) H (t, Tj ) = u (t) H (t, T ) = vj (t) , ω1 (t) = ¡ ¢ PM −1 j j i=0 δ i FT t, T i+1 j=0 j=0 ¤ N −1 £ PN−1 X ¤ £ j=0 δ j FT (t, Tj+1 ) a (Tj ) − µj = uj (t) a (Tj ) − µj . ω2 (t) = ¡ ¢ PM −1 i=0 δ i FT t, T i+1 j=0

eT is now the swaprate measure equivalent to PT induced by W fT (t), where If P fT (t) = dWT (t) + dW

M−1 X p ¢ ¡ V (t) ui (t) b t, T, T i+1 dt, i=0

eT -martingales then the weights uj (t) and ui (t) will be low variance P

" #∗ M−1 X ¢ ¡ duj (t) p fT (t) , = V (t) −b (t, T, Tj+1 ) + ui (t) b t, T, T i+1 dW uj (t) i=0 " #∗ X ¢ M−1 ¢ ¡ ¡ dui (t) p fT (t) . = V (t) −b t, T, T i+1 + ui (t) b t, T, T i+1 dW ui (t) i=0

Hence the approximation (which is the same as before) for the shift ω2 (t) ∼ = ω2 (0) = =

N −1 X j=0

¤ £ δ j FT (0, Tj+1 ) a (Tj ) − µj ¡ ¢ PM−1 i=0 δ i FT 0, T i+1

PN −1 j=0

¤ £ uj (0) a (Tj ) − µj = α (0) .

eT -martingales In a similar fashion, the vj (t) will also be P

¸∗ · ξ (t, Tj ) − b (t, T, Tj+1 ) dvj (t) p fT (t) , ¢ ¡ P = V (t) dW + M−1 vj (t) i=0 ui (t) b t, T, T i+1

(16.2)

154

Engineering BGM

for which a more stable SDE can be obtained using uj (t) ∼ = uj (0) like vj (t) = uj (t) H (t, Tj ) = uj (t) [K (t, Tj ) + a (Tj )] ∼ uj (t) K (t, Tj ) + uj (0) a (Tj ) = dvj (t) ∼ d [uj (t) K (t, Tj )] K (t, Tj ) = vj (t) uj (t) K (t, Tj ) H (t, Tj ) ¶¸∗ µ · p K (t, Tj ) −b (t, T, Tj+1 ) fT (t) . ¢ ¡ PM−1 = V (t) ξ (t, Tj ) + dW H (t, Tj ) + i=0 ui (t) b t, T, T i+1 ⇒

Hence the following SDE for the stochastic part ω1 (t) of the swaprate N−1 X p dω1 (t) fT (t) , σ (t) = V (t) = σ (t) dW Aj (t) ξ (t, Tj ) , ω1 (t) j=0    K (t, T ) N −1   X w (t)   H (t, T ) Aj (t) = wj (t) − hj (t) PN −1   → k) −− u (t) k=0 K(t,T =j H(t,Tk ) wk (t)

and

δ j FT (t, Tj+1 ) H (t, Tj ) wj (t) = PN −1 . j=0 δ j FT (t, Tj+1 ) H (t, Tj )

SDEs for the weights wj (t) show they are low variance martingales well approximated by their initial values wj (0), giving the following approximate SDE for the stochastic part ω1 (t) of the swaprate ω (t) ( ) N−1 P dω1 (t) p fT (t) , = V (t) Aj ξ ∗ (t, Tj ) dW ω1 (t) j=0   ¶ N −1 N−1  Xµ K X K → , λ= w −− u λ w Aj = wj − hj   H H =j

wj = wj (0)

hj = hj (0)

(16.3)

=0

K = K (0, T )

H = H (0, T )

− → → u =− u (0) .

Hence in stochastic volatility BGM, caplets and swaptions can be priced by the same method, because given a variance process V (t) satisfying (16.1) the time t value Cpl (t) of the time T maturing caplet struck at κ and paying at T1 is ¯ o n +¯ Cpl (t) = δB (t, T1 ) ET1 [K (T, T ) − κ] ¯ Ft , where (16.4) H (t, T ) = K (t, T ) + a (T ) ,

dH (t, T ) p = V (t)ξ ∗ (t, T ) dWT1 (t) , H (t, T )

while from (16.2) and (16.3) the time t value of a swaption maturing at T and

Stochastic Volatility BGM

155

struck at κ will be pSwpn (t) =

(M −1 X i=0

ω1 (t) ∼ = ω (t) + α (0) ,

16.3

¡

δ i B t, T i+1

) ¢

eT E

n

dω1 (t) p = Vt ω1 (t)

¯ o +¯ [ω (T ) − κ] ¯ Ft ,

(

N−1 P j=0

(16.5)

)

fT (t) . Aj ξ (t, Tj ) dW ∗

Shifted Heston options

The solution to the next problem gives prices of caplets and swaptions in stochastic volatility BGM by inserting into (16.6) ¯ ¯ ¯ ¯N−1 ¯ ¯P Aj ξ ∗ (t, Tj )¯ for swaptions ξ (t) = |ξ (t, T )| for caplets and ξ (t) = ¯ ¯ ¯ j=0 Problem 1 (The shifted Heston option problem) Find ¯ o n ¯ H (t, κ) = E [K (T ) − κ]+ ¯ Ft ,

(16.6)

p dK (t) = V (t)ξ (t) dW (t) , K (t) + a p and dV (t) = λ [µ − V (t)] dt + γ V (t)dU (t) , with λ, µ, γ > 0 V (0) = 1.

when

Setting S (t) = ln (K (t) + a) yields the 2-dimensional aﬃne system p 1 dS (t) = − V (t) ξ 2 (t) dt + V (t)ξ (t) dW (t) , 2 p dV (t) = λ [µ − V (t)] dt + γ V (t)dU (t) , V (0) = 1,

which is Markov in terms of the state vector X (t) = (S (t) , V (t))∗ .

16.3.1

Characteristic function

Let pb (θ; t), where θ is a complex constant, be the characteristic function of S (T ) conditional on X (t) pb (θ; t) = E { exp (iθS (T )) | X (t)}

with

pb (θ; T ) = exp (iθS (T )) .

REMARK 16.1 To be totally consistent with our Fourier transform definitions (A.10) we should perhaps write 2πθ rather than θ, but that is not

156

Engineering BGM

how characteristic functions are usually defined. Thus a little care is required in applying pb below. Because the system is Markov, pb (·) must satisfy the backward PDE

1 1 1 pbt − ξ 2 (t) V (t) pbs + λ [µ − V (t)] pbv + ξ 2 (t) V (t) pbss + γ 2 V (t) pbvv = 0. 2 2 2

and, because the system is aﬃne, pb (·) must also have the aﬃne form pb (θ; t) = exp (A (t) + B (t) S (t) + C (t) V (t)) , A (T ) = C (T ) = 0, B (T ) = iθ,

where A, B, C are functions of t (and θ) only. Substituting pb (·) into the PDE and setting the coeﬃcients of S (t), V (t) and time dependent terms zero, gives · ¸ ¡ 2 ¢ 1 2 2 1 2 V (t) + At + λµ C = 0 Bt S (t) + Ct − λC + γ C + ξ (t) B − B 2 2 ¤ 1 1£ 2 B − B ξ 2 (t) = 0, At + λµC = 0, Bt = 0, ⇒ Ct − λC + γ 2 C 2 + 2 2 with A (T ) = C (T ) = 0, B (T ) = iθ.

So immediately B (t) = iθ, and with the substitution C = − γ22 D ¤ 1 £ Dt = D2 + λD − γ 2 iθ + θ2 ξ 2 (t) , D (T ) = 0, 4 2λµ At = 2 D (t) , A (T ) = 0. γ This non-linear ODE for D (t) has an analytic solution only when ξ (t) is constant. So assume ξ (t) = ξ j is piecewise constant on (tj , tj+1 ], and solve for D (t) analytically by backward induction using the solution on each interval as the boundary value for the next iteration. For t ∈ (tj , tj+1 ], set ¤ £ ¤ 1 £ Dt = D2 + λD − γ 2 iθ + θ2 ξ 2j , ∆j = λ2 + γ 2 iθ + θ2 ξ 2j , 4 p ´ p ´ 1³ 1³ + dj = = ∆j , d− −λ + ∆j , −λ − j 2 2 and identify two cases depending on whether or not ∆j is zero:

Stochastic Volatility BGM 16.3.1.1

157

Case ∆ 6= 0

Using partial fractions, on t ∈ (tj , tj+1 ] p ∆j (tj+1 − t) =

Z

t

= ln

tj+1

Ã

d+ j giving

D (t) =

dD − D − d+ j

D−

D−

−

d− j

·

t

tj+1

dD D − d− j ! +

Dj+1 − dj

Dj+1 − d− j

Dj+1 −d+ j

Dj+1 −d− j

¸ ¡ p ¢ exp − ∆j (tj+1 − t)

, ¡ p ¢ exp − ∆j (tj+1 − t) Z tj+1 + dj − d− 2λµ 1 j y dy, A (t) = Aj+1 − 2 p γ (1 − y) y ∆j t ( + ) dj y (tj+1 ) − 1 2λµ y (tj+1 ) p − ln ln = Aj+1 − 2 , γ y (t) y (t) − 1 ∆j 1−

and then

where 16.3.1.2

d− j d+ j

Z

y (t) =

Dj+1 −d+ j

Dj+1 −d− j

Dj+1 − d+ j

Dj+1 −

d− j

³ p ´ exp − ∆j (tj+1 − t) .

Case ∆ = 0

In this case integrating directly on t ∈ (tj , tj+1 ] Z

tj+1

1 1 dD giving ¡ ¢ = + − +, + 2 D − d D j+1 − dj t D − dj j ¡ ¢ + Dj+1 + d+ j Dj+1 − dj (tj+1 − t) ¢ ¡ D (t) = , and then 1 + Dj+1 − d+ j (tj+1 − t) Z tj+1 Dj+1 + d+ 2λµ 1 j y ¢ dy, A (t) = Aj+1 + 2 ¡ + γ 1+y Dj+1 − dj t ¢ £ ¡ ¤ª 2λµ © + = Aj+1 − 2 d+ . j (tj+1 − t) + ln 1 + Dj+1 − dj (tj+1 − t) γ

(tj+1 − t) =

Hence both D and A can be found by backward induction from T to t using A (T ) = D (T ) = 0, yielding (recall V (0) = 1) ¶ µ 2 pb (θ; t) = exp A (t) − 2 D (t) V (t) + iθS (t) , γ µ ¶ 2 and pb (θ; 0) = exp A (0) − 2 D (0) + iθS (0) . γ

158

16.3.2

Engineering BGM

Option price as a Fourier integral

Letting k = ln (κ + a) be the log-shifted strike and p (s) = p (s, T ; X (0) , 0) be the transition density of S (T ) given X (0), then the time t = 0 option price H (k) = H (0, κ) is ¯ o Z ∞ n +¯ (exp s − exp k) p (s) ds. H (k) = E [K (T ) − κ] ¯ X (0) = k

Our first approach to deriving H (k) as a Fourier integral is standard and included for completeness, but the second is numerically superior. 16.3.2.1

First Solution

The result follows if for real c, we can find Z ∞ p (s) exp (cs) ds ⇒ H (k) = U (k; 1) − exp k U (k; 0) . U (k, c) = k

b (·), and its inverse back to U (·) are given by U (·)’s Fourier transform U ¶ Z ∞ µZ ∞ b (x, c) = p (s) exp (cs) ds exp (2πixk) dk, U −∞ k ·Z s ¸ Z ∞ = p (s) exp (cs) exp (2πixk) dk ds, −∞ −∞ ¶ µ Z ∞ exp (2πixs) 1 δ (x) + ds; = p (s) exp (cs) 2 2πix −∞ Z ∞ b (x, c) exp (−2πixk) dx, U (k, c) = U −∞ R ) ( 1 ∞ 2 −∞ p (s) exp (cs) ds i h R∞ R∞ , = dx + −∞ −∞ p (s) exp (i [2πx − ic] s) ds exp(−2πixk) 2πix Z Z ∞ exp (−2πixk) 1 ∞ pb (2πx − ic) dx, p (s) exp (cs) ds + = 2 −∞ 2πix −∞ Z Z ∞ 1 ∞ exp (−2πixk) = pb (2πx − ic) dx. p (s) exp (cs) ds + Re 2 −∞ πix 0

The last step, halving the integration range and taking real parts, is because ) Z ∞ ( exp(−2πixk) Z ∞ exp (−2πixk) p b (2πx − ic) 2πix pb (2πx − ic) dx = dx, 2πix + exp(2πixk) b (−2πx − ic) −∞ 0 −2πix p while

pb (2πx − ic; t) = E { exp (i [2πx − ic] S (T )) | X (t)}

implies

pb (2πx − ic; t) = E { exp (−i [2πx + ic] S (T )) | X (t)} = pb (−2πx − ic) .

Hence the call option value is given by the Fourier integral

Stochastic Volatility BGM

1 H (k) = [K (T ) − κ] + 2

Z

∞ 0

·

1 Re πix

µ

159

exp (−2πixk) pb (2πx − i) − exp (k − 2πixk) pb (2πx)

¶¸

dx.

The numerical shortcomings in this expression for H (k) include: 1. The integrand is singular at x = 0.

2. Technically U (k, c) is not integrable in k because U (−∞, c) 6= 0 Z ∞ lim U (k; 1) = K (0) + a, lim U (k; 0) = p (s) ds = 1. k→−∞

k→−∞

−∞

3. U (k, c) integrates over a discontinuity, contributing to inaccuracy. 16.3.2.2

Second Solution

Introduce a modified H (k); for real b > 0 define Hb (k) = exp (bk) H (k) , and note that Hb (k) now satisfies the growth condition at infinity because lim H (k) = 0,

k→∞

lim H (k) = K (0) + a

k→−∞

⇒

lim Hb (k) → 0.

|k|→∞

b b (·), and inverse back to Hb (k), are given by Hb (·)’s Fourier transform H µZ ∞ ¶ Z ∞ b b (x) = exp (bk) p (s) (exp s − exp k) ds exp (2πixk) dk, H −∞ k ¾ Z ∞½ Z s = p (s) [exp (s + bk + 2πixk) − exp (k + bk + 2πixk)] dk ds, −∞ −∞ ¶ µ Z ∞ 1 1 − ds, = p (s) exp (s + bs + 2πixs) b + 2πix 1 + b + 2πix −∞ pb (2πx − i (1 + b) , 0) = ; b + b2 − 4π2 x2 + 2πix (2b + 1) Z +∞ b b (x) exp (−2πikx) dx, H (k) = exp (−bk) H = 2 exp (−bk)

Z

0

+∞

−∞

b b (x) exp (−2πikx) dx. Re H

b b (x), and because As long as b > 0, there is no singularity in H ¸ · 2 pb (2πx − i (1 + b) ; 0) = exp A (0) − 2 D (0) + (1 + b) S (0) exp [i2πxS (0)] γ

it should stay bounded for finite b (a satisfactory default value for b is b = 12 ).

160

Engineering BGM

16.4

Simulation

The Glasserman approach of Section-9.1 translates well to stochastic volatility BGM. Relationships between his Z (t, T ) and V (t, T ) variables (please distinguish between Glasserman’s V (t, T ) and the stochastic variance V (t) = Vt ), forwards p K (·) and bonds B (·) remain the same, but, as above, there is an extra V (t) term in volatilities that will require simulation. Under the terminal measure Pn clearly Z (t, Tn ) = 1 and SDEs for the Z (t, T ) for j = @ (t) , .., n − 1 are dZ (t, Tj ) p = V (t)b∗ (t, Tj , Tn ) dWn (t) Z (t, Tj )   ¾ n−1 X ½ p Z (t, T ) +1 ξ ∗ (t, T ) dWn (t) , = V (t)  φ 1−λ Z (t, T ) =j

while V (t, Tn ) = 1 and SDEs for the V (t, T ) for j = @ (t) , .., n − 1 are   n−1 X ½ V (t, T ) ¾ ∗ dV (t, Tj ) p ∗ = V (t) ξ (t, Tj ) + ξ (t, T ) dWn (t) , φ V (t, Tj ) Z (t, T ) =j+1

Z (t, T ) = V (t, T ) + Π V (t, T +1 ) ... + Πn−1 V (t, Tn ) , P in which n−1 =n [·] = 0 when j = n − 1. Under the spot measure P0 SDEs for the Z (·) are dZ (t, T@ ) = 0 and for j = @ (t) + 1, .., n p dZ (t, Tj ) = − V (t)b∗ (t, T@ , Tj ) dW0 (t) Z (t, Tj )   ¾ ½ j−1 X p V (t, T ) ∗ ξ (t, T ) dW0 (t) , = − V (t)  φ Z (t, T ) =@(t)

while corresponding SDEs for the V (·) for j = @ (t) , .., n are   ¾ ½ j X ) dV (t, Tj ) p V (t, T = V (t) ξ ∗ (t, Tj ) − ξ ∗ (t, T ) dW0 (t) , φ V (t, Tj ) Z (t, T ) =@(t)

Z (t, T ) = V (t, T ) + Π V (t, T

16.4.1

+1 ) ...

+ Πn−1 V (t, Tn ) .

Simulating V (t)

The behavior of square-root diﬀusions like our variance process V (t) p dV (t) = λ [µ − V (t)] dt + γ V (t)dU (t) , λ, µ, γ > 0 V (0) = 1,

Stochastic Volatility BGM

161

is well known, see Glasserman’s book [46] Section-3.4 for background and references. For example, the mean and long-term mean of V (t) are given by dEV (t) = λµ − λEV (t) dt

⇒

EV (t) = µ + (V (0) − µ) exp (−λt) → µ

while, after Ito plus manipulation, the variance and long-term variance are h i p d [V (t)]2 = 2V (t) λ (µ − V (t)) dt + γ V (t)dWt + γ 2 V (t) dt ¢¾ ¡ ½ γ2µ γ 2 2V (0) e−λt − e−2λt 2 2 ¢2 ¡ . → Var V (t) = E [V (t)] − [EV (t)] = −λt 2λ 2λ +µ 1 − e More specifically V (t) given V (u) behaves like the random variable ! Ã ¡ ¢ γ 2 1 − e−λ(t−u) 4µλ 4λe−λ(t−u) 2 ¡ ¢ V (u) V (t) = χν ν= 2 , 2 −λ(t−u) 4λ γ γ 1−e

where χ2ν (λ) is a non-central chi-square random variable with non-centrality parameter λ and ν degrees of freedom (which need not be an integer). Glasserman gives detailed methods of simulating V (t) whatever the (positive) values of λ, µ and γ, that is, whether the degree of freedom ν is integral or not. But if ν is an integer, the square root process V (t) can be expressed as the sum of squares of ν independent OU processes Xj (t) as follows. Starting with 1 1 dXj (t) = − λ Xj (t) dt + γdWi (t) , j = 1, .., ν 2 2 let V (t) = X12 (t) + X22 (t) + .. + Xν2 (t) , ⇒ ¸ · 2 ν X γ ν − V (t) dt + γ dV (t) = λ Xj (t) dWi (t) , ⇒ 4λ j=1 p f (t) , dV (t) = λ [µ − V (t)] dt + γ V (t)dW ν X γ2ν X (t) f (t) = pj . where dW dWi (t) and µ = 4λ V (t) j=1

So restrictions on ν can be exchanged for the relative ease of simulating several OU processes (rather than Gamma and Poisson distributions, see [46] Section3.4). In practice ν = 3, or somewhat larger, seems the only reasonable choice, as ν = 1 or ν = 2 make V (t) = 0 attainable (see Section-A.6). REMARK 16.2 Because rapid mean reversion may be required to get a suitable convexity profile, the time steps used to simulate V (t) may need to be quite small (for methods of taking large accurate steps see Broadie and Kaya [34]). Note that if the stochastic volatility parameters do not change frequently, simulated trajectories of V (t) can be prepared at leisure, tabulated and reused, because they are independent of the yieldcurve drivers.

162

16.5

Engineering BGM

Interpolation, Greeks and calibration

We very briefly mention some of the practical problems in making stochastic BGM operational.

16.5.1

Interpolation

As in Section-8.1, for 0 ≤ t ≤ T0 < T < T1 suppose K (t, T0 ) and K (t, T1 ) are known and K (t, T ) is required. Interpolate ξ (t, T ) on maturity T , defining 1 θ (T ) {(T1 − T ) ξ (t, T0 ) + (T − T0 ) ξ (t, T1 )} , δ = α (T ) ξ (t, T0 ) + β (T ) ξ (t, T1 ) , θ (T0 ) = 1 = θ (T1 ) ,

ξ (t, T ) = with

which preserves correlation between forwards at nodepoints and allows θ (·) to be chosen to satisfy some auxiliary condition. Recall Z tp Z 1 t K (t, T0 ) ∗ = V (s)ξ (s, T0 ) dWT1 (s) − V (s) |ξ (s, T0 )|2 ds, ln K (0, T0 ) 2 0 0 so if the forward measures PT and PT1 are identified with PT0 , clearly

ln

K (t, T0 ) K (t, T1 ) K (t, T ) ∼ + β (T ) ln = α (T ) ln K (0, T ) K (0, T0 ) K (0, T1 ) Z h i 1 t V (s) α (T ) |ξ (s, T0 )|2 + β (T ) |ξ (s, T1 )|2 − |ξ (s, T )|2 ds. + 2 0

The convexity term in this interpolation for K (t, T ) in terms of K (t, T0 ) and K (t, T1 ) is now, however, stochastic and must either be jointly simulated with the Glasserman V (t, T ) variable or made deterministic by approximating the variance process V (s) with its expected value EV (s) = µ + (V (0) − µ) exp (−λs) or long-term mean µ.

16.5.2

Greeks

Assume that the stochastic volatility parameters λ, µ, γ have been fixed to return a desired convexity profile along the caplet or swaption implied volatility surface, and they remain stable for a period of several months so they do not have to be hedged (though reserves against possible longer term changes may be needed). With the stochastic parameters λ, µ, γ constant, it is easy to see that the pathwise delta and vega methods already developed for shifted BGM in Chapters 11 and 13 work well with stochastic volatility BGM.

Stochastic Volatility BGM

163

For pathwise deltas, the relevant spot measure equation (11.2) to be partially diﬀerentiated becomes H (t, Tj ) =E H (0, Tj )

½Z

0

t

(0)

µj (t) dt +

¾ Z tp Vt ξ ∗ (t, Ti ) dW0 (t) , 0

(0)

µj (t) = Vt ξ ∗ (t, Tj )

where

j X

hj1 (0) ξ (t, Tj1 ) ,

j1=@

while the equivalent equation (11.3) under the terminal measure becomes ½Z t ¾ Z tp H (t, Tj ) (n) =E µj (t) dt + Vt ξ ∗ (t, Ti ) dWn (t) , H (0, Tj ) 0 0 (n)

µj (t) = −Vt ξ ∗ (t, Tj )

where

n−1 X

hj1 (0) ξ (t, Tj1 ) .

j1=j+1

In either case, partial diﬀerentiation of H (t, Tj ) with respect to H (0, T ), or equivalently K (0, T ) proceeds similarly to (11.4), while subsequent formulae for partial derivatives of swaps and options in terms of the forwards in Chapter-11 remain the same. For vegas, in (16.5), separately perturb by small ∆θ the shift α (Tj , TN ) α (Tj , TN )

→

(1 + ∆θ) α (Tj , TN ) ,

and by small ∆ε the instantaneous volatility NP −1 j=0

Aj ξ (t, Tj )

→

PN −1

(1 + ∆ε)

j=0

NP −1

Aj ξ (t, Tj )

Aj ξ (t, Tj )

j=0

of just the j th swaption pSwpn (t, κ, Tj , TN ). Corresponding changes in the swaption value are given by ∆θ pSwpn (0, κ, Tj , TN ) = pSwpn ((1 + ∆θ) α (Tj , TN )) − pSwpn (α (Tj , TN )) ,  ³ ´  pSwpn (1 + ∆ε) PN −1 Aj ξ (t, Tj )  j=0 ³P ´ , ∆ε pSwpn (0, κ, Tj , TN ) = N −1   − pSwpn A ξ (t, T ) j j j=0

in which the swaption values pSwpn (·) must now of course be computed by the Heston option methods of Section-16.3. Afterward proceed as in Chapter-13.

16.5.3

Caplet calibration

On the understanding that λ, µ, γ are fixed, we now outline a procedure to fit at-the-money caplet volatility and skew on a daily basis. Start by fitting

164

Engineering BGM

a shifted BGM model (that is, one without stochastic volatility) to the atthe-money volatilities β (T ) and the at-the-money skews ∂β (T ) of the target caplet implied volatility surface to obtain a shift a (T ) and forward volatility ξ (t, T ) calibration. If we perturb these functions in the following way that aﬀects only the Tj -maturing caplet a (Tj ) → (1 + ε) a (Tj )

ξ (t, Tj ) → (1 + θ) ξ (t, Tj ) ,

then clearly increasing ε increases the caplet’s at-the-money skew, while increasing θ increases its at-the-money volatility, and vice-versa. Step-1: Using a (T ), ξ (t, T ) and the fixed λ, µ, γ already determined, find the resultant secondary caplet implied volatility surface with at-the-money volatility β 1 (T ) and the at-the-money skew ∂β 1 (T ). Step-2: If ∂β 1 (Tj ) > ∂β (Tj ) reduce ε, but increase it if ∂β 1 (Tj ) < ∂β (Tj ), and if β 1 (Tj ) > β (Tj ) reduce θ, but increase it if β 1 (Tj ) < β (Tj ). Step-3: Iterate by returning to Step-1 with new perturbed functions a (T ) and ξ (t, T ) but the same fixed λ, µ, γ.

16.5.4

Swaption calibration

The process is similar to that for caplets. Start by fitting a shifted BGM model to the at-the-money volatilities β (T, TN ) and the at-the-money skews ∂β (T, TN ) of the target swaption implied volatility surface to obtain a shift a (T ) and forward volatility ξ (t, T ) calibration. If we perturb these functions using the technique developed for computing swaption vegas in Chapter-13 that aﬀects only the Tj -maturing swaption shift and implied volatility but leaves others unchanged, that is α (Tj , TN ) → (1 + ε) α (Tj , TN )

σ (t, Tj , TN ) → (1 + θ) σ (t, Tj , TN ) ,

then clearly increasing ε increases the swaption’s at-the-money skew, while increasing θ increases its at-the-money volatility, and vice-versa. Step-1: Using a (T ), ξ (t, T ) and the fixed λ, µ, γ already determined, find the resultant secondary swaption implied volatility surface with at-the-money volatilities β 1 (T, TN ) and the at-the-money skews ∂β 1 (T, TN ). Step-2: If ∂β 1 (Tj , TN ) > ∂β (Tj , TN ) reduce ε, but increase it if ∂β 1 (T, TN ) < ∂β (T, TN ), and if β 1 (T, TN ) > β (T, TN ) reduce θ, but increase it if β 1 (T, TN ) < β (T, TN ). Step-3: With the perturbed swaption shifts α (T, TN ) and volatilities σ (t, T, TN ) compute the new perturbed a (T ) and ξ (t, T ) using the vega ‘inversion technique’. Step-4: Iterate by returning to Step-1 with new perturbed functions a (T ) and ξ (t, T ) but the same fixed λ, µ, γ.

Chapter 17 Options in Brazil

This extra chapter, which addresses some of the theoretical aspects of Brazilian options, can be justified by some BGM applications, but has been added mostly because the content appeals to the author! At the outset, it should be emphasized that it may contain mistakes and biased interpretations due to the author’s relative unfamiliarity with the area (details of some instruments are a struggle, in particular, the system of contract dates). For more information, readers might like to refer to the English language website [26] for the interest rate section of the Bolsa de Mercadorias & Futuros (BMF), which is the main Brazilian Mercantile & Futures Exchange.

17.1

Overnight DI

Without ever having lived in Brazil, the author nevertheless imagines that frequent financial crises combined with a relatively benign (for South America) government attitude to financial markets, have naturally lead to the invention of safe and flexible ways of increasing, or at least maintaining, wealth over short time horizons. For that reason he is not surprised that the foundation for most Brazilian interest rate derivatives is the CDI rate or overnight DI (Deposito Interbancario) rate D (t), which is an annualized rate paying 1

[1 + D (t)] 252 over a one day period at time t. It is calculated and published daily, and represents the average rate of all inter-bank overnight transactions in Brazil: banks usually express their cost of funding as a percentage of the published CDI terms. The IDI index IDI (t) accumulates those daily payments from some start date n−1 Y 1 [1 + D (ti )] 252 , n = b252 × tc IDI (t) = i=0

165

166

Engineering BGM

resetting to 100,000 from time-to-time. The IDI index therefore behaves like a bank account, and so it can be reasonably modelled in HJM style by setting 1

[1 + D (t)] 252 = 1 + r (t) ∆ (t) ∼ = exp r (t) ∆ (t) , Z t IDI (t) = β (t) = exp r (s) ds,

(17.1)

0

where ∆ (t) is one day and r (t) is the spot interest rate, or, in BGM fashion, by rolling it up into consecutive zero coupon bonds that mature daily.

17.2

Pre-DI swaps and swaptions

These are standard over-the-counter deals tailored to the needs of the counterparties, with the convention (the author’s understanding) that the length of swaps is always a whole number of months (that is, deals begin and end on the same day of the month). By definition the Pre-DI payer swap struck at K and accumulated over the interval [T, T1 ], where T1 = δ + T , pays β (T1 ) IDI (T1 ) −K = −K IDI (T ) β (T ) at time T1 . The time t swaprate ω (t) = ω (t, T, T1 ) is that value of K which makes the swap’s time t value zero · ¸¯ ¾ ½ ¯ β (T1 ) 1 − ω (t) ¯¯ Ft = 0, E0 β (T1 ) β (T ) namely

1 B (t, T ) = , B (t, T1 ) FT (t, T1 ) where E0 is expectation under the spot measure P0 , B (t, T ) is the time t value of a zero coupon bond maturing at time T , and FT (t, T1 ) is a T -forward contract on the zero maturing at T1 . To make numbers comparable with the CDI rate, the convention is that the actual market quoted swaprate f (t, T, T1 ) and actual market quoted strike κ are obtained from ω (t, T, T1 ) and K by the unique one-to-one formulae ω (t) =

ω (t) = ω (t, T, T1 ) = [1 + f (t, T, T1 )]

T1 −T 252

,

K = [1 + κ]

T1 −T 252

.

(17.2)

Also, to develop models, we will need the notion of forward accrual A (t) over the interval [T, T1 ] A (t) = A (t, T, T1 ) = [1 + f (t, T, T1 )] 1 − 1, = ω (t) − 1 = FT (t, T1 )

T1 −T 252

−1

(17.3)

Options in Brazil

167

which is virtually (give or take a scaling factor) the simple forward over [T, T1 ]. Substituting back, the time t value of a payer swap (paying fixed and receiving floating) is therefore

(17.4) pSwap (t) = pSwap (t, T, T1 ) ¯ ¾ · ¸¯ ¾ ½ ½ ¯ ¯ β (T1 ) 1 1 − K ¯¯ Ft = E0 [ω (t) − K]¯¯ Ft , = E0 β (T1 ) β (T ) β (T1 ) = B (t, T1 ) [ω (t) − K] = B (t, T ) − K B (t, T1 ) , = B (t, T1 ) [A (t) − (K − 1)] ,

which has exactly the same form as any standard swap, except that there is just one exchange. The time t value of the corresponding payer swaption can be expressed in several ways

pSwpn (t) = pSwpn (t, T, T1 ) ¯ ¾ ½ ¯ B (T, T1 ) [ω (T ) − K]+ ¯¯ Ft = β (t) E0 β (T ) ¯ o n ¯ = B (t, T ) ET [1 − KFT (T, T1 )]+ ¯ Ft , ¯ o n +¯ = B (t, T1 ) ET1 [A (T ) − (K − 1)] ¯ Ft ,

(17.5)

where ET (respectively ET1 ) is expectation under the T -forward measure PT (respectively PT1 ). Similar expressions hold for receiver swaptions (17.6) rSwap (t) = rSwap (t, T, T1 ) = K B (t, T1 ) − B (t, T ) , = B (t, T1 ) [(K − 1) − A (t)] , ¯ o n ¯ rSwpn (t) = rSwpn (t, T, T1 ) = B (t, T ) ET [KFT (T, T1 ) − 1]+ ¯ Ft , ¯ o n ¯ = B (t, T1 ) ET1 [(K − 1) − A (T )]+ ¯ Ft . Note that these results are model independent, and that the options can be regarded as either caps/floors or swaptions because they are on swaps with just one exchange.

168

Engineering BGM

17.2.1

In the HJM framework

An SDE for the forward FT (t, T1 ) under PT is dFT (t, T1 ) = −FT (t, T1 )

ZT1

σ ∗ (t, u) du dWT (t)

⇒

T

FT (T, T1 ) = FT (t, T1 ) E

 Z 

T t

ZT1

 

σ ∗ (s, u) du dWT (s)

T



,

giving, from (17.5), (17.6) and the Black B (·) formula (A.2.3), the following HJM style swaption formulae pSwpn (t) = B (t, T ) B (1, K FT (t, T1 ) , ζ) , rSwpn (t) = B (t, T ) B (K FT (t, T1 ) , 1, ζ) , ¯2 ¯ ¯ Z T ¯ZT1 ¯ ¯ 2 ¯ ¯ ζ = ¯ σ (s, u) du¯ ds. t ¯ ¯

(17.7)

T

Note that in the flat Ho & Lee case that produces a Black option formula with implied volatility (T1 − T ) σ.

17.2.2

In the BGM framework

The SDE for the reciprocal of the forward under PT1 yields the following SDE for the accrual A (t) d

µ

1 FT (t, T1 )

¶

=

µ

1 FT (t, T1 )

¶ ZT1

σ ∗ (t, u) du dWT1 (t)

⇒

T

dA (t, T, T1 ) = γ ∗ (t, T, T1 ) dWT1 (t) , 1 + A (t, T, T1 ) if, in BGM style, the stochastic HJM volatility is chosen to satisfy (1 + A (t, T, T1 ))

ZT1

σ (t, u) du = A (t, T, T1 ) γ (t, T, T1 ) ,

T

where γ (t, T, T1 ) is deterministic. From (17.5), (17.6) and the Black B (·) formula (A.2.3), that gives the following BGM style formulae for the swaptions pSwpn (t) = B (t, T1 ) B (A (t, T, T1 ) , K − 1, ζ) , rSwpn (t) = B (t, T1 ) B (K − 1, A (t, T, T1 ) , ζ) , Z T 2 2 ζ = |γ (s, T, T1 )| ds. t

(17.8)

Options in Brazil

17.3

169

DI index options

These are an exchange traded options, maturing at the beginning of the months Jan, April, July, Oct and also the month following the current month. But as we will see, the volatilities of these options steadily decrease (to zero at maturity), which seems to make them diﬃcult to use in practice; consequently they are neither heavily traded nor used to hedge swaptions. The payoﬀ at time T for a DI Index option is the accumulated index starting from some reference time T ∗ , which is generally take to be zero. Hence DI call and put options struck at K will respectively have time t value ¯ ¾ β (t) +¯ [β (T ) − K] ¯¯ Ft , β (T ) " !#+ ¯¯  Ã Z  T ¯  ¯ Ft . = E0 r (s) ds β (t) − K exp − ¯   t ¯ " ! #+ ¯¯  Ã Z  T ¯  DIput (t, T ) = E0 r (s) ds − β (t) ¯¯ Ft . K exp −  t ¯ 

DIcall (t, T ) = E0

½

Note that the reference time eﬀectively scales the contract, because a later β(T ) reference time T ∗ would result in an option payoﬀ of β(T ∗ ) instead of β (T ). Getting an option formula is easy in the HJM case, but messy in the BGM framework and not attempted here.

17.3.1

In the HJM framework

An SDE for the zero coupon bond B (t, T ) is "

dB (t, T ) = B (t, T ) r (t) dt −

Z

#

T ∗

σ (t, u) du dW0 (t) ,

t

(17.9)

which has time T solution  

⇒

 ³R ´ T  B (t, T ) exp t r (s) ds ³ R R ´ , (17.10) B (T, T ) = 1 =  ×E − T T σ∗ (s, u) du dW0 (s)  t s Ã Z ! Ã Z Z ! T T T ∗ exp − r (s) ds = B (t, T ) E − σ (s, u) du dW0 (s) . t

t

s

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Engineering BGM

Hence, from (A.2.3), the time t values of the options are " #+ ¯¯    β (t) ³ R R ´ ¯¯ DIcall (t, T ) = E0 F T T ∗ ¯ t ,  −K B (t, T ) E − t s σ (s, u) du dW0 (s) ¯ = B (β (t) , K B (t, T ) , ζ) DIput (t, T ) = B (K B (t, T ) , β (t) , ζ) ¯2 Z T ¯¯Z T ¯ ¯ ¯ 2 ζ = σ (s, u) du¯ ds. ¯ ¯ t ¯ s

which are Black-Scholes expressions with volatilities contracting to zero. In the flat Ho & Lee case they yield the corresponding Black-Scholes implied volatility 1 √ (T − t) σ. 3

17.4

DI futures contracts

Futures contracts and options on them, are exchange traded instruments that mature at the beginning of the months Jan, April, July, Oct and also the month following the current month; hence underlying contracts may have 1,3,6 or 12 months to run. The options are often used to hedge OTC swaptions, because their dynamics are similar (give or take diﬀerences in maturity) and their volatilities well behaved. The time T maturing futures contract can be entered or exited at any time without cost, its numerical value equals that of a zero coupon bond B (t, T ) maturing at the same time, and the daily margin payments ∆MT (t) associated with it are, from (17.1) with ∆t one day, 1

∆MT (t) = B (t + ∆t, T ) − B (t, T ) [1 + D (t)] 252 , = B (t + ∆t, T ) − B (t, T ) [1 + r (t) ∆t] . In the continuous case the dynamics (17.9) of a zero coupon contract imply dMT (t) = dB (t, T ) − B (t, T ) r (t) dt Z T σ ∗ (t, u) du dW0 (t) , = −B (t, T )

(17.11)

t

that is, the daily margin payment equals the daily change in the stochastic part of the zero coupon bond. Hence the time t value of a DI futures contract (exiting at an arbitrary time T1 ≤ T ) being equal to the present value of the

Options in Brazil

171

margin payments E0 = −E0

(Z

t

T1

(Z

B (s, T ) β (s)

T1 t

Z

s

T

¯ ) dMT (s) ¯¯ ¯ Ft β (s) ¯

¯ ) ¯ ¯ σ (s, u) du dW0 (s)¯ Ft = 0, ¯ ∗

must be zero. This must be true, of course, because of the zero cost of entering and leaving the contract. REMARK 17.1 Although the DI contract delivers nothing (in contrast to a standard futures contract which delivers something even if cash settled), it is a futures contract in the sense that it costs nothing to enter or leave and settles at the margin. Also the margin payments can be duplicated by borrowing to purchase a zero coupon and financing it at the overnight rate. An alternative approach is to take the given properties of the contract and posit an index B (t, T ) which: 1. Is positive and converges to unity at its maturity T , that is B (T, T ) = 1. 2. Can be entered and exited freely with margin payments dMT (t) = dB (t, T ) − B (t, T ) r (t) dt that have zero present value. For the margin payments to have zero present value whatever entry and exit times MT (t) must be a P0 martingale (see Section-A.7.1), and so B (t, T ) satisfies an SDE like dB (t, T ) = r (t) dt + ξ ∗ (t, T ) dW0 (t) B (t, T ) for some (possibly stochastic) volatility function ξ (t, T ). The solution at maturity T ! Ã Z ! ÃZ T T ∗ r (s) ds E − ξ (s, T ) dW0 (s) , B (T, T ) = 1 = B (t, T ) exp ⇒

Ã Z B (t, T ) E −

t

T ∗

ξ (s, T ) dW0 (s)

t

⇒

!

B (t, T ) = E0

(

Ã Z exp −

t

T

t

Ã Z = exp −

t

T

r (s) ds ,

!¯ ) ¯ ¯ r (s) ds ¯ Ft , ¯

which, according to (1.7), is a zero coupon bond.

!

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Engineering BGM

17.4.1

Hedging with futures contracts

The futures contract can be used to directly hedge pre-DI swaps, because from (17.4), (17.9) and (17.11) d pSwap (t) = pSwap (t) r (t) dt + dMT (t) − K dMT1 (t) , and this result is independent of the model (no assumptions made about the HJM volatility which may be stochastic). In the HJM framework, using (A.3.8) we can hedge swaptions via d pSwpn (t) = pSwpn (t) r (t) dt + B (1, K FT (t, T1 ) , ζ) dMT (t) +∆B (1, K FT (t, T1 ) , ζ) [dMT1 (t) − FT (t, T1 ) dMT (t)] , and DI Index options via ¢ ¡ d DIcall (t, T ) = DIcall (t, T ) r (t) dt+∆B β (t) , K B (t, T ) , ζ 2 K dMT (t) . Similarly in the BGM framework we can hedge swaptions via

d pSwpn (t) = pSwpn (t) r (t) dt + B (A (t, T, T1 ) , K − 1, ζ) dMT1 (t) +∆B (A (t, T, T1 ) , K − 1, ζ) (17.12) × [dMT (t) − (1 + A (t, T, T1 )) dMT1 (t)] .

17.5

DI futures options

This is an option on a zero coupon bond maturing at time T1 with payoﬀs at time T like ¸+ ¸+ · · 1 1 = B (T, T1 ) − , payer (T ) = B (T, T1 ) − T1 −T K [1+κ] 252 · ¸+ · ¸+ 1 1 − B (T, T1 ) , receiver (T ) = = T1 −T − B (T, T1 ) K [1+κ] 252 T1 −T 252

. Hence from (17.3) ( · ¸+ ¯¯ ) 1 1 ¯ B (T, T1 ) − payer (t) = B (t, T1 ) ET1 ¯ Ft , B (T, T1 ) K ¯ ¯ o n 1 +¯ = B (t, T1 ) ET1 [(K − 1) − A (T )] ¯ Ft , K 1 = B (t, T1 ) B (K − 1, A (t, T, T1 ) , ζ) . K 1 receiver (t) = B (t, T1 ) B (A (t, T, T1 ) , K − 1, ζ) . K

where K = [1 + κ]

Options in Brazil

173

These formulae are almost the same as those for swaptions (17.8), and so hedges as similar to those given by (17.12).

Appendix A Notation and Formulae

A.1

Swap notation

This is a summary of the notation introduced in Section-2.2

j Tj (Tj−1 , Tj ] ∆j K (t, Tj ) L (t, Tj ) H (t, Tj ) δj µj N j0, .., jN i T i ¡ ¤ T i−1 , T i ∆i κi δi M i0, ..., iM r j = J (i)

floating index j = 0, 1, 2, ..... floating period time nodes j th floating side interval = Tj − Tj−1 width of j th interval cash forward rate = K (t, Tj ) + µj Libor = K (t, Tj ) + b (Tj ) shifted cash rate = δ (Tj ) = ∆j+1 coverage of K (t, Tj ) = µ (Tj ) margin for L (t, Tj ) number floating payments in a swap floating side index inside a swap fixed index i = 0, 1, 2, ..... fixed period time nodes ith fixed side interval = T i¡ − ¢T i−1 width of ¡ith interval ¤ = κ ¡ T i¢ coupon over T i , T i+1 = δ T i = ∆i+1 coverage for κi number fixed payment dates in a swap fixed side index in a swap the regular roll N = rM floating j ∼ = i on fixed side

175

176

Engineering BGM

A.2

Gaussian distributions

A.2.1

Conditional expectations

If the multi-dimensional vectors X and Y are jointly normally distributed, then (see [118]) X given Y is also normally distributed (X |Y ) ∼ N [E (X |Y ) , var (X |Y )] where ∗

cov (X, Y ) = E (X − EX) (Y − EY )

var (X) = cov (X, X)

and E (X |Y ) = E (X) + cov (X, Y ) var−1 (Y ) [Y − E (Y )] , var (X |Y ) = var (X) − cov (X, Y ) var−1 (Y ) cov (Y, X) .

A.2.2

Density shift

For a pair of jointly distributed normal random vectors X and Y with zero mean the following density shift formulae hold ¶ ¸ · µ 1 E exp b∗ X − var b∗ X f (Y ) = Ef (Y + cov (Y, X) b) , 2

(A.1)

and in particular if Y = X ∼ N (0, ∆)

¶ µ 1 E exp b∗ X − var b∗ X f (X) = Ef (X + ∆b) . 2

PROOF

(A.2)

The left hand side of (A.2) is

¶ µ 1 E exp b∗ X − var b∗ X f (X) 2 µ ¶ µ ¶ Z 1 1 1 ∗ −1 = x exp b∗ x − b∗ ∆b f (x) exp − ∆ x dx. 1 n 2 2 Rn |∆| 2 (2π) 2 Remembering that ∆ is symmetric, the exponential part simplifies like 1 1 1 ∗ − x∗ ∆−1 x + b∗ x − b∗ ∆b = − (x − ∆b) ∆−1 (x − ∆b) , 2 2 2

Notation and Formulae and so

177

¶ µ 1 E exp b∗ X − var b∗ X f (X) 2 µ ¶ Z 1 1 ∗ −1 = (x − ∆b) f (x) exp − ∆ (x − ∆b) dx, 1 n 2 Rn |∆| 2 (2π) 2 µ ¶ Z 1 1 ∗ −1 f (x + ∆b) − x ∆ x dx, = 1 n exp 2 Rn |∆| 2 (2π) 2 = Ef (X + ∆b) .

Conditioning on Y , the left hand side of (A.1) becomes · µ ¶ ¸ 1 ∗ ∗ E exp b X − var b X f (Y ) 2 ¾¸ · ½ 1 ∗ ∗ = E f (Y ) E exp (b X |Y ) − var (b X |Y ) . 2 But from Section-, b∗ X given Y is also normally distributed ( b∗ X| Y ) ∼ N [E (b∗ X |Y ) , var (b∗ X |Y )] where E (b∗ X |Y ) = b∗ cov (X, Y ) var−1 (Y ) Y, £ ¤ var (b∗ X |Y ) = b∗ var (X) − cov (X, Y ) var−1 (Y ) cov (Y, X) b, cov (b∗ X, Y ) = Eb∗ XY ∗ = b∗ cov (X, Y ) , so the inside conditional expectation simplifies to ½ ¾ ½ µ ¶ ¾ 1 1 E exp (b∗ X |Y ) − var (b∗ X |Y ) = E exp b∗ X − b∗ var (X) b |Y , 2 2 ½ ¾ 1 = exp b∗ cov (X, Y ) var−1 (Y ) Y − b∗ cov (X, Y ) var−1 (Y ) cov (Y, X) b , 2 ½ ¾ ¡ ∗ ¡ ¢ ¢ 1 = exp b cov (X, Y ) var−1 (Y ) Y − var b∗ cov (X, Y ) var−1 (Y ) Y . 2 Hence, applying (A.2) · µ ¶ ¸ 1 ∗ ∗ E exp b X − var b X f (Y ) 2 ¢ ¡ ∗ ¾ ¸ · ½ (X, Y ) var−1 (Y ) Y b cov ¡ ¢ f (Y ) , = E exp − 12 var b∗ cov (X, Y ) var−1 (Y ) Y n o £ ¤∗ = Ef Y + var (Y ) b∗ cov (X, Y ) var−1 (Y ) , = Ef (Y + cov (Y, X) b) .

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Engineering BGM

A.2.3

Black formula

Let X, Y be jointly normally distributed with zero mean and set ζ 2 = var (X − Y ). The Black formula is ¶ µ ¶¸+ · µ 1 1 , B = B (K, L, ζ) = E K exp X − var X − L exp Y − var Y 2 2 ¡ ¢ 1 2 ln K L + 2ζ , = KN (h) − LN (h − ζ) where h = ζ with its Greeks given by the partial derivatives ∂B ∂B ∂B = N (h) , = −N (h − ζ) , = KN0 (h) = LN0 (h − ζ) , ∂K ∂L ∂ζ 1 0 ∂2B 1 0 ∂2B N N (h − ζ) , = (h) , = 2 2 ∂K Kζ ∂L Lζ 1 1 0 ∂2B = − N0 (h) = − N (h − ζ) . ∂K∂L Lζ Kζ PROOF Using (A.1) ¶ µ ¶¸+ · µ 1 1 E K exp X − var X − L exp Y − var Y 2 2 © ª ¶· µ ¸+ 1 Kexp (X − Y ) − 12 var (X − Y ) = Eexp Y − var Y −L , × exp (var Y − cov (X, Y )) 2 © ª · ¸+ Kexp (X − Y ) − 12 var (X − Y ) =E −L , × exp (var Y − cov (X, Y ) − var Y + cov (X, Y )) ¾ ¸+ · ½ 1 = E Kexp (X − Y ) − var (X − Y ) − L , 2 and it is trivial to show this is equal to B (K, L, ζ) as defined above. The Greeks follow either by partially diﬀerentiating B (K, L, ζ) directly, or in a similar fashion to the derivation of the Black formula by diﬀerentiating partially under the expectation and simplifying using Section-A.2.2 and Heaviside and Dirac functions, see Section-A.5. Thus at time-t a Black caplet or floorlet fixed at T , paid at T1 , struck at κ with implied volatility σ, have the same zeta ζ given by ζ 2 = σ 2 (T − t) and their values are respectively

³ ´ √ cpl (t) = B (t, T1 ) B K (t, T ) , κ, σ T − t , ³ ´ √ flt (t) = B (t, T1 ) B κ, K (t, T ) , σ T − t .

Notation and Formulae

A.2.4

179

Gaussian density derivatives m

1

If Z :R × R 2 m(m+1) 7→ R is a Gaussian density in Rm µ ¶ 1 1 ∗ −1 x Z {x, R} = exp − R x , 1 m 2 |R| 2 (2π) 2 x = (xi ) R = (ri,j ) ri,j = rj,i , then derivatives of the density satisfy ∂Z {x, R} 1 ∂ 2 Z {x, R} = (i = j) ∂ri,i 2 ∂x2i

∂Z {x, R} ∂ 2 Z {x, R} = (i 6= j) . ∂ri,j ∂xi ∂xj

Note that we are not assuming Z {x, R} is a standard Gaussian density (though it can be), that is, we are not assuming the diagonal terms on R are 1s. −m

PROOF Introduce c = (2π) 2 , let A = R−1 with A = (aij ) be the corresponding covariance, and write µ ¶ 1 1 ∗ 2 Z = c |A| exp − x A x . 2 For any i, j, on or oﬀ the diagonal, partial diﬀerentiation yields ( ¡ ¡ ¢ ¡ ¢ ¢) ∂ 12 x∗ A x ∂ 12 x∗ A x ∂ 2 12 x∗ A x 1 ∂2Z = − , Z ∂xi ∂xj ∂xi ∂xj ∂xi ∂xj ( ¡ ¢) 1 ∂ 12 x∗ A x 1 ∂ |A| 2 1 ∂Z = − . 1 Z ∂ri,j ∂ri,j |A| 2 ∂ri,j The proof is completed by showing ¡ ¢ 1 ∂ 2 12 x∗ A x 1 ∂ |A| 2 = λi,j − 1 ∂xi ∂xj |A| 2 ∂ri,j ¡ ¡ ¢ ¡ ¢ ¢ ∂ 1 x∗ A x ∂ 12 x∗ A x ∂ 12 x∗ A x =− 2 λi,j , ∂xi ∂xj ∂ri,j λi,i = 2

where

and

λi,j = 1 if

(A.3) (A.4)

i 6= j. 1

1

Clearly the left hand side of (A.3) is −ai,j . Also, because |A| 2 = |R|− 2 1

1 1 1 ∂ |A| 2 1 ∂ |R| − 3 ∂ |R| = − |R| 2 = − |A| 2 . ∂ri,j 2 ∂ri,j 2 |R| ∂ri,j

But, if Ri,j is the signed conjugate of ri,j in R ( R i,j 2 |R| = 2ai,j if i 6= j 1 ∂ |R| , = Ri,i |R| ∂ri,j i=j |R| = ai,i if

180

Engineering BGM

and (A.3) follows. The left hand side of (A.4) is just the ith row of Ax multiplied by the j th row of Ax. On the other side, start with AR = Im , diﬀerentiate, multiply by A, and get ∂R ∂A = −A A = −A Ii,j A, ∂ri,j ∂ri,j where Ii,j is the m × m matrix with 1 in the (i, j) and (j, i) positions and 0 elsewhere. Hence   0 ¢ ¡  row i contains j th row Ax   ∂ 12 x∗ A x 1 ∗ 1 ∗ , 0 = x A Ii,j Ax = (xA)  −   ∂ri,j 2 2  row j contains ith row Ax  0 ½ (Ax)i (Ax)j if i 6= j, , = 2 1 if i = j. 2 (Ax)i and (A.4) follows. This result has a lot of unexpected applications. For example, in the next Section-A.2.5 it gives a connection between the gammas and vegas of a multiasset European option. Another example is a power series expansion for the bivariate Gaussian density in terms of univariate densities Z2 (x, y; ρ) =

∞ X Z(n) (x) Z(n) (y) n ρ . n! n=0

(A.5)

Repeated application of the result to the bivariate density Z2 (x, y; ρ) with x1 = x, x2 = y and r1,2 = ρ gives ∂ 2n ∂n Z (x, y; ρ) = Z2 (x, y; ρ) , 2 ∂ρn ∂xn ∂y n from which it follows, setting ρ = 0, that ¯ ¯ ¯ ¯ ∂n ∂ 2n ¯ Z2 (x, y; ρ)¯ = Z2 (x, y; ρ)¯¯ = Z(n) (x) Z(n) (y) . n n n ∂ρ ∂x ∂y ρ=0 ρ=0

A standard Taylors series expansion of Z2 (x, y; ρ) in terms of ρ, then yields (A.5). Note that this result can be easily generalized to multi-variate Gaussian densities.

Notation and Formulae

A.2.5

181

Gamma and vega connection

For a European option with payoﬀ g : Rm 7→ R and present value · ¶ ¶¸ µ µ 1 1 C = Eg S1 exp X1 − var X1 , .., Sm exp Xm − var Xm , (A.6) 2 2 where X = (Xi ) ∼ N (0, R) , R = (ri,j ) , ¶ ¶ µ µ 2 ∂C ∂ C if Λ = (Λi,j ) = Γ = (Γi,j ) = ∂ri,j ∂Si ∂Sj 1 2 Λi,j = Si Sj Γi,j (i 6= j) . then Λi,i = Si Γi,i (i = j) 2 Thus, for example, a standard Black-Scholes option struck at K, exercising at T , on a stock with present value S and volatility σ in the presence of constant interest rates r, would be given by ¶ ¸+ · µ 1 C = E S exp X − var X − K exp (−rT ) , 2 where

X ∼ N (0, R) ,

R = σ 2 T.

So from Section-A.2.3 setting ζ 2 = R, K = S and L = K exp (−rT ) ∂B = KN0 (h) , ∂ζ

Λ=

∂C = SN0 (h) , ∂R

Γ=

∂2B 1 N0 (h) . = ∂K 2 SR

Note that while our gammas are quite standard, our vegas are slightly different from the conventional ones in that they include oﬀ diagonal covariance terms, they include the time dependency or theta, and there are as many individual vegas Λi,j as gammas Γi,i . For an option over a single index it is quite easy to prove that Λ = 12 S 2 Γ, and for multiple indices this generalizes to Case i 6= j. Take e (xi ) to mean ¶ µ 1 e (xi ) = exp xi − ri,i , 2 and note that the ri,i do not enter calculations in this case. Starting from the definition (A.6), integrating twice by parts with respect to xi and xj , and using Section-A.2.4, we get Z Γi,j = g (i,j) [S1 e (x1 ) , S2 e (x2 ) , .., Sm e (xm )] e (xi ) e (xj ) N (x, R) dx, Rm Z 1 g [S1 e (x1 ) , S2 e (x2 ) , .., Sm e (xm )] N(i,j) (x, R) dx, = S S m i j R Z ∂ 1 g [S1 e (x1 ) , S2 e (x2 ) , .., Sm e (xm )] N (x, R) dx, = Si Sj Rm ∂ri,j 1 Λi,j . = Si Sj

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Engineering BGM

Case i = j. In this case the ri,i in the e (xi ) must be considered. Integrating twice by parts with respect to xi , the gamma is Z Γi,i = g (i,i) [S1 e (x1 ) , S2 e (x2 ) , .., Sm e (xm )] e2 (xi ) N (x, R) dx, m ZR 1 (i,i) (x, R) dx = 2 g [S1 e (x1 ) , S2 e (x2 ) , .., Sm e (xm )] N Rm Si Z 1 (i) g [S1 e (x1 ) , S2 e (x2 ) , .., Sm e (xm )] e (xi ) N (x, R) dx, − Rm Si while the vega is Z ∂ Λi,i = g [S1 e (x1 ) , S2 e (x2 ) , .., Sm e (xm )] N (x, R) dx ∂r m i,i R Z 1 g (i) [S1 e (x1 ) , S2 e (x2 ) , .., Sm e (xm )] Si e (xi ) N (x, R) dx, − 2 m R 1 2 = Si Γi,i . 2

A.2.6

Bivariate distribution

From Abramowitz [1], the bivariate normal distribution is defined by µ ¶ µµ ¶ µ ¶¶ X 0 1ρ ∼N , , Y 0 ρ1 Z a Z b N2 (a, b; ρ) = EI {X ≤ a} I {Y ≤ b} = Z2 (x, y; ρ) dxdy, −∞ −∞ ¾ ½ ¤ £ 2 1 1 2 p . x Z2 (x, y; ρ) = exp − − 2ρxy + y 2 (1 − ρ2 ) 2π 1 − ρ2

Using the properties of conditional expectations

N2 (a, b; ρ) = EI (X ≤ a) I (Y ≤ b) = E {I (X ≤ a) E [ I (Y ≤ b)| X]} , ( Ã Ã !) Z ! a b − ρX b − ρx = E I (X ≤ a) N p N p = N0 (x) dx. 1 − ρ2 1 − ρ2 −∞ The bivariate generator L2 is given by

dX (t) = ρdW1 (t) , dY (t) = ρdW1 (t) + µ ¶ ∂2 1 ∂2 ∂2 . + ρ + L2 = 2 ∂x2 ∂y 2 ∂x∂y

A.2.7

p 1 − ρ2 dW2 (t) ,

Ratio of cumulative and density distributions

This result is essential for inverting normal distributions in the tails; for example, when finding implied volatilities of short-dated options away from

Notation and Formulae

183

the money. Setting 1

F (x) = x +

,

2

x+

3

x+ x+

4 x + ...

then, see Abramowitz [1], N1 (x) = F (x) 1 − N (x)

A.2.8

x>0

N1 (x) = F (−x) N (x)

and

x < 0.

Expected values of normals

Let N (·) denote the standard normal cumulative distribution function, and introduce ¶ µ 1 dn 1 N(n) (x) = n N (x) , N0 (x) = N(1) (x) = √ exp − x2 , 2 dx 2π Z x N (u) du. Φ (x) = N(−1) (x) = −∞

Note that

¶ µ 1 1 N(2) (x) = −x √ exp − x2 = −xN(1) (x) and 2 2π Z x Z x N (u) du = xN (x) − uN(1) (u) du, Φ (x) = −∞ −∞ Z x = xN (x) + N(2) (u) du = N0 (x) + xN (x) . −∞

Combinations and adaptions of the following formulae are often useful. If X ∼ N (0, 1) then ¶ µ b √ , E N (aX + b) = N a2 + 1 ¶n ¶ µ µ b 1 (n) (n) √ E N (aX + b) = √ , N a2 + 1 a2 + 1 ¶ µ b a (1) (1) √ EX N (aX + b) = aEN (aX + b) = √ , N a2 + 1 a2 + 1 ¶ µ p b E Φ (aX + b) = a2 + 1Φ √ . 2 a +1 PROOF

To establish the first formula partially diﬀerentiate I (a, b) = EN (aX + b)

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Engineering BGM

with respect to b, simplify Z ∞ ∂b I (a, b) = N0 (ax + b) N0 (x) dx −∞ · Z ∞ o¸ 1 1n = exp − (ax + b)2 + x2 dx, 2π −∞ 2  )2  ( · ¸ Z ∞ p 1 b2 1 ab 1  dx, exp − 2 exp − (a2 + 1)x + p = 2π 2 a + 1 −∞ 2 (a2 + 1) ¸ · 1 1 b2 , =p exp − 2 2a +1 2π (a2 + 1)

and integrate back to get

¸ · 1 1 u2 p du + function (a) I (a, b) = exp − 2 (a2 + 1) 2π (a2 + 1) −∞ ¶ µ 1 b , because I (a, 0) = =N √ ⇒ function (a) = 0. 2 2 a +1 Z

b

The second formula follows by partially diﬀerentiating the first formula n times with respect to b. For the third formula EX N (aX + b) Z ∞ Z ∞ = N (ax + b) xN(1) (x) dx = − N (ax + b) N(2) (x) dx, −∞ −∞ Z ∞ = a N(1) (ax + b) N(1) (x) dx = aEN(1) (aX + b) . −∞

The final formula requires the second and third EΦ (aX + b) n o = E N(1) (aX + b) + aXN (aX + b) + bN (aX + b) , ¶ ¶¾ µ ½ µ p b b b (1) 2 √ +√ . = a +1 N N √ a2 + 1 a2 + 1 a2 + 1

Notation and Formulae

A.3 A.3.1

185

Stochastic calculus Multi-dimensional Ito

If f : R+ × Rn → R has a continuous first order partial derivative in t ∈ R+ and continuous second order partial derivatives in x ∈ Rn , and Xt is an ndimensional Ito process, then df (t, Xt ) =

∂ 1 ∗ f (t, Xt ) dt + [∇f (t, Xt )] dXt + trace ∇∇∗ f (t, Xt ) d hXit ∂t 2

where ∇ is the gradient operator, and hXit is the quadratic variation matrix ³D E´ hXit = X (i) , X (j) , i, j = 1, ..., n. t

A.3.2

Brownian bridge

Let Mt be a continuous Gaussian martingale with absolutely continuous quadratic variation qt Z t Z t Mt = γ (u) dWu , qt = γ 2 (u) du. 0

0

If 0 < s < t, then ( Ms | Mt ) is normally distributed µ ¶ qs qs Mt , [qt − qs ] , ( Ms | Mt ) ∼ N qt qt and the conditional expectation E { E (Ms )| Mt } = exp

A.3.3

½

qs 1 qs2 Mt − qt 2 qt

¾

.

Product and quotient processes

If Xt and Yt are positive Ito processes with SDEs dXt = µX dt + σ∗X dWt , Xt

dYt = µY dt + σ ∗Y dWt , Yt

their product and quotient processes are also Ito processes with SDEs d (XY )t ∗ = (µX + µY + σ ∗X σ Y ) dt + (σ X + σ Y ) dWt , (XY )t ¡ ¢ d X ¡ XY¢ t = [µX − µY − σ∗Y (σ X − σ Y )] dt + (σ X − σ Y )∗ dWt . Y

t

186

Engineering BGM For the product process, we have

PROOF

d (XY )t = Yt dXt + Xt dYt + d hX, Y it , so dividing by Xt Yt the result follows from d (XY )t dXt dYt d hX, Y it = + + = µX dt+σ ∗X dWt +µY dt+σ∗Y dWt +σ ∗X σ Y dt. (XY )t Xt Yt Xt Yt Applying Ito d ³

´

on dividing by

1 Yt .

d

1 Yt 1 Yt

µ

1 Yt

=−

¶

=−

1 1 dYt + 3 d hY it , 2 Yt Yt

⇒

¢ d hY it ¡ dYt + = −µY + σ 2Y dt − σ ∗Y dWt . 2 Yt Yt

The quotient result follows on writing µ ¶ X 1 = Xt Y t Yt

and using the product process result.

A.3.4

Conditional change of measure

Let P and Q be equivalent measures with respect to a σ-algebra F (Q =ZP or EQ (Y ) = EP (ZY )), and suppose Y is F-measurable. If G is a sub σalgebra of F. (G ⊂ F), then EQ { Y | G} = PROOF A∈G

A.3.5

EP { ZY | G} . EP { Z| G}

(A.7)

The right-hand side of (A.7) is G-measurable, and so for any ¾ ¾ ½ ½ EP { ZY | G} EP { ZY | G} = EP Z I (A) EQ I (A) EP { Z| G} EP { Z| G} = EP {I (A) EP ( ZY | G)} = EP {Z I (A) Y } = EQ {I (A) Y } = EQ {I (A) EQ { Y | G}} .

Girsanov theorem

dQ = ZT and Zt = EP {ZT |Ft } = E {Yt } (so dZt = Zt dYt , EP Zt = 1), dP then Mt is a P-martingale iﬀ Mt − hM, Y it is a Q-martingale. If

Notation and Formulae

187

ft = Mt − hM, Y i , from (A.7) and using Ito’s lemma Setting M t

PROOF

n ¯ o E { Z [M − hM, Y i ]| F } t t s t ft ¯¯ Fs = P EQ M EP { Zt | Fs } 1 = EP { Zt [Mt − hM, Y it ]| Fs } , Zs ¯ ) ( Rt ¯ 1 Z [dM − d hM, Y i ] ¯ u u u Rt 0 = EP ¯ Fs , Zs + 0 [Mu − hM, Y iu ] dZu + hZ, M it ¯ ¯ ¾ ½Z t Z t ¯ 1 EP Zu dMu + [Mu − hM, Y iu ] dZu ¯¯ Fs , = Zs 0 ·Z0 s ¸ Z s 1 = Zu dMu + [Mu − hM, Y iu ] dZu , Zs 0 0

because both Zt and Mt are P0 -martingales, and therefore so are stochastic integrals with respect to them. Hence EQ

n

Rs · ¸ ¯ o 1 Zu [dMu − d hM, Y iu ] ¯ 0 f R Mt ¯ Fs = s Zs + 0 [Mu − hM, Y iu ] dZu + hZ, M is 1 fs , Zs [Ms − hM, Y is ] = M = Zs

ft is a Q-martingale. A similar argument establishes necessity. showing M

If Wt is P-Brownian motion and Q =E ft where then Q-Brownian motion is W À ¿ Z · f Hs dWs Wt = Wt − W, 0

nR T

o Rt Hs dWs P so Yt = 0 Hs dWs ,

0

ft = dWt − Ht dt, dW

or

t

ft = dWt − Ht dt. and we can talk about the change of measure defined by dW Also, if Wt is n-component Brownian Dmotion, with E correlation between the i (i) (j) and j components given by ρi,j dt = d Wt , Wt , and Yt =

Z

t

Hs dWs =

0

Z tX n

Hs(i) dWs(i) ,

0 i=1

ft then the correlation ρi,j will appear in the expression for W ft (t) = dWt (t) + dW (i)

(i)

n X j=1

(j)

ρi,j Ht

dt

(i = 1, ..., n) .

188

A.3.6

Engineering BGM

One-dimensional Ornstein Uhlenbeck process

This is the one and only one-dimensional stationary Gauss Markov process. Constants Initial values SDE Solution Mean Autocorrelation Variance Conditional expectation Conditional variance

A (t) = −λ a (t) = 0 σ (t) = σ m (0) = x V (0) = 0 X0 = x dXt = −λX t ³t + σdW ´ R t λs −λt Xt = e x + 0 σe dWs m (t) = e−λt x ¡ ¢ 2 ρ (s, t) = σ2λ e−λt eλs − e−λs ¡ ¢ 2 V (t) = σ2λ 1 − e−2λt Us = E (Xt |Xs ) = e−λ(t−s) Xs ¢ 2 ¡ Var (Xt |Xs ) = σ2λ 1 − e−2λ(t−s)

The best way to estimate λ is by linear regression of ∆Xt against Xt .

A.3.7

Generalized multi-dimensional OU process

The multi-dimensional Ornstein Uhlenbeck process is the one and only Gauss Markov process. OU SDE dXt = [A (t) Xt + a (t)] dt + σ (t) dWt , 0 ≤ t < ∞ Introduce Φ0 (t) = A (t) h Φ (t)R, Φ (0) = In i Rt t Solution Xt = Φ (t) X0 + 0 Φ−1 (s) a (s) ds + 0 Φ−1 (s) σ (s) dWs h i Rt Mean m (t) = EXt = Φ (t) m (0) + 0 Φ−1 (s) a (s) ds m0 (t) = A (t) m (t) + a (t) Autocorr ρ (s, t) =h E {Xs − m (s)} {Xt − m (t)}∗ i R s∧t ∗ = Φ (s) ρ (0, 0) + 0 Φ−1 (u) σ (u) σ ∗ (u) Φ−1 (u) du Φ∗ (t) Variance V (t) = ρ (t, t) V 0 (t) = A (t) V (t) + V (t) A∗ (t) + σ (t) σ ∗ (t) Dist Xt ∼ N {m (t) , V (t)} Again the best way to estimate a constant mean reversion parameter A (t) = A, is by multi-linear regression of ∆Xt against Xt .

A.3.8

SDE of a discounted variable

Let f (t, Xt ) be an arbitrage free instrument in which the driver Xt is a diﬀusion with SDE dXt = µ (t, Xt ) dt + σ (t, Xt ) dW0 (t)

Notation and Formulae under the spot measure P0 . Because β (t) = exp

³R t 0

189 ´ r (s) ds is a finite varia-

t) tion process, f (t,X β(t) is a P0 martingale, and stochastic and drift parts on both sides of an equation must match, that is ¾ ½ f (t, Xt ) 1 ∂ = (expression) dW0 (t) = f (t, Xt ) σ (t, Xt ) dW0 (t) d β (t) β (t) ∂x ∂ f (t, Xt ) σ (t, Xt ) dW0 (t) . ⇒ df (t, Xt ) = f (t, Xt ) r (t) dt + ∂x

A.3.9

Ito-Venttsel formula

Let Wt be multi-dimensional Brownian motion, and suppose F (t, u) is twice diﬀerentiable with respect to the parameter u and has an SDE of form dF (t, u) = A (t, u) dt + B ∗ (t, u) dWt . If ut satisfies the SDE dut = C (t, ut ) dt + D∗ (t, ut ) dWt , then (see [127] and [110]) an SDE for F (t, ut ) is dF (t, ut ) = A (t, ut ) dt + B ∗ (t, ut ) dWt ∂ 1 ∂2 2 F (t, ut ) dut + F (t, ut ) |D (t, ut )| dt ∂u 2 ∂u2 ∂ ∗ B (t, ut ) D (t, ut ) dt. + ∂u

+

A.4 A.4.1

Linear Algebra Cholesky decomposition

This is a convenient way of constructing a set of normally distributed random variables with given variances and correlations, and is very useful for simulation. We give the 3-dimensional version; simply truncate to the first two components for two dimensions. Let U ∼ N (0, I3 ), then the vector X = ΓU , where   0 q 0 σ1  σ ρ σ 1 − ρ2  0  2 1,2 2  1,2 q   ¡ ¢ Γ= , 1 − ρ21,3 − ρ21,2 − ρ22,3 + 2ρ2,3 ρ1,3 ρ1,2    ρ2,3 − ρ1,3 ρ1,2  σ 3 ρ1,3 σ 3 q  q¡ σ3 ¢ 1 − ρ21,2 1 − ρ21,2

190

Engineering BGM

is distributed N (0, ∆), with covariance matrix    1 ρ1,2 ρ1,3 σ1 0 0 σ1 0 0 ∆ = ΓΓ∗ =  0 σ 2 0   ρ1,2 1 ρ2,3   0 σ2 0  . 0 0 σ3 0 0 σ3 ρ1,3 ρ2,3 1 

A.4.2

Singular value decomposition

A square n × n matrix V is orthogonal if V T V = In ; that is, its columns are pairwise orthogonal and form an orthogonal basis for Rn . It follows that its rows are also pairwise orthogonal V V T = In and form a basis for Rn , because ¡ ¢ V T V = In ⇒ V T V V T = V T ⇒ V T V V T − In = 0 ⇒ V V T − In = 0.

Let A be any real m × n matrix; that is, any one of m < n or m = n or m > n is permissible. If Av = wu, then w is called a singular value of A with corresponding singular vectors v and u. The singular value decomposition (SVD) algorithm produces the U, V and W matrices in the following theorem (see either [47], or [98] with a slightly diﬀerent notation for U and W ). THEOREM A.1 If A is a real m × n matrix, there exists orthogonal matrices U = [u1 , u2 , ..., um ] ∈ Rm×m ,

V = [v1 , v2 , ..., vn ] ∈ Rn×n

and a diagonal matrix W such that A = U W V T,

W = diag (w1 , w2 , ..., wp ) ∈ Rm×n ,

p = min {m, n} ,

where w1 ≥ w2 ≥ ..... ≥ wp ≥ 0. Some very practical uses for the SVD stem from the next theorem. THEOREM A.2 Suppose A = U W V T is the SVD of A ∈ Rm×n with r = rank A. If b ∈ Rm , then r X uTj b vj (A.8) x e= wj j=1

minimizes kAx − bk2 and has the smallest 2-norm of all minimizers. Moreover 2

kAx − bk2 =

m X ¡

j=r+1

¢2 uTj b .

Notation and Formulae

191

If A is not rank deficient, that is, r = rank A = min {m, n} (interestingly the SVD can identify and remedy such deficiencies, see [47]), then all the wj are strictly positive and we can form an n × m matrix ¶ µ 1 1 1 ∈ Rn×m , , ..., W −1 = diag w1 w2 wp by transposing W and inverting non-zero elements, so that m≤n

m≥n

⇒

⇒

W W −1 = Im , W −1 W = In .

The point here is that if m < n then W −1 W will have zeroes on the diagonal, and similarly for W W −1 when m > n. With this notation the solution (A.8) can be rewritten more revealingly as min{m,n}

x e=

X j=1

uTj b vj = A−1 b wj

where

A−1 = V W −1 U T ,

and the second theorem then says 1. If m = n then A−1 is both a left and right inverse to A AA−1 = U W V T V W −1 U T = U W In W −1 U T = U In U T = In , A−1 A = V W −1 U T U W V T = V W −1 In W V T = V In V T = In , and x e = A−1 b is the unique solution to Ax = b.

2. If m < n then A−1 is a right but not left inverse to A AA−1 = U W V T V W −1 U T = U W In W −1 U T = U Im U T = Im , A−1 A = V W −1 U T U W V T = V W −1 W V T 6= In , and x e = A−1 b is the particular solution to Ax = b with minimum 2-norm.

3. If m > n then A−1 is a left but not right inverse to A

AA−1 = U W V T V W −1 U T = U W W −1 U T 6= Im ,

A−1 A = V W −1 U T U W V T = V W −1 Im W V T = V In V T = In , so while x e does not solve Ax = b we can still compute A−1 Ax = x = A−1 b

which is the least squares best solution to Ax = b.

192

Engineering BGM

REMARK A.1 When A is an n × n positive definite matrix the SVD and eigenvalue decomposition coincide. To see that, note that A now has a Cholesky decomposition A = G GT where G is a lower triangular matrix. So taking the SVD of GT GT = U W V T ⇒

A.4.3

⇒

G = V W UT

A = G GT = V W U T U W V T = V W 2 V T .

Semidefinite programming (SDP)

Karmarkar’s 1984 polynomial time algorithm concentrated attention on interior point methods for linear programming (LP), and led to extensions to more general convex programs, see [130], [82], [125], [84] and [129]. Instead of tracing the edges of the feasible region to a minimum vertex as in the simplex method, interior point algorithms pursue the central path through a convex region to a solution, and so can be generalized to semidefinite programming (SDP), in which the variables are symmetric matrices lying in the convex cone of positive semi-definite matrices (in contrast to real numbers lying in the non-negative orthant in LP). Here are the basic concepts. Denote the space of real symmetric n × n matrices by © ª Sn = X ∈ Rn×n : X T = X ,

with the Frobenius inner product and F-norm k · kF defined by X • Y = trace(X T Y ) =

n X n X

Xij Yij

i=1 j=1

and

kXkF =

√ X • X.

Note that all matrices in Sn have real eigenvalues. That is because for any T eigenvalue λ with eigenvector X such that X X = 1 clearly A being real implies AX = λX,

AX = λX

⇒

T

X AX = λ,

X T AX = λ, T

and then transposing the second equation (A symmetric) ⇒ X AX = λ and subracting from the first gives λ = λ. A matrix X ∈ Sn is positive semidefinite, written X º 0, if any one of the following equivalent conditions hold: 1. For every vector w ∈ Rn , wT Xw ≥ 0. 2. Every eigenvalue of X is nonnegative: λi (X) ≥ 0, i = 1, . . . , n. 3. The Cholesky factorization X = LLT , where L ∈ Rn is lower triangular, exists.

Notation and Formulae

193

4. X has a square root Y ∈ Sn such that X = Y Y T . 5. X • Y ≥ 0 for every Y ∈ Sn , Y º 0. Importantly, a matrix X ∈ Sn is a covariance if and only if X is positive semidefinite. Suﬃciency comes from Condition-1 because if ζ ∈ Rn is a vector of correlated random variables with X = cov ζ, then n³ ´o ¯ ¯2 T = E ¯wT [ζ − Eζ]¯ ≥ 0. ∀ w ∈ Rn , wT Xw = E wT [ζ − Eζ] [ζ − Eζ] w Necessity comes from Condition-4 because if X is semidefinite matrix, the normal random variable Y ζ where X = Y Y T,

ζ ∼ N (0, Ir ) ,

r = rank (Y )

has covariance X. Semidefinite programming deals with optimization problems of the form find to minimize subject to and

X ∈ Sn C •X Ak • X = bk X º0

k = 1, . . . , K

for given C ∈ Sn and Ak ∈ Sn , bk ∈ R (k = 1, . . . , K). Note the objective function C • X and the K constraints Ak • X = bk are all linear functions of the variables X. The dual problem is find to maximise subject to

y ∈ Rm yT b P C− K k=1 yk Ak º 0

If both primal and dual have strictly feasible points (a stronger condition than is required in linear programming), then the primal and dual objective values are equal at a solution. LP is just a special case of SDP in which the matrices are diagonal X = diag(x) and Aj = diag(aj ) with x, aj ∈ Rm and X • Aj = aTj x for j = 1, .., J. Note also that several diﬀerent semidefinite matrices X1 , X2 , . . . , Xp can be accumulated into one larger block diagonal semidefinite matrix X = diag(X1 , X2 , . . . , Xp ). A standard problem which explicitly includes both symmetric positive semidefinite matrix variables and vectors of non-negative variables, therefore has form find to minimize subject to and

X ∈ Sn x ∈ Rm T C •X +c x Ak • X = bk k = 1, . . . , K aTj x = β j j = 1, . . . , J X º0 x≥0

(A.9)

194

Engineering BGM

where X,C and Ak k = 1, . . . , ms are symmetric Pp block diagonal matrices, each with p blocks of size n1 , . . . , np and n = i=1 ni , and aj ∈ Rm , β j ∈ R j = 1, . . . , J. SDP is closely related to eigenvalue optimization problems. If λ1 , λ2 , . . . , λn are the eigenvalues of X ∈ Sn , the simple objective function In • X = trace(X) =

n X

λi ,

i=1

where In is the n × n identity matrix, can be used to minimize the largest eigenvalue of X º 0 (and hence the size of X) subject to the constraints Ak • X = bk , k = 1, . . . , K, by solving find to minimize subject to and

X ∈ Sn ζ∈R ζ ζIn − X º 0 Ak • X = bk k = 1, .., K X º 0, ζ ≥ 0

This works because the eigenvalues of ζIn − X are ζ − λi (X), so ζIn − X º 0 is equivalent to ζ ≥ λi (X) ≥ 0 for i = 1, . . . , n (so really the constraint ζ ≥ 0 is redundant). This can then be converted into the standard form (A.9) by introducing a dummy matrix D = ζI − X º 0, constructing a 2n × 2n block diagonal matrix diag {X, D} ∈ Sn and then adding n2 standard equality constraints of type Ak • X = bk to ensure corresponding elements of D and ζI − X are the same. Now consider the problem of making a variable covariance matrix X close (in some sense) to a target covariance matrix G. The diﬀerence X − G is not generally positive semidefinite, so objective functions like trace(X − G) or largest eigenvalue of X − G are useless. Moreover, neither the Frobenius norm nor its square, can be used directly to force X close to G because kX − Gk2F = (X − G) • (X − G) is not linear in X. Nevertheless, X − G being symmetric still allows the 2−norm kX − Gk2 = max |λi (X − G)| i=1,...,n

to be minimized by adding two extra constraints and solving find to minimize subject to

and

X ∈ Sn ζ∈R ζ ζIn − (X − G) º 0 ζIn + (X − G) º 0 Ak • X = bk k = 1, .., K X º 0, ζ ≥ 0

Notation and Formulae

195

(convert to the standard form (A.9) by introducing dummy matrices D1 = ζI −(X −G) and D2 = ζI +(X −G)). This works because the extra constraints imply ζ ≥ λi (X − G) and ζ ≥ −λi (X − G) for i = 1, . . . , n ⇒ ζ ≥ max {λi (X − G), −λi (X − G)} = kX − Gk2 . i=1,...,n

Note that the constraint ζ ≥ 0 is in fact implied by the positive semidefinite constraints. A more diﬃcult problem is to find the best covariance matrix X with rank(X) specified.

A.5

Some Fourier transform technicalities

Some useful Fourier Transform results are presented in this section, along with some non-rigorous outlines for their derivation. Using probablistic convention, the Fourier transform fb of the function f and the inverse Fourier transform are defined by the pair Z ∞ Z ∞ b f (y) exp (2πixy) dy, f (y) = fb(x) exp (−2πixy) dx. f (x) = −∞

−∞

(A.10) Standard Fourier theory (involving an isometry plus Cauchy sequences, etc) demonstrates that if fb exists then its inverse must be f . Moreover, transforms and inverses are interchangeable in the sense that Z ∞ Z ∞ f (y) exp (2πixy) dy = g (x) ⇒ f (x) exp (−2πixy) dx = g (−y) . −∞

−∞

Hence to establish a Fourier transform pair, we need only obtain fb from f , or f from fb. Two important functions in Fourier analysis are the Dirac delta function δ (·) (defined as a probability measure concentrated at 0, which makes it even), and the Heaviside function I (·) (which steps from 0 to 1 at x = 0)  ½ Z b  1 when x > 0 1 when 0 ∈ (a, b) δ (y) dy = , I (x) = 12 when x = 0 . 0 when 0 ∈ / (a, b)  a 0 when x < 0 (A.11) Integrating these functions yields expressions for their derivatives Z

x −∞

+

I (u) du = (x) ⇒

d (x)+ = I (x) , dx

Z

x

−∞

δ (u) du = I (x) ⇒

dI (x) = δ (x) . dx

196

Engineering BGM

Note I (0) =

1 2

is consistent with δ (·) being even, and diﬀerentiating (x)+ = x I (x) ⇒ I (x) = I (x) + xδ (x) ⇒ xδ (x) = 0,

that is, the zero in x so to speak dominates the infinity in δ (x) around x = 0. [1] Some common Fourier transform pairs are f (y) fb(x) R∞ R∞ b = −∞ f (x) exp (−2πixy) dx = −∞ f (y) exp (2πixy) dy 1

δ (y) 1 1 δ (y) + 2 2πiy

I (x)

1 [δ (y + a) + δ (y − a)] 2 1 i [δ (y + a) − δ (y − a)] 2 exp (−2πia y) 1 a 2 2 r π y µ+ a 2 2 ¶ π y π exp − a a

cos (2πax) sin (2πax) δ (x − a)

exp (−2πa |x|) ¢ ¡ exp −a x2

[2] A useful contour integral is Z

∞ −∞

1 exp (2πixy) dx = I (y) − . 2πix 2

(A.12)

PROOF Integrating counterclockwise around©the contour C consisting of ª iθ θ ≤ 0 , part of part the real axis (−R, −r), the small semicircle re : −π ≤ ª © the real axis (r, R), and the large semicircle Reiθ : 0 ≤ θ ≤ π , we have I

C

exp (2πizy) dz = 0. 2πiz

When y = 0, breaking down the integral into its component parts, the result follows on letting r → 0 and R → +∞ in Z

−r −R

1 dx + 2πix

Z

0

π

1 ireiθ dθ + 2πi reiθ

Z

R r

1 dx + 2πix

Z

π 0

1 iReiθ dθ = 0. 2πi Reiθ

When y > 0, on the large semicircle exp (2πizy) dz exp (2πiy cos θ R − 2πy sin θ R) R eiθ dθ = lim = 0, R→∞ R→∞ z R eiθ lim

Notation and Formulae

197

so breaking down the integral into its component parts, the result follows on letting r → 0 in ¢ ¡ Z ∞ Z 0 Z −r exp 2πiy reiθ exp (2πixy) exp (2πixy) dx + dx + ireiθ dθ = 0. 2πix 2πix 2πi reiθ −∞ r π

When y < 0, changing the variable in the integral by setting s = −x and then integrating similarly to the case y > 0, clearly gives Z ∞ Z ∞ 1 exp (2πixy) exp (−2πisy) dx = − ds = − . 2πix 2πis 2 −∞ −∞

[3] Another useful contour integral is Z ∞ Z ∞ exp (−2πixy) dx = exp (2πixy) dx = δ (y) , −∞ −∞ Z ∞ ⇒ I0 (y) = exp (2πixy) dx = δ (y) .

(A.13)

−∞

PROOF Using the definition δ (·) and integrating over (a, b) with respect to y, we need to show ¸ ½ Z ∞· exp (2πixb) exp (2πixa) 1 if a < 0 < b, − dx = , 0 otherwise 2πix 2πix −∞ which follows directly from (A.12). [4] This integral is useful for options: Z s exp (2πiy s) 1 . exp (2πixy) dx = δ (y) + 2 2πiy −∞ PROOF gives

(A.14)

Partial diﬀerentiation of the integral (A.14) with respect to s exp (2πisy) = exp (2πisy) ,

and shows that for y 6= 0 the integral must have form Z s exp (2πiy s) , exp (2πixy) dx = e (y) + io (y) + 2πiy −∞ and it remains to identify the functions e (·) and o (·). Equating real and imaginary parts Z s sin (2πy s) , cos (2πxy) dx = e (y) + 2πy −∞ Z s cos (2πy s) , sin (2πxy) dx = o (y) − 2πy −∞

198

Engineering BGM

indicates that e (·) must be an even and o (·) an odd function for y 6= 0. The changes of variables x=u+

1 , 4y

s=r+

1 , 4y

(y 6= 0) ,

in the second integral, yield Z r sin (2πy r) , cos (2πyu) du = o (y) + 2πy −∞ which on comparison with the first integral establishes o (y) = e (y) = 0 for y 6= 0. Hence the left hand side of (A.14) has form Z s exp (2πiy s) , exp (2πixy) dx = h (y) + 2πiy −∞

(A.15)

where h (·) is an even function concentrated at 0. But reversing the signs of x, y and s in this integral (A.15) gives Z ∞ exp (2πiy s) , exp (2πixy) dx = h (−y) − 2πiy s which, on addition to the original integral (A.15) and use of (A.13), yields the result because Z ∞ exp (2πixy) dx = h (y) + h (−y) = 2h (y) = δ (y) . −∞

[5] An integral along the positive real line yields the reciprocal function: Z ∞ 1 1 . I (x) exp (2πixy) dx = δ (y) − 2 2πiy −∞ PROOF

A.6

Set s = 0 in (A.14) and subtract from (A.13).

The chi-squared distribution

For ν a positive integer, the central χ2 -distribution is defined by the random variable ν X χ2ν = Xi2 where the Xi ∼ N (0, 1) are IID, i=1

Notation and Formulae

199

and the non-central χ2 -distribution by the random variable χ2ν (λ) =

ν X

(Xi + ai )

2

where λ2 =

i=1

and the

ν X

a2i

i=1

Xi ∼ N (0, 1)

are IID.

Note that these definitions can be extended to non-integer values of ν, for example see [46]. 1. The distribution of χ2ν is given by ¡ ¢ P χ2ν ≤ y =

1 ¡ ¢ 2 Γ ν2 ν 2

Z

y

z

ν

e− 2 z 2 −1 dz, 0

(go to spherical polar coordinates in ν-dimensions to prove that). 2. Only the norm λ = |a| is relevant (and not the components ai of a) in the distribution of χ2ν (λ) because Ã ν ! X ¡ 2 ¢ 2 P χν (λ) ≤ y = EI (Xi + ai ) ≤ y Ã ν X

i=1

!

Ã

! ν 1X 2 = I (xi + ai ) ≤ y exp − x dv ν 2 i=1 i (2π) 2 Rν i=1 Ã ν ! Ã ! Z ν X 1X 1 2 2 I xi ≤ y exp − (xi − ai ) dv = ν 2 i=1 (2π) 2 Rν i=1 µ ¶ Z √y Z ¢ √ 1 1¡ 2 2 + r = exp − + λ λr.n drdS, ν 2 (2π) 2 r=0 U 1

Z

2

where U is the surface of the unit sphere, and n is a unit vector in the direction of a = (a1 , .., aν ). Hence, rotating axes to a new x1 -axis that lies along n µ ¶ Z √y Z ¢ √ ¢ ¡ 1 1¡ 2 2 + r exp − + λ λx drdS P χ2ν (λ) ≤ y = ν 2 (2π) 2 r=0 U ! Ã µ³ ¶ ν ³ √ ´2 X √ ´2 2 2 Xi ≤ y = P X1 + λ + χν−1 ≤ y . = EI X1 + λ + i=2

3. Cumulative non-central 2-degrees of freedom. The distribution is given by µ³ ¶ √ ´2 ¢ ¡ P χ22 (λ) ≤ y = P X + λ + Y 2 ≤ y , 1 = 2π

Z

√ y

r=0

µ ¶ ½Z 2π ¾ ³ √ ´ ¢ 1¡ 2 exp − r + λ exp r cos θ λ dθ rdr, 2 θ=0

200

Engineering BGM producing the density ¢ d ¡ 2 P χ2 (λ) ≤ y dy µ ¶ Z 2π ³p ´ 1 1 exp − (y + λ) exp yλ cos θ dθ. = 4π 2 0 f (y) =

Hence during simulation, χ22 (λ) may frequently approach 0 because µ ¶ 1 1 f (0) = exp − λ > 0, 2 2 indicating ν = 2 is not a good choice for the χ2ν (λ) process. 4. Cumulative non-central 3-degrees of freedom. The distribution is given by µ³ ¶ √ ´2 ¢ ¡ 2 2 2 P χ3 (λ) ≤ y = P X + λ + Y + Z ≤ y , =

1

3

(2π) 2

Z

√ y

r=0

µ ¶ √ ¢ 1¡ 2 2 exp − r + λ + r λ cos θ r2 sin θdrdθdφ, 2 U

Z

going to polar coordinates and making U the surface of the unit sphere. Hence ¢ ¡ P χ23 (λ) ≤ y ¢¶ µ 1¡ 2 Z √y Z π 2π − 2 √r + λ2 r2 sin θdrdθ = P (y, λ) exp = 3 λ cos θ +r 2 (2π) r=0 0 ³ √ ´  ¶ µ Z √y − exp −r λ ¢ ¡ 1 1 ³ √ ´  r dr, exp − r2 + λ2  =√ 2 2π λ r=0 + exp r λ ´ ´ ³ ³ √ √ 1 1 √ √ y + λ − √ N0 y− λ = √ N0 λ λ ³√ ³√ √ ´ √ ´ +N y+ λ +N y − λ − 1,

with density

¢ d ¡ 2 P χ3 (λ) ≤ y f (y) = dy ³ ³√ √ ´ √ ´o 1 n 0 √ y + λ + N0 y− λ . = √ −N 2 λ So f (0) = 0 indicating ν = 3 is a better choice for a χ2ν (λ) to simulate.

Notation and Formulae

A.7 A.7.1

201

Miscellaneous Futures contracts

A contract settled at the margin (like, for example, a futures contract) is entered into at zero cost, is settled daily as the underlying index Gt changes, and can generally be exited at zero cost. Hence the present value of the payout stream from some time s to any time t generated by Gt ½Z t ¾ dGu |Fs = 0, E0 s β (u) must be zero in order for the arrangement to be arbitrage free. So for any s, t Z t dGu ⇒ E0 {Mt |Fs } = Ms Mt = β (u) 0 making Mt , and any integral with respect to it, a P0 -martingale. In particular the underlying index Z t

Gt =

β (u) dMu

0

must be a P0 -martingale. Hence the time t value GT (t) of a futures contract settling to physical g (T ) at time T (that is, GT (T ) = g (T )) is GT (t) = E0 {g (T ) |Ft } .

A.7.2

Random variables from an arbitrary distribution

Suppose the random variable Y has density g (y) on the interval [a, b]. Then G : [a, b] 7→ [0, 1] defined by the cumulative density function Z y g (u) du, G (y) = a

is one-one and onto and so invertible. To generate samples of Y , first generate samples X from the uniform distribution X ∼ U [0, 1], and then set Y = G−1 (X). The point is that £ ¤ P [Y ≤ y] = P G−1 (X) ≤ y = P [X ≤ G (y)] = G (y) .

A.7.3

Copula methodology

Consider two random variables X and Y whose marginal distributions e and Ye deFX (·) and FY (·) are known. Introduce new random variables X fined by e = N−1 FX (X) Ye = N−1 FY (Y ) . X

202

Engineering BGM

e and Ye have Gaussian distributions because By construction both X h i £ ¤ £ ¤ e ≤ x = P N−1 FX (X) ≤ x = P X ≤ F −1 (N (x)) P X X ¡ −1 ¢ (N (x)) = N (x) . = FX FX

e and Ye is The Gaussian copula hypothesis is that the joint distribution of X also Gaussian with density Z2 (•, •; ρ) in which the correlation ρ can be chosen to satisfy some other modelling requirement (for example, transform the X, Y e Ye data). That allows an expectation data and estimate ρ on the derived X, involving both X and Y to be computed according to ³ ³ ´´ ³ ³ ´´´ ³ −1 e , F −1 N Ye N X Ef (X, Y ) = Ef FX Y ³ ´ Z e Ye = g (x, y) Z2 (x, y; ρ) dxdy = Eg X, R2

Obviously, distributions other than Gaussian can be used in a similar way to produce diﬀerent copulas.

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Index

absolute maturities, 46 aﬃne system, 155 alive (T dependent function), 19 att @(·) function, 19

Black formula B(), 178 Greeks, 178 zeta, 178 bonds, 11 bond volatility diﬀerence, 13 forward measure, 13 real world measure, 12 zero coupon, 11 Brazilian options, 165 Bolsa de Mercadorias & Futuros (BMF), 165 CDI rate, 165 DI futures contracts, 170 DI futures options, 172 DI index options, 169 IDI index, 165 overnight DI, 165 overnight DI (Deposito Interbancario) rate, 165 Pre-DI swaps and swaptions, 166 Brownian bridge, 185

backward construction, 24 basic assumption, 24 terminal measure, 26 backward recursion, 104 alternative method, 106 continuation value, 105 discounted Bermudan, 105 intrinsic value, 105 Bermudans, 103 backward recursion, 104 callable exotics, 104 callable swap, 103 deltas, 111 exotic coupons, 103 lower bounds, 106 payer Bermudan, 104 payers and receiver, 103 upper bounds, 110 Big-step methods, 79 big-step simulation, 81 drift approximation, 82 under a tailored spot measure, 84 under the terminal measure, 84 volatility approximation, 81 bivariate normal, 182 Black caplet, 6, 22 Black caplet, first correct, 6 heat equation, 7 lognormal type, 6 Miltersen, Sandmann and Sondermann, 6 probablistic proof, 7

calibration techniques, 55 absolute strike, 55 bi-exponential function, 60 caplet skew, 57 cascade, 69 fitting the skew, 57 homogeneous spines, 59 implied volatility matrix, 55 interpolated swaption volatilities, 55 market objects, 55 maturity only fit, 58 Pedersen’s method, 66 piecewise linear, 59 Rebonato’s function, 60 relative delta strike, 55

212

References semidefinite programming, 71 separable multi-factor, 63 separable one-factor, 61 stripped caplet implied volatilities, 55 sum of exponentials, 60 swaption skew, 57 volatility cube, 55 callable range rate accruals, 63 cash forward, 4, 14 central interest rate model, 28 chi-squared distributions, 198 central, 198 non-central, 199 Cholesky decomposition, 189 CME futures contract on inflation, 146 conditional change of measure, 186 consumer price index, 141 correlation function, 46, 134 correlation matrix, 46 coupon, 15 coupon bond, 11 covariance estimator, 48, 50, 52, 133 coverage, 15 cross-economy BGM, 121 correlation, 131 correlation functions, 134 cross-economy covariance, 133 cross-economy models, 126 distant maturity FX volatility, 136 forward contracts, 123 FX spot volatility deterministic model, 128 HJM preliminaries, 121 implied forward correlation, 134 implied forward covariance, 134 interest rate parity, 123 measure change, 126 Pedersen objective function, 137 Pedersen type calibration, 135 volatility parity in BGM, 126, 127 volatility parity in HJM, 124

213

cross-economy correlation, 131 cross-economy covariance, 133 current interval, 19 current swaprate, 17 current time, 19 dead forwards, 76 default calibration, 58 deltas of barrier caps and floors, 100 deltas of forwards, 96 deltas of vanilla caplets and swaptions, 99 deltas of zeros and swaps, 97 DI futures contracts, 170 DI futures options, 172 DI index options, 169 die (T dependent functions), 19 die-at-exercise convention, 116 diﬀerentiating option payoﬀs, 98 Dirac delta function, 98 Heaviside function, 98 postive value function, 98 Dirac delta function, 195 discounted option deltas, 95 displaced diﬀusion, 21 distant maturity FX volatility, 136 drift approximation, 88 Fisher equation, 143 fixed side, 15 fixed time nodes, 15 floating side, 14 floating time nodes, 14 forward BGM construction, 8 Black formula, 9 forward constructed, 9 terminal measure, 9 forward contract, 4 SDE for contract, 5, 14 SDE for reciprocal, 5, 14 forward contract on foreign zero, 123 forward contract on FX, 123 forward contract on inflation, 143 forward inflation curve, 143 forward measure, 4, 13

214

References

forward swap rate, 17 forward value factor, 42 forwards alive, 9 forwards dead, 9 Fourier transforms, 195 common transform pairs, 196 inverse Fourier transform, 195 futures contract on inflation, 145 futures contracts, 201 FX spot volatility deterministic model, 128 simulation, 129 Gaussian copula, 202 Gaussian density, 179 Gaussian distributions, 176 bivariate distribution, 182 bivariate generator, 182 Black formula, 178 conditional expectations, 176 density shift, 176 expected values of normals, 183 gamma and vega connection, 181 Gaussian density derivatives, 179 ratio of cummulative and density distributions, 182 generic calibration, 56, 66, 69 Girsanov theorem, 186 Glasserman type methods, 79 Glasserman type simulation, 79 under the spot measure, 80 under the terminal measure, 80 Heaviside function, 195 hedge pair for vega and shift, 114 historical correlation, 45 average shift, 47 bestfit, 45 filmshow, 45 flapping of the long end, 45 implied forward covariance, 52 lack of smoothness, 45 oﬀ forwards, 48 oﬀ swaprates, 50 oﬀ yield-to-maturity, 49

phantom principle components, 47 super-smooth, 45 swap-world, 47 target, 45 historical correlation function, 56 HJM cross-economy, 121 domestic dollars, 121 foreign arbitrage free measure, 123 foreign bank account, 122 foreign forward rate, 121 foreign spot rate, 122 foreign zero coupon bonds, 122 foreign zlotty, 121 spot FX rate, 121 HJM domestic model, 2 arbitrage free measure, 2 instantaneous forward rate, 2 model free result, 4 numeraire bank account, 2 present value, 3 spot rate, 2 volatility function, 2 zero coupon, 2 homogeneous volatility, 46 implied forward correlation, 134 implied forward covariance, 134 indicative prices, 56 inflation, 141 CME futures contract, 146 consumer price index, 141 Fisher equation, 143 forward contract on inflation, 143 forward inflation curve, 143 futures contracts, 145 inflation rate, 143 TIPS and the CPI, 141 zero coupon CPI swap rate, 142 inflation rate, 143 ingredients of HJM, 2, 121 initial discount vector, 15 instantaneous correlation, 46, 56

References instantaneous forward FX volatility, 124 interest rate parity, 123 interpolating between nodes, 75 consistent volatility, 78 dead forwards, 76 interpolating forwards, 75 interpolation of discount factors, 77 stochastic volatility BGM, 162 interpolating forwards, 75 interpolation of discount factors, 77 intial cash forward vector, 15 intuition behind BGM, 8 Ito-Venttsel formula, 189 jumping measure, 44 level function, 16, 35 Libors over several periods, 18 Longstaﬀ’s LB technique, 106 comments, 109 continuation value, 108 regression technique, 108 when to exercise, 107 margin, at the, 201 Markovian projection, 27 measure change, 22, 126, 130, 150, 187 multi-dimensional Ito, 185 multi-period caplet, 35 Ornstein Uhlenbeck multi-dim, 188 Ornstein Uhlenbeck one-dim, 188 overnight DI, 165 pathwise deltas, 95, 163 barrier caps and floors, 100 bump and grind, 95 deltas of forwards, 96 deltas of zeros and swaps, 97 diﬀerentiating option payoﬀs, 98 discounted option deltas, 95 option deltas, 95 pathwise method, 95

215

swap deltas in terms forward deltas, 98 vanilla caplets and swaptions, 99 payer swap, 15 payer swaption, 34 Pedersen, 66 cross-economy fit, 135 exact fit by cascade, 71 objective function, 67, 137 pseudo-homogeneous volatility function, 66 Pre-DI swaps and swaptions, 166 forward accrual, 166 payer swap, 167 payer swaption, 167 receiver swaptions, 167 swaprate, 166 product and quotient processes, 185 properties of measures, 39 forward and swaprate measure changes, 40 jumping measure, 44 spot Libor measure, 42 terminal measure, 41 pseudo-bank account, 42 receiver swap, 16 relative forward, 45 relative FX forward contract, 133 relative maturity, 45 relative maturity zero coupon bond, 50 relative swaprate, 50 SDP calibration, 71 homogenous by layer, 71 implied correlation, 71 optimization problem, 73 pseudo-homogenous volatility function, 71 semidefinite programming SDP, 192 central path, 192 convex cone, 192 covariance, 193 F-norm, 192

216

References

Frobenius inner product, 192 interior point methods, 192 positive semi-definite matrices, 192 positive semidefinite, 192 real symmetric, 192 shift part hedge, 115 shifted BGM, 21 Black caplet, 22 drift term, 22 driver, 21 measure change, 22 shift, 21 shifted forward, 21 shifted heston options, 155 simulation, 79 big step, 81 Glasserman type, 79 singular value decomposition, 190 orthogonal, 190 singular value, 190 singular vectors, 190 spot Libor measure, 42 pseudo-bank account, 42 stochastic volatility BGM, 149 backward construction, 149 caplet calibration, 163 caplet value, 154 characteristic function, 155 Greeks, 162 interpolation between nodes, 162 measure change, 150 option price as a Fourier integral, 158 pathwise deltas, 163 shifted heston options, 155 simulating the variance, 160 simulating under spot measure, 160 simulating under terminal measure, 160 swaprate dynamics, 153 swaprate measure, 153 swaption calibration, 164 swaption value, 154

terminal measure, 152 variance process, 149 vegas, 163 swap notation, 175 swap rate, 17 swaprate covariance estimator, 52 swaprate dynamics, 27 approximation to shift part, 30 in stochastic volatility BGM, 153 level function, 35 multi-period caplets, 35 present value swaption, 35 shift part of swaprate, 28 stochastic part, 31 stochastic part approximation, 34 stochastic part of swaprate, 28 swaprate measure, 30 swaprate models, 36 swaprate SDE, 33 swaprate volatility, 32 swaption implied volatility, 35 swaption shift, 35 swaption values, 34 swaption zeta, 35 swaprate measure, 30, 153 swaprate models, 36 coterminal swaps, 37 Jamshidian’s models, 36 total tenor structure, 37 swaps, 14 att @(·) function, 19 coupon, 15 coverage, 15 current interval, 19 current swaprate, 17 current time, 19 fixed side, 15 floating side, 14 forward starting, 15 forward swap rate, 17 index mapping function, 18 initial discount vector, 15 intial cash forward vector, 15 level function, 16 Libor, 14

References Libors over several periods, 18 map fixed side indices, 18 margin, 14 notation, 14 paid in arrears, 14 payer, 15 receiver, 16 roll, 18 standard swap form, 17 swap rate, 17 tenor intervals, 14 terminal node, 15 swaption implied volatility, 35 swaption shift, 35 swaption zeta, 35 tenor intervals, 14 closed on the right, 15 coverage, 15 open on the left, 15 width, 15 terminal measure, 26, 41, 152 forward value factor, 42 timeslicers, 87 drift approximation, 88 separable, 87 spot measure problematical, 90 technical points cubics against Gaussian density, 92 node placement, 91 splining the integrand, 92 two-dimensional, 93 under intermediate measure, 89 under terminal measure, 88 Treasury Inflation Protected Securities TIPS, 141 vega and shift hedging, 113, 114 die-at-exercise convention, 116 hedge pair, 114 into liquid swaptions, 118 perturbing shift and volatility, 113 shift part hedge, 115 volatility part hedge, 116

217

vegas, 163 volatility parity, 124 volatility parity in BGM, 126, 127 volatility parity in HJM, 124 volatility part hedge, 116 zero coupon, 2, 11 zero coupon CPI swap rate, 142