Proceedings of the 5th Ritsumeikan International
STOCtlA5TIC
Editors
Jiro Akahori Shigeyoshi Ogawa Shinzo Watanabe
a
Proceedings of the 5th Ritsumeikan International Syrnposlurn
I T O ( t l A I T I ( PROCfIIfI A I D APPLlCATlOllS T O MATtIFMATlCAL F I I A I C f
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Ritsumeiltan Univevsity, Japar~
3
-
6 M a r c h 2005
Editors
J i r o Alcahori Shigeyoshi Ogawa Shinzo Watanabe R i t s u m e k a n University, Japan
\b world Scientific
NEW JERSEY
. LONDON . SINGAPORE . BEIJNG . SHANGHAI . HONG KONG
TAIPEI
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STOCHASTIC PROCESSES AND APPLICATIONS TO MATHEMATICAL FINANCE Proceedings of the 5th Ritsumeikan International Symposium Copyright Q 2006 by World Scientific Publishing Co. Pte. Ltd.
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PREFACE The international colloquium on Stochastic Processes and Applications to Matlzematical Filrarzce was held at Biwako-Kusatsu Campus (BKC) of Ritsumeikan University, March 3-6,2005. If counted from the first symposium on the same title held in the year 2001, this colloquium was the fifth one of that series of symposia. The colloquium has been organized under the joint auspices of Research Center for Finance and Department of Mathematical Sciences of Ritsumeikan University, and financially supported by MEXT (Ministry of Education, Culture, Sports, Science and Technology) of Japan, the Research organization of Social Sciences (BKC),Ritsumeikan University, and Department of Mathematical Sciences, Ritsumeikan University. The aim of this research project has been to hold assemblies of those interested in the applications of theory of stochastic processes and stochastic analysis to financial problems, in wluch several eminent specialists as well as active young researchers have been jointly invited to give their lectures. In the organization of this colloquium, the committee chaired by Mr. Shigeyoshi Ogawa aimed to organize it as a Winter School especially for those younger researchers who intend to join in or just begin the research activity on the relevant subjects. For this reason we asked some of the invited speakers to give introductory talks composing of two or three unified lectures on the same themes (cf. the program cited below). As a whole we had about eighty participants with nine invited lecturers. The present volume is the proceedings of this colloquium based on those invited lectures. We, members of the editorial committee of this proceedings listed below, would express our deep gratitude to those who contributed their works to this proceedings and to those who kindly helped us in refereeing them. We would express our cordial thanks to Professors Toshio Yamada and Keisuke Hara at the Department of Mathematical Sciences, of Ritsumeikan University, for their kind assistance in our editing this volume. We would thank also Mr. Satoshi Kanai for his works in editing TeX files and Ms. Chelsea Chin of World ScientificPublishing Co. for her kind and generous assistance in publishing this proceedings. December, 2005, Ritsumeikan University (BKC) Jiro Akahori Shigeyoshi Ogawa Shinzo Watanabe
PROGRAM March, 3 (Thursday) 9:50-10:OO Opening Speech, by Shigeyoshi Ogawa (Ritsumeikan University) 10:00-10:50 Shinzo Watanabe (Ritsumeikan University, Kusatsu) Martingale representation and chaos expansion I 11:10-11:50 Monique Jeanblanc (Universite d'Evry, Val dlEssonne) Hedging defaultable claims I (joint work with T. Bielecki and A. M. Rutkowski) 12:00-13:30 Lunch time 13:30-1420 Paul Malliavin (Academie des Sciences, Paris) Stochastic calculus of variations in mathematical finance I 1430-1520 Yoshio Miyahara (Nagoya City University) Geometric Levy process models in finance I 16:00-16:50 Arturo Kohatsu-Higa (Universitat Pompeu Fabra, Barcelona) Insider modelling in financial market I 1230- Welcome party
March, 4 (Friday) 10:00-10:50 Paul Malliavin (Academie des Sciences, Paris) Stochastic calculus of variations in mathematical finance I1 11:10-11:50 Shinzo Watanabe (Ritsumeikan University, Kusatsu) Martingale representation and chaos expansion I1 12:00-13:30 Lunch time 13:30-1420 Monique Jeanblanc (Universitk dlEvry, Val d'Essonne) Hedging defaultable claims I1 (joint work with T. Bielecki and A. M. Rutkowski) 14:30-1520 Arturo Kohatsu-Higa (Universitat Pompeu Fabra, Barcelona) Insider modelling in financial market I1 16:00-16:50 Hideo Nagai (Osaka University) A family of stopping problems of certain multiplicative functionals and utility maximization with transaction costs.
March, 5 (Saturday) 10:00-10:50 Monique Jeanblanc (Universitk dfEvry, Val dlEssonne) Hedging defaultable claims I11 (joint work with T. Bielecki and A. M. Rutkowski) 11:10-11:50 Paul Malliavin (Academie des Sciences, Paris) Stochastic calculus of variations in mathematical finance I11 12:00-13:30 Lunch time 13:30-1420 Shinzo Watanabe (Ritsumeikan University, Kusatsu) Martingale representation and chaos expansion 111 14:30-1320 Makoto Yamazato (University of Ryukyus) Levy processes in mathematical finance 15:30-16:OO Break 16:00-16:50 Yoshio Miyahara (Nagoya City University) Geometric Lkvy process models in finance I1 18:30- Reception (at Kusatsu Estopia Hotel)
March, 6 (Sunday) 10:00-10:50 Toshio Yamada (Ritsumeikan University, Kusatsu) On stochastic differential equations driven by symmetric stable processes (joint work with H. Hashimoto and T. Tsuchiya) 11:00-1220 Short Communications; 1. Romuald Elie (Centre de Recherche en ~conomieet Statistique, France) Optimal Greek weights by kernel estimation 2. Kiyoshi Kawazu (Yamaguchi University) The recurrence of product stochastic processes in random environment
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CONTENTS
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Program.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vi Harmonic Analysis Methods for Nonparametric Estimation of Volatility: Theory and Applications E. Barucci, P. Malliavin and M . E. Mancino . . . . . . . . . . . . . . . . . . . . . .
1
Hedging of Credit Derivatives in Models with Totally Unexpected Default T R. Bielecki, M . Jeanblanc and M. Xutkowski . . . . . . . . . . . . . . . . . . . . 35
A Large Trader-Insider Model A. Kohatsu-Higa and A. Sulem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101 [GLP & MEMM] Pricing Models and Related Problems Y.Miyahara . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 . Topics Related to Gamma Processes M . Yamazato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .157 On Stochastic Differential Equations Driven by Symmetric Stable Processes of Index a H. Hashimoto, T. Tsuchiya and T. Yamada . . . . . . . . . . . . . . . . . . . . . . . . 183 Martingale Representation Theorem and Chaos Expansion S. Watanabe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .195
Harmonic Analysis Methods for Nonparametric Estimation of Volatility: Theory and Applications Emilio ~arucci',Paul Malliavin2, and Maria Elvira Mancino3
' Dipartimento di Matematica, Politecnico di Milano, Italy email:
[email protected] '10 rue Saint Louis en l'Isle, 75004 Paris, France
email:
[email protected] 3DIMAD,Universiti di Firenze, Italy email: mariaelvira.mancino8dmd.unifi.it
Key words: Volatility, Fourier analysis, time series, hedging. 1. Introduction
We have proposed in [41] a method to compute the volatility of a semimartingale based on Fourier series (Fourier method). The method allows to compute both the instantaneous volatility and the volatility in a time interval (integrated volatility). The method is well suited to employ high frequency data and therefore to compute volatility of financial time series. Since the method has been proposed, it has been extendedlapplied in several directions. This paper aims to review these contributions. The benchmark to compute the volatility of a financial time series in a time interval with high frequency data (e.g. daily volatility) is provided by the cumulative squared intraday returns (realized volatility), see [2,11]. In the limit, as the time interval between two consecutive observations converges to zero the realized volatility converges to the quadratic variation of the process and its derivative provides the instantaneous volatility of the process, e.g. the coefficient in front of the Brownian motion in case of a semimartingale. The major novelty of the Fourier method is that it allows to reconstruct the instantaneous volatility as a series expansion with coefficients gathered from the Fourier coefficientsof the price variation, by using an integration procedure, instead of a differentiation one. The key point of Fourier approach is the realization of the volatility as a function of time; this fact makes possible to iterate the volatility functor and, for instance, to compute the volatility of the volatility function. As we will show below, this feature is useful in several applications: a double application of
the volatility functor will lead to an effective computation of the leverage effect, which is a first order effect; by iterating three times the volatility functor we obtain the feedback volatility rate effect. The contribution is organized as follows. Section 2 presents the Fourier methodology and the main theoretical results. We present the method to compute the volatility both in an univariate and in a multivariate setting and we show consistency of the estimator and a central limit theorem. In Section 3 we show how the method has been implemented to compute the volatility of financial time series and we compare its performance to the realized volatility. In Section 4 some applications of the Fourier methodology are presented in order to illustrate the potentiality of the method. In Section 5 we obtain an estimator of the instantaneous volatility using Laplace transform. In Section 6 we generalize the Fourier methodology to obtain a non-parametric estimation of the Heath-Jarrow-Mortongenerator for the interest rate curve and of the Lie bracket of the diffusion driving vectors. A key fact is that all the infinitesimal generators of risk free measures have a drift which is completely determined by their second order terms: it vanishes for assets in Black-Scholes type model and for the interest rate curve the drift is fully determined by the HJM model. The second order terms can be computed in real time, and model free, by a volatility analysis made on a single trajectory of the market. By consequence pathwise volatility analysis gives access to a pathwise computation and model free of the infinitesimal generator of the risk free measure; by iterating this procedure the Greek Delta can be computed pathwise and model free; in the same spirit the hypoelliptic structure underlying the HJM infinitesimal generator can be computed pathwise and model free. 2. Fourier Methods for Volatility Computation Let p be the asset price process, we will make the following assumption: p(t) is a continuous Brownian semi-martingale satisfying the stochastic differential equation
where W is a Brownian motion on a filtered probability space (Q, (%)tE[o,Tll P), (T and b are stochastic processes such that
We suppose that (T is adapted and b is not necessarily adapted.
The semi-martingale satisfyinghypothesis (H) is the most familiar semimartingale in econometrics and in finance. Note that this class includes stochastic volatility models. The Fourier method for estimating the instantaneous (and integrated) volatility are based on an exact mathematical formula relating the Fourier transform of the price process p(t) to the Fourier transform of the volatility process u2(t).This identity is obtained in [43] and is proposed in Theorem 2.1. Different methods which address the problem of estimating the instantaneous volatility have been proposed in [25, 20,461. They are based on the quadratic variation formula and use a double asymptotic in order to perform both the numerical derivative and the approximating procedure. Before presenting the result, we recall some definitions from harmonic analysis theory (see for instance [38]). By change of the origin of time and rescaling the unit of time we can always reduce ourselves to the case where the time window [0, TI becomes [O, 2x1. Then given a function @ on the circle S1 its Fourier transform is defined on the group of integers Z by the formula F(+)(k):=
&
2n
+(9) exp(-ik9) d9, for k
E
Z.
We define F(d@)(k):=
2n
S
exp(-ik9) d@(s).
10,2n[
We emphasize that in the above formula we considered the open interval ]0,2n[, which means that the Dirac mass appearing in 272 by the jump p(2n) - p(0) is not taken into account. Using integration by parts, we have
Given two functions @, Y on the integers Z, we say that the Bohr convolution product exists if the following limit exists for all integers k ( 0 *B Y)(k) := lim N-W
tO(s)V(k-s), 2N + 1 s=-N
The subindex B is the initial of Bohr, who developed similar ideas in the context of the theory of almost periodic functions. Theorem 2.1. Consider a process p satisfyingassumption (H). Then we have: 1
(2)
7(a2)= 4,*s @, 271
where 0(k) = F(dp)(k)
for any integer k. Convergence of the convolution product (2) is attained in probability. The proof is based on the It6 energy identity for stochastic integrals (see for instance [39]),after having observed that the drift component b(t) in the semimartingale p(t) does not give contribution to the formula (2). The reconstruction of the function (r2(t)from its Fourier coefficients computed by Theorem 2.1, can be obtained as follows. Define
%(k) = 7 ( d p ) ( k )for Ikl I 2N and 0 otherwise
Then the Fourier-Fejer summation gives the instantaneous volatility function (3)
(r2(t)= lim N-m
exp(ikt) for almost all t
E
(0,277).
Remark 2.2. We note that the approximating trigonometric functions appearing in the sequence of polynomials (3) are positive (see [43]). It is possible to implement Theorem 2.1 in real terms, by expanding the Fourier transform of the volatility function as a linear combination of sines and cosines (see [41]).This procedure allows us to determine a formula for the volatility (r2(t)analogous to (3). The formalism is a little cumbersome and less conceptual than working with complex exponentials; nevertheless the numerical implementation of the real Fourier series is easier. Therefore we recall this procedure. Suppose that the function p(t) satisfies (H). Define
then
We define the prolongation to all integersk by parity for ak and by imparity for bk;more precisely let
(4) a; = b; = 0, a' -
and b. k
(
bk(dp)
- -b-k(dp)
for k > 0 for k < 0.
The following Theorem provides the formula of the Fourier coefficients of u2(t)in terms of a series of combination of the Fourier coefficients of dp, previously defined. Theorem 2.3. Consider a process p satisfying hypothesis (HI.Define, for 0 I
q 5 2N,
Denote by aq(a2),bq(02)the Fourier coeficients of 02(t). Then, for all fixed
q 2 0, thefollowing convergence in probability holds: lirn y (N) =
N++rn
1 ; aq(02),
:Irn
1
pa(N)= bq(of). 7-L
The above result provides an alternative formula to compute the so called instantaneous volatility. h fact the reconstruction of the function 02(t)from its Fourier coefficients, derived by (5), can be obtained for instance by the classical Fejer inversion formula: N k (6) of(f) = lim C ( 1 - -)(ak(02) cos(kf) + bk (02)sin(kt)), N+m N k=O
for almost all t E [0,2n]. 2.1 Integrated volatility As a byproduct of formula (2), we can also compute the integrated volatility in the time interval [O,2nl through the following identity
Using (2) we have
= lim N+W
2N + 1 s=-N
This can be expressed using the Fourier coefficients, because (7) is equal to
where a; and b; are defined in (4). Finally
Therefore (9) provides a measure of the integrated volatility, see [41]. 2.2 Asymptotic analysis The asymptotic properties of the Fourier estimator, i.e., consistency and central limit theorem, have been obtained in [43]. Apart from the statistical importance of these results, they are necessary to understand how the estimator behaves for very small time intervals, due to market frictions. In fact, the presence of market frictions makes this limiting argument not really accurate for very small time intervals. Such difficulties with limiting arguments are present in almost all area of econometrics and the development of a central limit theorem helps in understanding the behavior of the Fourier estimator for relatively small time intervals. We consider the following discrete unevenly spaced sampling of the price process p. We fix a sequence Sn of finite subsets of [0,2n], let S, := (0 = to,, Itl,, I . . . 2 tk,,,, = 2711 for any n I 1 and denote
Remark that the Fourier estimation method allows to consider nonsynchronous and random observations. Therefore we could choose the observations randomly in an independent way of W. Nevertheless by splitting the probability space, this kind of sampling reduces to a deterministic sampling. Therefore all our study will be made in the deterministic context. We use the following interpolation formula
where lLt ,,,,tj+,,,,[ denotes the indicator function of the interval [ti,,, t j + ~ , ~ [ . Then for any integer k, with Ikl I 2N compute
Consider now the estimator of the volatility function based on (3) defined by
where for any Ikl I N
The following theorem shows consistency in probability uniformly in t of the estimator (10). Theorem 2.4. Let p(n) + 0 as n + w , and consider the estimator of the volatilityfunction defined by (10). Then thefollowing convergence in probability holds: lim sup I;;ftN(t) - a2(t)l= 0. n,N+mtc[0,2n]
We consider now the distribution of the error. The Central Limit Theorem is obtained as p(n) + 0 and the interval [O,2n] remains fixed. For the existence of a limiting distribution it is necessary that the number of the Fourier coefficients N and the number of observations n increase with a suited order. Therefore let 9(n)be a function such that 9(n) -+ w as n + w . Consider the estimator
of 02(t)as defined in (lo), but in (12) we have highlighted dependence between the number of Fourier coefficients 9(n)and the number of observations n. Then: Theorem 2.5. Assume that as n + w then p(n) + 0,9(n) -+ and ~ ( n ) ( 9 ( n+ ) )m. ~ Assumefurthermore that
supj(fj+l,n- fj,n) lim . = 1. n-'m lnfj(fj+l,n - fj,n) Then,for anyfunction h E L2(0,277)satishing the condition
m,
p(n)S(n) -+0
thefollowing result holdsfor;i2,(t)defined in (12):
converges in law to a mixture of Gaussian distributions with random variance h2(t)04(t)dt. equal to 2
$
Remark 2.6. Condition (13)is a measure of the regularity of the partition. A different condition called €-balance, with E E (0,I ) , is considered in [9]. Our condition implies &-balancecondition for any E E (0,l). In [46] a general study of central limit result is done in the case where prices are recorded at irregular time intervals. 2.3 Cross-volatility computation
The Fourier method was proposed in [41]having in mind difficulties arising in the multivariate setting when applying the quadratic variation methodology because of non-synchronicity of prices observed for different assets. Assume that p(t) = (pl(t),. . .,pn(t))are Brownian semi-martingales satisfying the following It6 stochastic differential equations
where W = (W1,.. .,wd)are independent Brownian motions, and o: and tf are random processes satisfying hypothesis H. There is a large literature in which cross-volatilities are estimated through the quadratic covariation formula. However this formula is not well suited to provide a good estimate of cross-volatilities. Difficulties arise from the absence of synchronous observations. The non-synchronicity tradingproblem has been studied for quite a long time in empirical finance, e.g. see [52,37]. The bias caused by non-synchronicity and random sampling for the cross-correlations estimation has been recently highlighted in [29].The Fourier methodology proposed in [41]is immune from these difficulties due to its own definition, being based on the integration of "all" data. We recall now the Fourier method for computing multivariate volatilities. From the representation (16)we define the volatility matrix, which in our hypothesis depends upon time:
The Fourier method reconstructs C'.'(t) on a fixed time window (which we can reduce always to [0,2n] by change of origin and rescaling) using the Fourier coefficients of dp8(t).First we compute the Fourier coefficients of dpj for j = 1,. . .,n defined by
We then consider the Fourier coefficients of the cross-volatilities ..
zi,j(t)dt, ak(xifj)= -
(19) ao(Z91)=
71
bk(zi,j)= TI
S
cos(kt)~"(t)d f , 10,2n[
j-10,2n[sin(kt)Pi(t)dt.
By a polarization argument in [41]it is proved the following Theorem. Theorem 2.7. Fix an integer no > 0, thefunctions Ci,j(t)of the volatility matrix
have the following Fourier coejicients
N
ak(zi,j)= lim
N-tm
n C(as(dpi)as+k(dpj) + as(dpj)as+k(dpi)). N + 1 - no s=no
Finally using the Fourier-F6jer inversion formula we can reconstruct zi,j(t)from its Fourier coefficients:
..
(24)
Ci,i(t)= N+m lim ~ z ( t )
where for any t E (0,277)
3. Time Series Analysis
This Section has two main goals: to show how the Fourier method can be implemented to compute both volatility and cross volatility; to compare the method with other methods proposed in the literature to compute the integrated volatility.
3.1 Volatility computation Efficiency of the Fourier method to compute the integrated volatility of a stochastic process representing the asset price has been analyzed in several papers, see [15,16,33,50,48]. To implement the method, [15, 161 proceed as follows. Given a time series of N observations (ti, ti)), i = 1,. . . N, data are compressed in the interval [O,2n] and integrals are computed through integration by parts:
To compute the integrals, we need an assumption on how data are connected. As a matter of fact, high frequency data are not equispaced and therefore there is no constant time length between two consecutive observations. To handle this problem, a time grid with a fixed time interval is chosen (e.g., 30 seconds, 1 minute, 5 minute): tk, k = 1,2, . . .. We may not have an observation on a point of the grid. To cope with this problem, two methods have been proposed in the high frequency data literature: interpolation and imputation of data. In the first case ti) and p(ti+1)are connected Instead through a straight line: if f k E [ti, ti+l[,then p(tx)= (fk- ti)-. according to the interpolation method p(tk)= p(t;) (piecewise constant). In [15,16] the imputation method has been employed, then the integral in (25) in the interval [ti,ti+l]becomes
thus avoiding the multiplication by k which amplifies cancellation errors when k becomes large. The methodology has been applied to compute volatility in a standard GARCH setting. Let p(t) = log S(t),where S(t) is a generic asset price, and rt = p(t) - p(t - 1) is the logarithmic return. It is assumed that the asset price follows the continuous time GARCH(1,l)model proposed in [47]:
where W I ,W2 are independent Brownian motions. This model is closed under temporal aggregation in a weak sense, see [22], and its discrete time analogous is given by:
where et are i.i.d. Normal distributed random variables. The exact relation between (q,a, p) and (8,w , A) is derived in [22]. The task is to assess the capability of the Fourier method to reproduce the theoreticalvolatility of the GARCH(1,l)model. The analysis is based on Monte Carlo simulations. High frequency unevenly sampled observations have been generated as follows: one day of trading [0,1] has been simulated by discretizing (27) with a time step of one second, for a total of 86.400 observations a day. Then observation times have been extracted in such a way that time differences are drawn from an exponential distribution with mean equal to T = 45 seconds, which corresponds to the average value observed for many financial time series. As a result, we have a dataset (tk,p(tk), k = 1,. . .,N) with tk unevenly sampled. The most used way to compute the integrated volatility is to exploit 1 the quadratic variation and therefore to compute u2(.r)d.ras the sum of squared intra-day returns (realized volatility), see [36] for the relation with the Fourier method. Provided a grid with m points . . .I), volatility in a day is computed as follows:
L
(i, i,
Theoretically, thanks to the Wiener theorem, by increasing the frequency of observations, an arbitrary precision in the estimate of the integrated volatility can be reached. Note that the Fourier method uses all observations, instead the sum of squared intraday returns uses only a fraction of observations, i.e., for low m some observations are lost. In most of the papers estimating volatility with high frequency data, e.g. see [I], (29)is computed with m = 288 corresponding to five-minute returns. In the simulation settingit is also considered rn = 144, corresponding to ten-minute returns, and m = 720 corresponding to two-minutes returns. In the literature the interpolation technique is employed. The performance of the Fourier method is compared to that of (29)with m = 144,288,720 by the statistics: fi2(s)ds - 82 /'=
% C(s)ds
, std =
[I% 12]' u2(s)ds- b2
I
&(s)ds
where P is the estimate and J' 02(s)dsis the integrated volatility generated in a simulation, whose value is known in the simulation setting. We also evaluate the forecasting performance of the model (28), when ex-post volatility is measured by a2. This is done by means of the R~ of the
linear regression
We recall that without manipulating the data, we should observe smaller p and std when increasing the frequency. Figure 1 shows the results on simulated time series with a = 0.25,p = 0.7, q(1- a - p) = 1. First let us consider the realized volatility. The ten-minute estimator provides a downward biased estimate of the integrated volatility with a standard deviation larger than the bias. The five-minute estimator is also downward biased with a standard deviation of the same order of the bias in mean. Increasing further the frequency, the estimator is characterized by a smaller variance but a larger bias is observed. This effect can only be due to the interpolation scheme described above and therefore it can be linked to non-synchronous trading, see also [37]. The Fourier estimator is characterized by the smallest bias in mean and by a variance smaller than that of the 5-10 minutes estimate and slightly larger than that of the 2 minutes estimate. To check the robustness of these results, we repeated the Monte Carlo experiments on a grid of values (a, p, $ = (1- a -/?)-I) with 2 and 5 minutes returns. Results, reported in Table 1,can be summarized as follows: the estimator (29)turns out to be downward biased (p > 0), with a bias increasing in m, while the bias of the Fourier estimator is almost null. If m is chosen in such a way that the bias of (29) is less than its standard deviation, then the Fourier estimate provides a smaller standard deviation. Analyzing the forecasting capability of the discrete time GARCH model (28) we have that the Fourier estimate renders a better performance than the classical estimator computed with 2 and 5 minute returns. The Fourier methods to compute volatility has been applied in several directions. Volatility is not constant over time; empirically it has been shown that days with high (low)volatility are followed by days with high (low)volatility, i.e., there is persistence in the volatility process. In the '90s, a large set of volatility models capturing a persistence component has been proposed in the financial time series literature. GARCH models provided the mile stone. The main problem with these models was that while their forecasting performance in sample was good, the forecasting performance out of sample using rough volatility proxies, e.g., square of the closing price minus the opening price, was very poor. [I]have shown that the problem was given by a poor ex post computation of volatility. Using high frequency data and the cumulative squared intraday returns to compute volatility they show that the forecasting performance of the standard GARCH(1,l) model in predicting the exchange rate volatility is quite good. [16]apply the Fourier method to compute volatility of the Deutsch Mark-Us dollar and of Yien-Us dollar exchange rate considering a one year of high frequency
Table 1 p,std, R~ (multiplied by 100) for the three estimators: (29) with m = 720 denoted by 2', (29) with m = 188 denoted by 5' and the Fourier estimator denoted by F, on a grid of values for (n,p)in (28), and J, . (1 - n - p) = 1. All the values are computed with 10000 "daily"replications.
Table 2 R~ for the two time series. Estimator
Fourier
0.470
0.143
observations (fromOctober, IS'1992to September 30th1993), i.e., the dataset analyzed in [I]. The dataset consists of 1.466.944quotes for the Deutsch Mark-Us dollar and 567.758 quotes for the Yien-Us dollar exchange rate. Performance of the GARCH(1,l) model has been evaluated when the integrated volatility is computed according to the Fourier method. The parameters of the model are those estimated in [I]. Table 2 provides the corresponding R2. We observe that the GARCH model performs well in forecasting when the Fourier method is employed to compute the integrated volatility. Its performance is better than that associated with the sum of squared intraday returns as an integrated volatility measure. On performance of volatility
models and volatility computation see also [31]. As the Fourier method provides an accurate estimate of the volatility, we can handle the volatility as an observable process. In the literature, time varying volatility has been handled as a latent process, e.g., the GARCH process. In [13], this idea has been applied to compute the volatility of the overnight interest rate in the Italian money market and to test for the martingale hypothesis correcting for heteroschedasticitywith a good proxy of the volatility and thus avoiding estimation problems connected with the use of a GARCH model. In [17] an autoregressive process for the volatility estimated according to the Fourier method is estimated to forecast oneday return volatility and to define the value at risk threshold. The method performs well in forecasting, better than the GARCH(1,l)model and the exponential smoothing proposed by RiskMetrics. In recent years other volatility models have been proposed to cope with empirical regularities observed for volatility of financial time series. Among the models, we have models with long memory in the volatility process considering a fractional Brownian motion W2 of order d indepen-d~(r). To capture sharp increase dent of WI in (27), i.e., W2(t)= J~ in volatility, jumps in the volatility process have been introduced adding a Poisson process in (27). In these more general models, the sum of squared intraday returns and the Fourier method do not provide consistent estimators of the volatility. [48] compare through Monte Carlo simulations the bias and the root mean square error of the Fourier method, of the cumulative squared intraday returns and of a wavelet estimator. They show that the Fourier method provides the lowest bias and root mean squared error. They also compare the three methods when a bid-ask bounce effect is inserted, i.e., as a random buy/sell order arrives in the market there is a liquidity effect on the price creating spurious serial correlation in returns and volatility. In this case the realized volatility and the Fourier method are no longer consistent for the integrated volatility. As far as the bias and the root mean square error is concerned, the Fourier method performs better than the wavelet and the cumulative squared intraday returns method. On the comparison between Fourier method and the wavelet method see also [33]. 3.2 Computation of cross volatility In this section we analyze the performance of the method in the bivariate case using Monte Carlo simulations of high frequency asset prices as studied in [44]. As in [51], we simulate two correlated asset price diffusions with the
Std -0.5
-0.4
=
-0.3
0.078 -0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
Std = 0.045 250 '-0.5
-0.4
-0.3
-0.2
-0.1
Meat-
O0 400
Figure 1. Distribution of
Std =
02(t)dt-e2
,where d2 are three different estimators of the integrated 'L o2it)dt
volatility: (a) estimator 69) kith rn = 144; @) estimator (29) with rn = 288; (c) estimator (29) with rn = 720; (d) Fourier estimator. The distribution is computed with 10,000 "daily" replications.
bi-variate continuous GARCH(1,l) model:
and all the other correlations between the Brownian motions set to zero. The choice of this particular model comes from the fact that it is the continuous time limit of the very popular GARCH(1,l) model, and it has been studied extensively in the literature, e.g. see [28]. We will use the parameter values estimated by [I]on foreign exchange rates, i.e. = 0.035,wl = 0.636,Al = 0.296,02 = 0.054,wz = 0.476,Az = 0.480. We will instead analyze two mirror cases for the correlation coefficient: p = 0.35 and p = -0.35. To get a representation of high-frequency tick-by-tick data, after discretizing (31) by a first-order Euler discretization scheme with a time step
Table 3 Average correlation measurement on 10,000 Monte Carlo replications of the model (31). Two generated values of the correlation are considered, p = 0.35 and p = -0.35. We compute the variance-covariancematrix via the Fourier estimator and via the realized volatility estimator (32). L.I. means Linear Interpolation, while l?T. means Previous Tick interpolation. Standard deviationsof in-sample measurementsare reported in the columns named Std. Estimator Fourier Realized 5', L.I. Realized 5', ET. Realized 15', L.I. Realized 15', P.T. Realized 30', L.I. Realized 30', P.T.
Generated correlation p = 0.35 Measured Std 0.350 0.204 0.181 0.338 0.329 0.345 0.342
0.039 0.058 0.060 0.090 0.091 0.127 0.127
Generated correlation p =-0.35 Measured Std -0.349 -0.203 -0.180 -0.337 -0.328 -0.344 -0.341
0.039 0.055 0.058 0.090 0.092 0.126 0.126
of one second, we extract observation times drawing the durations from an exponentialdistribution with mean 30 seconds and 60 seconds respectively. Observation times are drawn independently for the two time series. After simulating the process (31) we compute on it daily (86400 seconds, corresponding to 24 hours of trading, as for currencies) variance-covariance matrix according to the Fourier theory and according to the realized volatility measure of 131, given by:
The choice of rn in (32)comes from a tradeoff between increasing precision and cutting out microstmcture distortions. A typical choice is rn = 288, corresponding to five minute retums. As pointed out above, to obtain an equally spaced time series we can rely upon a linear interpolation (L.I.) or a previous tick interpolation (P.T.). Both methods have been applied for rn = 288,96,48 (correspondingrespectively to 5,15,30 minute retums), when measuring correlations on Monte Carlo experiments. Table 3 shows the results. First of all we notice that the Fourier estimator performs considerably better than the realized volatility, which is biased toward zero. The bias in the correlation measurement of realized volatility is more and more severe when the sampling frequency is increased. For the fiveminute estimator with the previous-tick interpolation, we get a mean value
of 0.181 (-0.180), which is quite far from the true value of 0.35 (-0.35); this bias is completely due to the non synchronicity of quotes. Realized volatility with linearly interpolated returns is closer to the right value, but this is because of the downward bias in the volatility measurement due to the linear interpolation documented in [15, 161. In these papers, it is shown that the spurious positive serial correlation induced by the linear interpolation technique lowers volatility estimates. Since variances are spuriously measured to be lower, correlations turn out to be spuriously higher, thus compensating in some way the bias due to non-synchronicity. This is also true, but to a much lower extent, for the 15-minute and 30-minute realized volatility estimator. The precision of the Fourier estimator, as measured by the standard deviation of measurement errors across Monte Carlo replications, is always better than the realized volatility estimator. We implemented the Fourier estimator with N = 500 coefficients for the first (30 seconds average spacing) time series, N = 160 coefficients for the second (60 seconds average spacing), and N = 160 coefficients for the computation of covariance. Increasing the number of coefficients would increase the variance measurement precision but not the covariance measurement precision, because of the Epps effect, see [51]. On the other hand, even the gain in precision of the realized covariance measurement obtained when increasing the sampling frequently is cancelled out by the bias. 4. Applications The potentiality of our method relies in the fact that it reconstructs instantaneous volatility and cross-volatility as functions of time. This feature of the Fourier method is essential when a stochastic derivation of volatility along time evolution is performed as in contingent claim pricinghedging. In this Section we present some developments of the multivariate Fourier estimation procedure which illustrate this point. 4.1 Leverage effect There is a vast empirical literature showing that the asset price and its volatility are related. Among other phenomena, it has been observed the so called leverage effect, i.e., a negative return sequence is associated with a volatility increase. In some recent papers, see [12,10], the integrated volatility estimation methodology based on the so called realized volatility has been extended to allow for a leverage effect. From the mathematical point of view the no leverage hypothesis means that o(t)is independent from the Brownian motion W. The no leverage hypothesis simplifies the study of the properties of the volatility estimator, but is not realistic for the analysis of equity returns. The asset price model H is very general and includes stochastic volatil-
ity as well as level dependent volatility models, i.e. volatility is a time independent function of the asset price, it allows for feedback effects of asset price on volatility and in particular the leverage effect. In this section it is proved that these effects can be non-parametrically estimated using the Fourier methodology without knowledge of the exact form of the evolution equation for the volatility process. A first order indicator of the stochastic dependence between the asset price process and the volatility process is obtained and we refer to it as leverage effect, we also consider a second order indicator which is called feedback effect rate. Consider the asset price model which satisfies hypothesis H. Moreover assume that the process o(t) is a random process satisfying
where a,3! , and y are predictable functions (for simplicity suppose they are almost sure bounded) and WI is another Brownian motion independent of W. Then the leverage effect is defined as
where
* denotes the It6 stochastic contraction divided by dt. Therefore
Nevertheless we have no knowledge of the random function a(t). Therefore it is interesting to find a formula for estimating B(t) starting from the price observations. By It6 calculus the following result holds:
Theorem 4.1. Let p and a satisfy H and Ti, then we have
where Vol(p k 02)means the volatilityfunctionof the stochastic process p k 02.
In order to estimate B(t) from the asset prices we see that the Fourier transform of p can be computed from price observations; the Fourier transform of o2is computed by Theorem 2.1; therefore the Fourier transform of p + o2 and p - o2 are known; a second application of Theorem 2.1 gives the Fourier transform of the volatility vol(p*02)and finally the usual inversion formula produces the computation of B(t).
4.2 Delta hedging (estimationof the gradient)
Suppose that the price process p ( t ) in logarithmic scale satisfies
where W is a Brownian motion and u is an unknown C1-function . No hypothesis on the shape of this function are done. In the following it is shown that is possible to estimate the pathwise gradient using Fourier transform methodology, therefore also the Greek Delta can be estimated non-parametrically. In financial applications the Greek Delta is the sensitive which allows to perform the so called Delta-hedging, that is to make the portfolio neutral with respect to small modifications of the initial value of the price. In model (33),the Greek Delta is defined as
where p E ( f )corresponds to the solution of (36)with pE(to)= p(to)+ e. Then It follows that the computation of A ( f )depends on the estimation of the volatility u(p(t))and the derivative of the volatility uf(p(t)).Denote by B(t) the function obtained in Theorem 4.1, then it holds
This first result expresses the derivative of the volatility function as the ratio of terms which can be estimated using prices data. By substitution of (35)in (34)and using (33),it follows
Note that all terms in the above equation are given or estimated by Fourier method from market asset prices. A more general Delta hedging result has been obtained in [I41 by using the Fourier method. 4.3 Feedback volatility rate In [14] we have constructed a second order indicator of the feedback effects of asset price on volatility and we have used the Fourier estimation method in order to compute this effect from asset price data. Working under this general assumption, we produced a time-dependent pricevolatility feedback effectrate function and in the multivariate case a timedependent matrix, which will be called elasticity matrix, implementable in real time from asset price observations.
The mathematical theory suggests that eigenfunctions associated with positive eigenvalues of the elasticity matrix correspond to instability directions of the market; eigenfunctions associated with negative eigenvalues correspond to stability directions of the market. The key mathematical tool to construct the elasticity matrix is a new methodology of transferring price perturbations through time, using the inertial frame transport; this transport is designed so that through time the variation is governed by a first order linear ordinary differential equation. The rate of variation through time of the initial perturbation is given by the elasticity matrix. Computation of the elasticity matrix is a three-steps outcome; at each steps it is necessary to compute volatilities of observed quantities or computed in the previous step. Therefore the Fourier methodology works out. We can write explicit expressions in terms of Fourier coefficients of asset prices leading to real-time determination of feedback rates. For simplicity we present the univariate case, but the method can be developed for any finite number of assets (see [14]). Let p(t) be the risky asset price at time t. Suppose it satisfies the stochastic differential equation in logarithmic scale:
where W is a Brownian motion, o is a fixed but unknown C 2 ( ~function. ) Let S(t)be the variation process which is solution of the linearized stochastic differential equation
We associate to S(t) the rescaled variation defined as
In Section 4.2 we have obtained the SDE driving the Delta propagation. It is a remarkable fact that this SDE can be reduced to an ODE at the price of a renormalization of the Greek Delta, as it is showed in the following result. Theorem 4.2. The rescaled variation is a differentiablefunctionwith respect to t; denote by A(t) its logarithmic derivative, then
Figure 2. Average estimate of daily volatility A(t) on IBM data. On the x axis, the
time window [0,2n]corresponds to one trading day (6.5 hours). A(t) displays the typical U-shape of volatility in stock markets.
where
l
Definition 4.3. We will call A(t) the price-volatilityfeedback ratefunction.
Note that in the standard Black-Scholes framework A = 0. We derive a nonparametric estimation of the feedback volatility rate by using Fourier methodology. We suppose that we do not know the explicit expression of the function a; we want to obtain a non parametric estimation in real time of A(t) from the observation of a single market evolution.
Theorem 4.4. Denoting by following cross-volatilities:
* the It6 contraction divided by dt, and define the
Then the price-volatilityfeedback effect ratefunction A can be expressed as
We stress the point that all terms A, B, C can be obtained from the asset price data through the Fourier cross-volatility estimation. Therefore the feedback rate A(t) can be empirically estimated. The theory suggests that the sign of A is associated with stability of the market: a negative A would witness a period of stability, a positive A would signal instability. Precise estimation of quadratic and higher order variation asks for huge quantities of data; high frequency data are a natural candidate for this
purpose. In [14] we have used two data sets: a data set containing quotes of the DEM-USD foreign exchange rates from October 1992 to September 1993and a data set of IBM quotes from January to December 1999. In the case of the IBM stock price, estimates of A(t),B(t),C(t)have been computed for a one day time window (6.5 hours); the larger number of Fourier coefficients produces a higher resolution of the plots. Figure 2 gives the volatility A(t) averaged over the full year; here we recognize the U-shape pattern which is typical of stock market intra-day volatility. Short-horizon estimates of A are the most important for traders. We present in Figure 3 as typical sample of daily (non-averaged)estimates the values for A(t),B(t),C(t) and A(t) for two days in 1999. It is noteworthy that taking the logarithm of the stock price mainly changes the scales of A(t),B(t) and C(t), but lets the shape of the curves more or less invariant. For this reason, in Figures 2,3 the scales have been chosen according to the stock price (without taking logarithms). On January4th, 1999, the beginning of the trading day reveals positivity of A which detects instability of the market and which is revealed by subsequent large picks of A. Over the whole day the positive values of A dominate and indicate an unstable trading day. On April 9th, 1999, the beginning of the day reveals negativity of A, which indicates stability of the market and which is revealed by a subsequent progressive damping of A. In contrast to January 4th, over the whole trading day, small and mainly negative values of A dominate and indicate stability. As argued above, precise estimation of A could, in principle, result in important consequences for trading strategies. In this context, it can be of some help to analyze the sign of A week-by-week or even day-by-day; our results show that, by using high frequency data, an estimate of this effects can be readily accomplished. 4.4 Dynamic principal component analysis The Fourier method of cross-correlation estimation has been applied in [44] to develop a dynamic principal component analysis. The theory differs from the statistical principal component analysis because we do not assume any linear functional dependence; moreover we describe a dynamic situation, while usually principal component analysis describes a static situation. Classical PCA provides a linear sub-manifold of smaller dimension which carries the essential information coming from the data. Dynamic PCA will produce an abstract curve, which we call the Core, which allows us to determine the eigenvectors of multivariate volatility in continuous time. The analysis is model free. Making mild assumptions on the multivariate prices behavior (Brownian semi-martingale hypothesis),we get by the Fourier method multivariate volatilities as a time dependent quadratic form, let C(t). We suggest a
23
-
- stock price I
I
I
I
I
I
Figure 3. Daily values of A(f),B(t), C(t),A(f) on IBM data. The time windows [O,2n]
correspond to two typical trading days in 1999 (6.5 hours each). Jan 4 (left-hand side) displays positivity of A with large picks of A (instable market); April 9 (right-hand side) displays small and mainly negative values of A along with a progressive damping of A (stable market). During the first two hours of trading, A has about the same shape on both days, but A develops dramatically different shapes; computing A in real time could give an indicator forecasting market instability.
method to construct an abstract curve which contains the essential information coming from multivariate volatilities. The analysis of the evolution in time of this curve can be used to decipher the stability of the market about an asset, or the degree of market integration of a given asset. We then introduce the concept of reference assets, via a geometric definition: a reference asset is one whose volatility is mainly due to the market volatility instead of its idiosyncratic noise. We omit mathematical details of the construction, while we illustrate the ideas via the analysis of two months, April and May 2001, of highfrequency data for 98 stocks, selected into the S&P 100 index, among the most liquid ones. The results point out the need of a time-dependent analysis versus a static one, since the variance-covariance eigenvalues structure turns out to be deeply time-varying. In particular, there are some days in the market in which correlations are widely distributed, thus less factors are needed to explain the variance-covariance structure. On the whole, we analyze 42 days; for each day, we compute the variance-covariance matrix using the Fourier method. We start by performing PCA for each day, after normalizing the variance-covariance matrix in order to have the variance of every stock equal to 1. The first factor explains, on average, 25.79% of the movements. However in some days the first factor's weight can be as large as 56.09% (April, 18'~)and 73.26% (May, 16'h).Moreover, and more interestingly for any financial application, this phenomenon seems to present some degree of persistence. In the second step of our analysis, we define the Core of the market as the vector subspace spanned by the first 30 eigenvectors, and we divide our 42 market days into 6 periods of 7 market days each. In each subperiod, we perform principal component analysis on the aggregate variance-covariancematrix, and we obtain the coordinates of the Core. In each subperiod, we define 15reference assets as those who have the largest projection onto the Core. We interpret these assets as those who are more correlated with the market itself, or alternatively as the basic constituents of the market. The list of the reference assets in any subperiod is shown in Table 4. Given the low value of the average correlation, we expect that the basket of reference assets is quite variable, given the unalienable noise in the correlation measurements. This is indeed the case. The month of April shows more persistence: 4 reference assets in the first period, out of 15, are in the second period too; and 6 of the second are still reference assets in the third. The month of May shows much more variability, or less "market integration"; only two stocks are reference assets in the third and fourth period, only one in the fourth and fifth period and none out of 15 in the fifth and sixth period. Loolung at individual stocks, AES Corporation is a reference asset in the whole sample, with the exception of the sixth period,
Table 4 Lists the reference assets by ticker name in the six subperiods considered. They are ranked according to their projection on the core, which is reported in brackets. Also the five assets with the smallest projection on the Core are listed. Each subperiod is composed of 7 market days. In bold face, we indicate those stocks who remain reference assets in the subsequent period. Periods 1
2
ccu (n2%)
AOL (86.1%) NK(84.1%) AA (82 m)
3
4
5
6
HIG (78.7%) HAL (76 7%) WMT (7I8'h) HWP (71.7%) TYC (70.9%) ONE (68.8%) HNZ (68.3%) F (68.1%) U'IX (67 3%) HCA (66.8%) ORCL(66.1%) HET (65.5%) DIS la.%)
GM (74.7%) DOW (72.1%) IBM (70 4%) GE (70.4%) LTD (70.4%) BUD (697%) FDX (Ml70) JNJ(67.6%) MAY (67.1%) MSm(66.4%) AIG (65.5%) ETR (65 3%) MMM (64.6%)
2 AOL (71.1%) INTC(70.0%) AES (69.5%) PHA (68.841.) 1P (64 6%) UTX (64.1%) CPB (64 0%) GE (63.7'14 TYC (63.6%) EK (63 2%) VZ (62.0%) BNI(62 m)
JPM IBM (T9.9%) IN1(78.5%) kB~(77 5%) AT1 (74 8%) MRK (70.5%) IP (70 PA) T (69.6%) MCD (67.7%) LU 167.5%)
WFC (n.~%) TOY (R.I%) AES (72.0%) All (n.9x) LU (734%) WMB (68 2%) CSC (67.7%) AEP (R.5%) XOM (672%) AW (72.2%) TXN (70.3%) MSFM66.3%) BUD (69.8%) PFE (65.8%) AIG (67.2%) PEP (65 1%) MRK (64.9%) CSCO(65.1%) BAX (64.4%) LID (64.8%) BMY (64.3%) AMGN(64.7%) MDT (63.5%) SLB (63.1%) IN1 (61.7%) WFC 163.1%)
and in three periods it is the asset with the largest absolute projection on the Core. Table 4 also shows the percentage projection on the Core, and the five assets, in any period, with the lowest projection onto it. For example, in period 2,93.3% of the variance of AES can be considered to be driven by the market, and 6.7% is explained by idiosyncratic fluctuation; thus AES essentially lies on the Core, which is the subspace which explains most of the variance of the whole market; on the other hand, in the same period for Coca-Cola (KO), only 35.1% of its variance is driven by the market, while 64.9% is independent fluctuation. Summarizing, out of the 98 assets, 32 are never reference assets; 46 are only once; 18 are twice, Johnson & Johnson (JNJ) is thrice and AES is five times. Then our analysis identifies nearly 20 assets which had a major role in market integration. In the set of this 20 assets, 8 are among the 20 most capitalized; thus capitalization plays an important role in defining leading assets, but it is not the only factor to be taken in account. For example, in our analysis AES turned out to be the most important stock, but its capitalization (measured as market value) is only about 0.5 of the capitalization of Microsoft, the most capitalized stock in our sample. 5. Laplace Transform Method for Volatility Computation The Fourier theory has been extended in [42] where it is shown that the Laplace transform is an appropriate tool to build estimators of the in-
stantaneous volatility based on a long time series of prices by smoothing past data and retaining recent price observations. The Laplace transform estimator has the same advantages of the Fourier estimation procedure in comparison with the quadratic variation methods, being based on the integration procedure of all the data as the Fourier estimation method. Moreover it has further good features: in [41] it was shown that the estimation efficiency is better in the center of the considered time window, while the use of Laplace transform estimator allows to obtain an estimator of instantaneous volatility which becomes less sensitive to boundary effects when approaching to the present time (say going from (t = -m) to (t = 0)). From a conceptual point of view the introduction of Laplace transform has two advantages: firstly to avoid the artificial modification into periodic functions subjacent to Fourier series; secondly to lead to formula (41) which constitutes a bridge between the two different methods of computation of volatility, the method based on quadratic variation and our approach by Fourier series. On the other side the Laplace method is based on the use of integrals instead of series; the discretization of these integrals will introduce a drawback from the numerical point of view. For simplicity consider
the log-price process, where W(t) is a Brownian motion on (-m, 01 and ~ ( t ) is a stochastic process adapted to the filtration generated by W. Consider the Laplace transform of dp on (-m, 0] :
Integrating by parts we have that @w(z)can be expressed as
We take z = a + is. In [42] it is proved that the Laplace transform of the volatility process 02(t)(which is expressed by the right hand side of (40)) can be exactly computed through the Bohr convolution product of the Laplace transform of dp(t), which is expressed by the function Qw defined in (39). Then the Laplace inversion formula allows to reconstruct the process 02(t) for any t E (-oo,O]. Later on we denote by $ the conjugate of any complex function $.
Theorem 5.1. Lef Ow(z)be the Laplace transform of dp as given by (39). Then the following convergence in probability holds
Remark 5.2. In formula (40)we have the choice of an arbitrary parameter a; we must consider always a > 0. If we want to damp quickly the effect of the past time we have to take la1 large. Using the exact formula (40) it is possible to derive an estimator of volatility given a discrete unevenly spaced sampling of the price process p, containing only an averaged sum of the jumps of the prices combined with universal kernels. Denote by ti the times at which prices are observed, let -m < . . . < < ti < . . . < to I 0 and let hi@):= - p ( f l ) + ~ ( f i - ~for ) , any i 2 1.
Let the integration interval in (40)be fixed equal to [-R, R],then the following approximation formula for the volatility function can be obtained:
where us is defined for 6 > 0 by
6. HJM Model Looked as an Hypoelliptic Operator
Consider the Heath-Jarrow-Morton model [32]for the interest rates in the Musiela parametrization, see [45]. In this parametrization a market state at time to is described by the instantaneous interest rate curve rto(S) to a maturity 5, defined for 5 E R+.The price P(t0, T ) of a default-free zero coupon bond traded at to with maturity to + T, has the value
The interest rate curve takes its value in the infinite dimensional space C of continuous functions on [O, m [ . A remarkable experimental fact is that the rank n of its volatility matrix is very low n 4, see [I81 . Then elliptic models are ruled out and hypoelliptic models are the most regular models still available.
-
By restricting the HJM model to the case of a finite number of scalarvalued driving Brownian motions Wl, . . .,W,,, then the HJM model is expressed by the following SDE:
where the drift of the risk-free process is completely determined under the no arbitrage condition by the volatility matrix B,(t, .). In fact it is known from [21] that the risk-free generator associated to the risk free measure has its infinitesimal generator fully determined by its second order terms. For the stock price the drift associated vanishes under the risk free measure. For the interest rate model the drift is determined by the no-arbitrage condition implemented in the HJM model. This key property implies that the infinitesimal generator of the HJM process can be pathwise and model free computed through the computation of the volatility matrix. Under the hypothesis of Markovian completeness of the market, the dependence of the volatility matrix B,(t, .) is factorizable through the final value rt(.), therefore it is possible to write
where Ak are "driving vector fields" defined on C. An appropriate notion of "smoothness" of vector fields is a necessary hypothesis in order to prove existence and uniqueness of solutions for (42), see [24]. In [40] a non-parametric estimation of HJM generator is obtained and a method similar to the methodology of the price-volatility feedback rate is developed for the interest rate curve. The question of the efficiency of the mathematical theory proposed in this section to decipher the state of the market has not yet confirmed by numerical computations. The first point is the possibility of measuring, in real time by high frequency market data, the full historical volatility matrix. We remark that, while the correlations between stocks prices at high frequency has no clear economic meaning, on the contrary the high frequency cross-correlations are clearly sigruficant in the HJM model. Assume that the HJM hypothesis for the infinitesimal generator is satisfied; no assumption on the explicit expression of the vector field Ak in (42) are done. It is proved in [40] that an estimation of the maps t H Ak(yt) is achieved by the observation of an unique evolution of the market.
Fix N different maturities ( N can be extremely large and in particular it can be n << N ) , then define the time-dependent N x N covariance matrix
*
where 5, q vary between all maturities and denotes the It8 contraction divided by dt. For simplicity of notation consider two maturities r.7 and the bidimensional process: t H (rt(<),rt(il)).Then it holds: Theorem 6.1. Given two dzJ7erent maturities E, 17 we consider the 2-dimensional process: t H (rt(0,rt(7));then
The vector A j ( r t )is defined as the jf" column of the matrix Note that the r.h.s. of (44)can be computed using the Fourier methodology for the estimation of volatility 6.1 Non parametric estimation of Hormander bracket Definition 6.2. Given two vector fields A1,A2 denote by A: the S-th component of Ak. The Lie bracket of A l and A2 is the vector field having the components:
The Lie algebra 3 generated by n vector fields A1,. . .,A, is defined as the vector space of all fields obtained as linear combinations with constant coefficients of
<
Given r E RN, let 3-f:= (5E RNJ = Z(r) for some Z
E
371.
Definition 6.3. The vector fields A1,. . . A , satisfy the Hormander criterion for hypoellipticity if A1,. . .,A,, are infinitely often differentiable and if 3 ( r ) = R~ for any r E R ~ . Due to the non-ellipticity of the market, the multivariate price-volatility feedback rate constructed in [14] cannot be applied, so [40] construct a pathwise econometric computation of the bracket of the driving vector of the diffusion.
Without any hypothesis on the explicit form of the vector fields Ak(rt), we estimate the bracket [AlrA2](rt) from a single trajectory. For simplicity we omit writing explicitly the dependence of A k on rt. In order to get this result, by the definition (45), the main point is the estimation of
This estimation is obtained by the relation
together with the fact that the 1.h.s. of (47) can be computed by Fourier methodology. Therefore we can state the result: Theorem 6.4. The brackets of the driving vectorfields of the HIM di@sion can be numerically computed from a single time series of market data, without any assumption on the model. 6.2 Global and pathwise compartmentage by maturities
Let C the space of continuous functions on [0, m[. Define the logarithm derivative of a measure p , ~depending on a parameter A as the function satisfying
for every test function f . Denote nt(ro, dr) the fundamental solution of the heat equation associated to the HJM model; define an Hilbert norm on tangent vectors at ro by
Definition 6.5. We say that the global compartmentage holds if the following condition holds:
the norms llzlls and llzlls~,s # s' are inequivalent. In [40]it is proved that the global compartmentage condition holds generically in the HJM setting. As a consequence, it is showed that it is impossible to hedge an European option at maturity T by trading options on maturities larger than T .
The operator Qh is defined in [40] (5.3) by solving the linearized HJM model:
where the function Ak, taking values in the space of N x N matrices, is defined by
Choose on C an Hilbert metric 11 * Ilc; by using the canonic Hilbert structure of L2([0,s];Rn), the adjoint (Q&)* is defined and it is possible to define the Malliavin covariance matrix:
Consider a fixed trajectory W of the Brownian motion. Denote u';(s) the covariance matrix computed on [0, s].
Theorem 6.6. The pathwise compartmentage property holds o$(s) is a strictly increasing function of s.
if the range of
Therefore it is possible to deduce the property of pathwise compartmentage from the behavior of Hormander brackets, which we showed can be estimated non-parametrically using Fourier methodology.
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Hedging of Credit Derivatives in Models with Totally Unexpected Default Tomasz R. BIELECKI1, Monique JEANBLANC2, and Marek RUTKOWSK13 'Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA 2D6partementde Mathematiques, Universite d'fivry Val d'Essonne, 91025 ~ v r y Cedex, France 3Schoolof Mathematics, University of New South Wales, Sydney, NSW 2052, Australia & Faculty of Mathematics and Information Science, Warsaw University of Technology,00-661 Warszawa, Poland The paper analyzes alternative mathematicaltechniques, which can be used to derive hedging strategies for credit derivativesin models with totally unexpected default. The stochastic calculus approach is used to establish abstract characterization results for hedgeable contingent claims in a fairly general set-up. In the Markovian framework, we use the PDE approach to show that the arbitrage price and the hedging strategy for an attainable contingent claim can be described in terms of solutions of a pair of coupled PDEs. Key words: arbitragepricing, hedging, credit risk, default intensity.
Introduction This paper presents some methods to hedge defaultable derivatives under the assumption that there exist tradeable assets with dynamics allowing for elimination of default risk of derivative securities. We investigate hedging strategies in alternative frameworks with different degrees of generality, an abstract semimartingale framework and a more specific Markovian set-up, and we use two alternative approaches. On the one hand, we use the stochastic calculus approach in order to establish rather abstract characterization results for hedgeable contingent claims in a fairly general set-up. We subsequently apply these results to derive closed-form solutions for prices and replicating strategiesin particular models.
On the other hand, we examine the PDE approach in a Markovian setting. In this method, the arbitrage price and the hedging strategy for an attainable contingent claim are described in terms of solutions of a pair of coupled PDEs. Again, for some standard examples of defaultable claims, we provide explicit formulae for prices and hedging strategies (for further examples of trading strategies involving tradeable credit derivatives, we refer to Laurent [31] or Bielecki et al. [B]). As expected, both methods yield identical results for some special cases considered in this work. For the sake of simplicity, we only deal with financial models with no more than three primary assets (models with an arbitrary number of primary assets were studied in Bielecki et al. [6]). Also, it is postulated throughout that the default time is the same for all defaultable securities. An extension of our results to the case of several (possibly dependent) default times is crucial if someone wishes to cover the so-called basket credit derivatives (in this regard, see Section 6 in Bielecki et al. [7]). Let us comment briefly on the terminology used in this work. Traditionally, credit risk models are classified either as structural models (also known as value-of-the-firm models) or as reduced-form models (also termed intensity-based models). In their original forms, the two approaches, structural and reduced-form, are extreme cases, in the sense that the default time is modelled either as a predictable stopping time (the first moment when the firm's value hits some bamer, as in Black and Cox [9]), or by a totally inaccessible stopping time (defined via its intensity, as in Jarrow and Turnbull [25]). However, as argued by several authors (see, for instance, Duffie and Lando [17], Giesecke [21]-[22], Jarrow and Protter [24], Jeanblanc and Valchev [28], or Guo et al. [23]), probabilistic properties of default time are directly related to the publicly available information (it is important, for instance, whether the value of the firm and/or the default triggering barrier are observed by the market with absolute accuracy). In fact, in several structural models the default time is no longer predictable, as it was the case in classic models with deterministic default triggering barrier and full observation of the firm value process (see, Merton [33] or Black and Cox [9]). For this reason, we decided to refer to credit risk models considered in this work as models with totally unexpected default (the strict mathematical term, totally inaccessible stopping time, seems to be rather cumbersome for a frequent use). For a more exhaustive presentation of mathematical theory of credit risk, we refer to Arvanitis and Gregory [I], Bielecki and Rutkowski [3], Bielecki et al. [4], Cossin and Pirotte [IS], Duffie and Singleton [18], Lando [30], or Schonbucher [36].
1. Totally Unexpected Default In this section, we describe briefly the fundamental features of the credit risk models with unexpected default. Also, we collect here few technical results that are used in subsequent sections. 1.1 General set-up We assume that we are given a probability space (Q, 6,P)and a nonnegative random variable T on this space. We always postulate that T is strictly positive with probability 1. Note that the probability measure P represents the historical probability reflecting the real-life dynamics of prices of primary traded assets, rather than some martingale measure for our financial model. We first focus on different definitions of default intensity encountered in the literature. 1.1.1 Intensity of a stopping time Suppose that ( f 2 , 8 , IF') is endowed with some filtration such that T is a E-stopping time. Let H be the default process, defined as Ht = 3LIttTl (note - T admits a E-intensity that H is a bounded c-submartingale). We say that if there exists a E-adapted, nonnegative process A such that the process
c
is a e-martingale (the second equality in (1) follows from the fact that the process H is stopped at T). Then M is called the compensated Emartingale of the default process H. In order for a E-stopping time T to admit a c-intensity it has to be totally inaccessible with respect to c , so that P(T= 0) = 0 for any c-predictable stopping time 8. The simplest example is the moment of the first jump a Poisson process. Note that the intensity necessarily vanishes after default.
x,
n
Remark 1.1. Some authors define the intensity as the process such that th't A, du is a E-martingale. In that case, the process r\ is not uniquely Ht defined after time T. 1.1.2 IF-intensity of a random time
We change the perspective, and we no longer assume that the filtration is given a priori. We assume instead that T is a positive random variable on some probability space (Q,G, P). Let M = (fit,t 2 0) be the natural filtration generated by the default process (Ht, t 2 O), and let F = (Ft,t 2 0) be some referencefiltrationin (CJ,G, P). We assume throughout that the information available to an investor is modeled by the filtration G = F V M. Consequently, we can reduce our
study to the case where the default intensity (if it exists) is G-adapted, meaning that the-process M given by (1)is a G-martingale for some Gadapted process A. In this setting, there exists a process A = (At, t 2 O), called the - default, - IF-intensity of T,which is IF-adapted and equal to xbefore so that A t l l t s r l = At31,t5,1 for every t E lR+. The existence of 11(and its uniqueness under some technical conditions) follows from the following result (see Dellacherie et al. [16], Page 186). Lemma 1.1. Let 43 = IF V M. Then for any G-predictable process 5 there exists an IF-predictable process rsuch that
If, in addition, the inequality Ft := P(T5 t I f i ) < 1 holdsfor every t E IR+ then the process csatisfying (2)is unique. Of course, we have that
Suppose that the reference filtration is chosen in such a way that the default events (T I t ]are not in IF. Then the IF-intensity A is uniquely defined after T and, typically, does not vanish after 7. 1.2 Hypothesis (H) In this section, we focus on the invariance property of the so-called hypothesis (H) under an equivalent change of a probability measure. Definition 1.1. We say that filtrations IF and G, with IF G G, satisfy the hypothesis (H) under P whenever any IF-local martingale L follows also a 6-local martingale. Remark 1.2. We emphasize that, in general, an IF-martingale may fail to follow a G-martingale. As a trivial example, consider a fixed date T > 0 and take Gt = FTfor every t E [0, TI. Then any IF-martingale L satisfies IEP(Lt IGs) = Lt for s I t, and thus L is not a G-martingale, in general. It is even possible, but more difficult, to produce an example of an IFmartingale, which is not a semi-martingale with respect to G. For other counter-examples, in particular those involving progressive enlargement of filtrations, we refer interested reader to Protter [35], or Mansuy and Yor [321.
The original formulations of the hypothesis (H) refer to martingales (or even square-integrablemartingales), rather than local martingales. We
shall show that in our set-up the definition given above is equivalent to the original definition. In fact, the hypothesis (H) postulates a certain form of conditional independence of a-fields associated with IF and G, rather than a specific property of IF-(local) martingales. In particular the following well known result is valid. Lemma 1.2. Assume that G = IF v M, where IF is an arbitrayfiltration and M is generated by the process Ht = ll,,t,. Then thefollowing conditions are equivalent to the hypothesis (H). (i) For any t, h E R+, we have
(i') For any t
E
IR+,we have
(ii) For any t E IR+,the 0-fields 7,and Gt are condifionally independent given 7; under IP, that is,
for any bounded, 9%-measurable random variable 5 and bounded, Gt-measurable random variable tl. (iii) For any t E IR+,and any u 2 t the o-jields Fuand Gt are conditionally independent given 7;. (iv) For any t E IR+ and any bounded, %-measurable random variable 5: EP(Z I Gt) = Ep(Z 155). (v) For any t E IR+, and any bounded, Gt-measurable random variable q: 17;)= Ep(f117,). Proof. The proof of equivalence of conditions (i')-(v) can be found, for instance, in Section 6.1.1 of Bielecki and Rutkowski [3] (for related results, see Elliott et al. [20]). Using monotone class theorem it can be shown that conditions (i) and (if)are equivalent. Hence, we shall only show that condition (iv) and the hypothesis (H) are equivalent. Assume first that the hypothesis (H) holds. Consider any bounded, 7,-measurable random variable E. Let Lt = IEp(<1 f i )be the martingale associated with 5. Then, (H) implies that L is also a local martingale with respect to G, and thus a G-martingale, since L is bounded (recall that any bounded local martingale is a martingale). We conclude that Lt = Ep(S 1 Gt) and thus (iv) holds. Suppose now that (iv)holds. First, we note that the standard truncation argument shows that the boundedness of 5 in (iv) can be replaced by the
assumption that 5 is P-integrable. Hence, any IF-martingale L is an 6martingale, since L is clearly G-adapted and we have, for every t Is,
Now, suppose that L is an IF-local martingale so that there exists an increasing sequence of IF-stopping times T,, such that lim,,, T , = w , for any n the stopped process LTtrfollows a uniformly integrable IF-martingale. Hence, LTnis also a uniformly integrable G-martingale, and this means that L follows a G-local martingale. 1.2.1 Hazard process
Let T be a random time on a space (Q,G, P) such that the filtrations IF and G = IF v M satisfy the hypothesis (H). Then, from (4), the process Ft = P(T 5 t I 7 t ) is increasing. We make the standing assumption that Ft < 1for every t E IR+,and we define the IF-hazard process T by setting Tt = - In (1-Ft). Let, in addition, the process F be absolutely continuous with respect to the Lebesgue measure, so that
l t
Ft =
f. du, V t E R+,
for some IF-progressively measurable (or IF-predictable) process f . Then the IF-hazard process satisfies
where the IF-intensity y is given by
From now on, we take (5) as the definition of the IF-intensity y, and we make the standing assumption that the hypothesis (H) holds under P. The following auxiliary result is standard (see, for instance, Elliott et al. [20] or Blanchet-Scalliet and Jeanblanc [lo]). Lemma 1.3. For any P-integrable, 7T-measumblerandom variable X we have,
for t E [O, TI, ~EIP(X~(T
We now describe the canonical construction of a random time with a given IF-hazard process. Let Y be an IF-adapted, increasing, nonnegative
process with Yo = 0 and limt,, variable T by setting
Yt = oo. We define a nonnegative random
where O is a random variable independent of IF, with the exponential distribution of parameter 1. Of course, the existence of O on the original probability space (fl, 6,P)is not guaranteed, so that we allow for an extension of the underlying probability space. We shall now find the process Ft = P{T t 151.Since clearly (T > t ] = {O> Yt},we get
Consequently,
and so F is an IF-adapted, continuous, increasing process. We conclude that for every t E R+
and thus Y coincides with the IF-hazard process r of T and the hypothesis (H) is valid. It is also not difficult to show that the process Mt = Ht - T t A T = Ht - YtA7follows a 6-martingale. The following result shows that under the hypothesis (H), for any random time T with continuous hazard process T, the auxiliary random variable O can be constructed on the original probability space, using T and r (see El Karoui [19] or Blanchet-Scalliet and Jeanblanc [lo]). Lemma 1.4. Let T be a random time on a probability space (0,G, P) such that the IF-hazard process r of T under P is continuous and the hypothesis (H) holds. Then there exists a random variable O on ( 0 , 6, P), independent of IF and with the exponential distribution of parameter 1, such that
Proof. It suffices to check that the random variable O = T, has the desired properties. Indeed, we have, for every t E IR,,
where A is the left inverse of
r, so that rAl= t for every t E R+.
1.3 Change of a probability measure Kusuoka [29] shows, by means of a counter-example, that the hypothesis (H) is not invariant with respect to an equivalent change of the underlying probability measure, in general. It is worth noting that his counterexample is based on two filtrations, IH1 and IH2,generated by the two random times ,rl and ,r2,and he chooses M1 to play the role of the reference filtration IF. We shall argue that in the case where IF is generated by a Brownian motion (or, more generally, by some martingale orthogonal to M under P), the above-mentioned invariance property is valid under mild technical assumptions. 1.3.1 Preliminary lemma Let us first examine a general set-up in which G = IF v M, where IF is an arbitrary filtration and I His generated by the default process H. We say that Q is locally equivalent to P if Q is equivalent to P on (R, Gt) for every t E R,. Then there exists the Radon-Nikodjhn density process q such that
Part (i) in the next lemma is well known (see Jamshidian [27]). We assume that the hypothesis (H) holds under P. Lemma 1.5. (i) Let Q be a probability measure equivalent to P on (R, Gt) for every t E IR,, with the associated Radon-Nikodtjm density process q. I f the density process q is IF-adapted then we have Q(7 I t 1 7;) = P(TI t I 7;)for every t E IR,. Hence, the hypothesis ( H ) is also valid under Q and the IF-intensities of c under Q and under P coincide. (ii) Assume that Q is equivalent to P on (fl,G) and dQ = q, dP, SO that ijt = IEp(q, I &). Then the hypothesis (H) is valid under Q whenever we have, for every t E R,,
Pro06 To prove (i), assume that the density process q is IF-adapted. We have for each t I s E IR,
where the last equality follows by another application of the Bayes formula. The assertion now follows from part (i) in Lemma 1.2. To prove part (ii), it suffices to establish the equality
Note that since the random variables qtl(rstland qt are P-integrable and (&-measurable,using the Bayes formula, part (v) in Lemma 1.2, and assumed equality (9),we obtain the following chain of equalities
We conclude that the hypothesis (H) holds under Q if and only if (9) is valid. Unfortunately, straightforward verification of condition (9) is rather cumbersome. For this reason, we shall provide alternative sufficient conditions for the preservation of the hypothesis (H)under a locally equivalent probability measure. 1.3.2 Case of the Brownian filtration Let W be a Brownian motion under Pwith respect to its natural filtration IF. Since we work under the hypothesis (H), the process W is also a Gmartingale, where G = IF V M. Hence, W is a Brownian motion with respect to G under IP. Our goal is to show that the hypothesis (H) is still valid under Q E Q for a large class Q of (locally) equivalent probability measures on (Q, G). Let Q be an arbitrary probability measure locally equivalent to IP on (a,@).Kusuoka [29] (see also Section 5.2.2 in Bielecki and Rutkowski [3]) proved that, under the hypothesis (H), any G-martingale under P can be represented as the sum of stochastic integrals with respect to the Brownian motion W and the jump martingale M. In our set-up, Kusuoka's representation theorem implies that there exist G-predictable processes B and > -1, such that the Radon-Nikodw density 11 of Q with respect to IP satisfies the following SDE
<
with the initial value 170 = 1. More explicitly, the process q equals
where we write
and
Moreover, by virtue of a suitable version of Girsanov's theorem, the following processes and 6i are 6-martingales under Q
Proposition 1.1. Assume that the hypothesis ( H ) holds under lP. Let Q be a probability measure locally equivalent to P with the associated Radon-Nikodym density process 11 given by formula (12). I f the process 8 is IF-adapted then the hypothesis (H) is valid under Q and the IF-intensity of .r under Q equals yt = (1+ C)yt, where c i s the unique IF-predictable process such that the equality Stllt
e.
We claim that the hypothesis (H)holds under From Girsanov's theorem, the process W follows a Brownian motion under with respect to both IF and 6. Moreover, from the predictable representation property of W under F, we deduce that any IF-local martingale L under @ can be written as a stochastic integral with respect to W. Specifically, there exists an IF-predictable process 5 such that
This shows that L is also a 6-local martingale, and thus the hypothesis (H) holds under F. Since
by virtue of part (i) in Lemma 1.5, the hypothesis (H) is valid under Q as well. The last claim in the statement of the lemma can be deduced from
the fact that the hypothesis (H)holds under Q and, by Girsanov's theorem, the process
We claim that the equality $ = IP holds on the filtration IF. Indeed, we have d F I .r; = Tjt d P I ' ~ ;where r we write Tjt = IEp(qt(2) I E),and
where the first equality follows from part (v) in Lemma 1.2. To establish the second equality in (17), we first note that since the process M is stopped at T, we may assume, without loss of generality, that = cwhere the process c i s IF-predictable (see Lemma 1.1). Moreover, in view of (7) the conditional cumulative distribution function of .r given 7, has the form 1- exp(-rt(ot)). Hence, for arbitrarily selected sample paths of processes 5 and r, the claimed equality can be seen as a consequence of the martingale property of the Doleans exponential. Formally, it can be proved by following elementary calculations, where the first equality is a consequence of (14)),
<
where the second last equality follows by an application of the chain rule.
1.3.3 Extension to orthogonal martingales Equality (17) suggests that Proposition 1.1can be extended to the case of
is convenient, arbitrary orthogonallocal martingales. Such a if we wish to cover the situation considered in Kusuoka's counterexample. Let N be a local martingale under P with respect to the filtration IF. It is also a G-local martingale, since we maintain the assumption that the hypothesis (H) holds under P. Let Q be an arbitrary probability measure locally equivalent to P on (C2,G). We assume that the Radon-Nikodym density process q of Q with respect to lP equals
<
for some G-predictableprocesses 0 and > -1 (theproperties of the process 8 depend, of course, on the choice of the local martingale N). The next result covers the case where N and M are orthogonal G-local martingales under P, so that the product MN follows a G-local martingale. Proposition 1.2. Assume that the following conditions hold: (a)N and M are orthogonal G-local martingales under P, (b)N has the predictable representation property under P with respect to IF, in the sense that any IF-local martingale L under P can be written as
S t
~t
= LO +
5. d ~ . ,
v t E R+,
for some IF-predictable process 5, is a probability measure on (a, G) such that (16) holds. (c) Then we have: (i) the hypothesis (H) is valid under F, (ii) if the process 6 is IF-adapted then the hypothesis (H) is valid under Q. The proof of the proposition hinges on the following simple lemma. Lemma 1.6. Under the assumptions of Proposition 1.2, we have: (i) N is a G-local martingale under @, (ii)N has the predictable representation propertyfor IF-local martingales under F. Proof. In view of (c), we have dF 1 g, = 1,:) dP 1 g,, where the density process q(2)is given by (14), so that dq?) = @Lt dMt. From the assumed orthogonality of N and MI it follows that N and i f 2 ) are orthogonal G-local martingales under P,and thus N ~ (is~a )G-local martingale under P as well. This means that N is a G-local martingale under F, so that (i) holds. To establish part (ii) in the lemma, we first define the auxiliary process q by setting 6 = 1~~(17j2) 1 7;).Then manifestly d@ I = 6 dP I E, and thus
-
in order to show that any IF-local martingale under F follows an IF-local martingale under IP, it suffices to check that = 1 for every t E JR+,so that 6 = IP on IF. To this end, we note that
5
where the first equality follows from part (v) in Lemma 1.2, and the second one can established similarly as the second equality in (17). We are in a position to prove (ii). Let L be an IF-local martingale under F. Then it follows also an IF-local martingale under P and thus, by virtue of (b),it admits an integral representation with respect to N under P and E. This shows that N has the predictable representation property with respect to IF under @. We now proceed to the proof of Proposition 1.2.
Proof of Proposition 1.2. We shall argue along the similar lines as in the proof of Proposition 1.1. To prove (i),note that by part (ii) in Lemma 1.6 we know that any IF-local martingale under @admitsthe integral representation with respect to N. But, by part (i) in Lemma 1.6, N is a G-local martingale under F. We conclude that L is a G-local martingale under 6, and thus the hypothesis (H) is valid under E. Assertion (ii)now follows from part (i) in Lemma 1.5. Remark 1.3. It should be stressed that Proposition 1.2 is not directly employed in what follows. We decided to present it here, since it sheds some light on specific technical problems arising in the context of modeling dependent default times through an equivalent change of a probability measure (see Kusuoka [29]).
Example 1.1. Kusuoka [29] presents a counter-example based on the two independent random times T I and ~2 given on some probability space (a,G, P). We write M; = H; - yi(u)du, where H,' = lltlri and yi is the deterministic intensity function of .ri under .'FI Let us set dQ I g, = qt dP I g,, where qt = qjl)qf) and, for i = 1,2 and every t E IR+,
gA"
for some 02-predictable processes I$'), i = 1,2, where 02 = IH1 v lH2. We set IF = IH1 and H = lH2. Manifestly, the hypothesis (H) holds under P.
Moreover, in view of Proposition 1.2, it is still valid under the equivalent probability measure 5 given by
It is clear that
='tI on IF, since
However, the hypothesis (H) is not necessarily valid under Q if the process <(I) fails to be IF-adapted. In Kusuoka's counter-example, the process was chosen to be explicitly dependent on both random times, and it was shown that the hypothesis (H) does not hold under Q. For an alternative approach to Kusuoka's example, through an absolutely continuous change of a probability measure, the interested reader may consult Collin-Dufresne et al. [13]. Semimartingale Model with a Common Default In what follows, we fix a finite horizon date T > 0. For the purpose of this work, it is enough to formally define a generic defaultable claim through the following definition.
2.
Definition 2.1. A defaultable claim with maturity date T is represented by a triplet (X, Z, T),where: (i) the default time T specifiesthe random time of default, and thus also the default events (T 5 t ) for every t E [0, TI, (ii) the promised payof X E 7;- represents the random payoff received by the owner of the claim at time T, provided that there was no default prior to or at time T; the actual payoff at time T associated with X thus equals X~IT<~I, (iii) the IF-adapted recovey process Z specifies the recovery payoff ZT received by the owner of a claim at time of default (or at maturity), provided that the default occurred prior to or at maturity date T.
In practice, hedging of a credit derivative after default time is usually of minor interest. Also, in a model with a single default time, hedging after default reduces to replication of a non-defaultable claim. It is thus natural to define the replication of a defaultable claim in the following way. Definition 2.2. We say that a self-financing strategy $ replicates a defaultT ,X1ll~,,l ,~ able claim (X, Z, T) if its wealth process V($) satisfies V T ( @ ) ~ ~ = and V ~ @ ) ~ I=BZ.~~~I( T > T J .
When dealing with replicating strategies, in the sense of Definition 2.2, we will always assume, without loss of generality, that the components of the process 4 are IF-predictable processes. 2.1 Dynamics of asset prices We assume that we are given a probability space (fl,G, P) endowed with a (possibly multi-dimensional) standard Brownian motion W and a random time .r admitting an IF-intensity y under IP,where IF is the filtration generated by W. In addition, we assume that T satisfies (4), so that the hypothesis (H) is valid under P for filtrations IF and G = IF v M. Since the default time admits an IF-intensity, it is not an IF-stopping time. Indeed, any stopping time with respect to a Brownian filtration is known to be predictable. We interpret T as the common default time for all defaultable assets in our model. For simplicity, we assume that only three primary assets are traded in the market, and the dynamics under the historical probability P of their prices are, for i = 1,2,3 and t E [0, TI,
or equivalently,
The processes (pi, oi, K ; ) = (pi,t,o;,t, li;,t, t 2 0), i = I, 2,3, are assumed to be 62-adapted, where G = IF v M. In addition, we assume that ~i 2 -1 for any i = 1,2,3, so that Yi are nonnegative processes, and they are strictly positive prior to T . Note that, according to Definition 2.2, replication refers to the behavior of the wealth process V(q5) on the random interval 10, T A T ] only. Hence, for the purpose of replication of defaultable claims of the form (X,Z, T), it is sufficient to consider prices of primary assets stopped at T A T. This implies that instead of dealing with G-adapted coefficientsin (19),it suffices to focus on IF-adapted coefficients of stopped price processes. However, for the sake of completeness, we shall also deal with T-maturity claims of the form Y = G(Y$, Y,: Y;, H T ) (see Section 5 below). 2.1.1 Pre-default values As will become clear in what follows, when dealing with defaultable claims of the form (X, 2, T), we will be mainly concerned with the so-called pre-default prices. The pre-default price ? of the ith asset is an IF-adapted, continuous process, given by the equation, for i = 1,2,3 and t E [0, TI,
with ?; = Y;. Put another way, ? is the unique IF-predictable process such that (see Lemma 1.1) ?I,,,,, = Yfl,,,,, for t E JR,. When dealing with the pre-default prices, we may and do assume, without loss of generality, that the processes pi, ai and K ; are IF-predictable. It is worth stressing that the historically observed drift coefficientequals pi,t - xi,tYt, rather than !lilt. The drift coefficient denoted by p;,t is already credit-risk adjusted in the sense of our model, and it is not directly observed. This convention was chosen here for the sake of simplicity of notation. It also lends itself to the following intuitive interpretation: if @ is the number of units of the ith asset held in our portfolio at time t then the gains/losses from trades in this asset, prior to default time, can be represented by the differential
$Id z = $jg(pbtd t + n i , t d ~ t-) $ : g ~ ; , dt. t ~ ~ The last term may be here separated, and formally treated as an effect of continuously paid dividends at the dividend rate x;,tyt. However, this interpretation may be misleading, since this quantity is not directly observed. In fact, the mere estimation of the drift coefficient in dynamics (21) is not practical. Still, if this formal interpretation is adopted, it is sometimes possible make use of the standard results concerning the valuation of derivatives of dividend-paying assets. It is, of course, a delicate issue how to separate in practice both components of the drift coefficient. We shall argue below that although the dividend-based approachis formally correct, a more pertinent and simpler way of dealing with hedging relies on the assumption that only the effective drift p;,t - h-ityt is observable. In practical approach to hedging, the values of drift coefficients in dynamics of asset prices play no essential role, so that they are considered as market observables. 2.1.2 Market observables To summarize, we assume throughout that the market observables are: the pre-default market prices of primary assets, their volatilities and correlations, as well as the jump coefficients ~ i , t(the financial interpretation of jump coefficients is examined in the next subsection). To summarize we postulate that under the statistical probability IP we have
where the drift terms lift are not observable, but we can observe the volatilities c~i,t(and thus the assets correlations), and we have an a priori assessment of jump coefficients ~ i , t . In this general set-up, the most natural assumption is that the dimension of a driving Brownian motion W equals
the number of tradable assets. However, for the sake of simplicity of presentation, we shall frequently assume that W is one-dimensional. One of our goals will be to derive closed-form solutions for replicating strategies for derivative securities in terms of market observables only (whenever replication of a given claim is actually feasible). To achieve this goal, we shall combine a general theory of hedging defaultable claims within a continuous semimartingale set-up, with a judicious specification of particular models with deterministic volatilities and correlations. 2.1.3 Recovery schemes It is clear that the sample paths of price processes Yi are continuous, except for a possible discontinuity at time T. Specifically, we have that AY,! := Y: - r7= lCi,,Y,-, -.
so that Y', = Y;-(l + ~ i , , ) = Y:-(1 + ~i,,). A primary asset Y' is termed a default-fveeasset (defaultableasset, respectively) if Ki = 0 ( ~ Zi 0, respectively). In the special case when Kj = -1, we say that a defaultable asset Y' is subject to a total default, since its price drops to zero at time T and stays there forever. Such an asset ceases to exist after default, in the sense that it is no longer traded after default. This feature makes the case of a total default quite different from other cases, as we shall see in our study below. In market practice, it is common for a credit derivative to deliver a positive recovery (for instance, a protection payment) in case of default. Formally, the value of this recovery at default is determined as the value of some underlying process, that is, it is equal to the value at time T of some IF-adapted recovery process Z. For example, the process Z can be equal to 6, where 6 is a constant, or to g(t, 6Yt) where g is a deterministic function and (Yt, t 2 0) is the price process of some default-free asset. Typically, the recovery is paid at default time, but it may also happen that it is postponed to the maturity date. Let us observe that the case where a defaultable asset Y' pays a predetermined recovery at default is covered by our set-up defined in (19). For instance, the case of a constant recovery payoff bi 1 0 at default time .r corresponds to the process ~ i , t= hi(?-)-' - 1. Under this convention, the price Y' is governed under P by the SDE
If the recovery is proportional to the pre-default value Y:-, and is paid at default time .r (this scheme is known as thefiactional recovery of market value), we have ~ i , t= 6i - 1 and
(24)
d ~=i Y ; - ( ~ ~dt, + ~ o,, dWf + (6, - 1)d ~ t ) .
2.2 Risk-neutral valuation To provide a partial justification for the postulated dynamics of the price of a defaultable asset delivering a recovery, let us study a toy example with two assets: a savings account with constant interest rate rand a defaultable asset Y represented by a defaultable claim (X, Z, T). In this toy model, the only source of noise is the default time, hence, the only relevant filtration is M (in other words, the reference filtration IF is trivial). We assume that by choosing today's prices of a large family liquidly traded defaultable assets, the market implicitly specifies a martingale measure Q, equivalent to the historical probability P.More precisely, the probability distribution of .r under an equivalent martingale measure (e.m.m.) Q can be inferred from market data. We are thus interested in the dynamics of the price process of (X, Z, 7)under Q. It is worth noting that in this subsection we adopt a totally different perspective than in the rest of the present paper. In fact, no attempt to replicate a defaultable claim is done in this section. We assume instead that the risk-neutral default intensity can be uniquely determined from prices of traded assets, and we postulate that the price of (X, Z, 7) is defined by the standard risk-neutral valuation formula. The argument that formally justifies the use of this pricing rule is that we obtain in this way an arbitragefree market model in which Q is a martingale measure, and a defaultable claim can be considered to be an additional traded asset. Since we do not assume here that a defaultable claim is attainable, its spot price (that is, the price expressed in units of cash) depends explicitly on the riskneutral default intensity. As was mentioned above, the arbitrage price of a defaultable claim, when expressed in terms of tradeable assets used for its replication, will be shown to not depend directly on real-world (or risk-neutral) default intensity. To conclude, the rationale for the calculations given below, is that we strive here to justify the dynamics of prices of primary assets in our model. The risk-neutral valuation considered in this subsection is not supported by replication-based arguments, and thus it is not surprising that it exhibits specific features that are not present in the replication-based valuation. We make the standing assumption that .r admits a continuous cumulative distribution function Funder Q. Hence, the hazard function T'is also continuous, and the process 6 i t = Ht - F(t A T) is an M-martingale under Q. The following result is standard (see, e.g., Proposition 4.3.2 in Bielecki and Rutkowski 131).
Proposition 2.1. Assume that the cumulative distribution function F of T is continuous. Let M" be an H-martingale given by M: = !EQ(h(.r)I 'Ift)for some Bore1 measurablefunction h : IR+ -, IR such that the random variable h ( ~is)
Q-integrable. Then
where we write
g(f) = er(" E Q ( (t,,th(r)). ~
Remark 2.1. Using the above proposition, it can be easily shown that on (0,GT)we have
for some M-predictable process 5. 2.2.1 Price dynamics of a survival claim (X,0, T ) . In what follows, we shall refer to a defaultable claim of the form (X,0, T ) as a survival claim. By virtue of the risk-neutral valuation formula, the price of the payoff that settles at time T equals, for every t E [0,TI,
Note that X is fi-measurable, and thus constant since the 0-field 7~ is trivial. To find the dynamics of the price process, it suffices to apply . the Q-martingale Proposition 2.1 to the function h(u) = l l u , T t e - r T ~For M! = ecrtYt,we thus get, for every t E [0, TI,
Suppose that R t ) = J~ ~ udu.)Then an application of ItB's formula yields
We deal here with an example of a defaultable asset that is subject to the total default, meaning that its price vanishes at and after default. 2.2.2 Price dynamics of a recovery claim (0,Z, T ) . Recall that our standard convention stipulates that the recovery Z is paid at the time of default. Hence, the price process Y of (0, Z, T) is given by the expression
We now have h(u) = ll(u
By applying ItB's formula, we conclude that the dynamics under Q of an asset that delivers Z(T)at time T are
2.2.3 Price dynamics of a defaultable claim (X, Z, T).
By combining the formula above with (26), and using Remark 2.1 together with Girsanov's theorem, we arrive at the following result.
Proposition 2.2. The price process Y of a defaultable claim (X, Z, T) satisfies under Q dyt = rYt- dt + ( ~ ( t-) ~ t - ) d G t with the initial condition
Under the statistical probability P,the price process Y satisfies
where the G-martingale M under P equals
Remark 2.2. Proposition 2.2 can be extended to the case when the recovery is random, and is given in the feedback form as Z(t) = g(t, Yt-) for some function g(t, y), which is Lipschitz continuous with respect to y. Assume, for instance, that the claim is subject to the fractional recovery of market value, so that Z(t) = 6Yf- for some constant 6. If, in addition, 5 and Fare constant, then we obtain (cf. (24))
l)x
Note that here the drift coefficient pt = r + l(t,,l(6 in dynamics of Y follows a G-predictable process, but it is not IF-predictable. However, the drift of the pre-default value Y is simply r.
3. Trading Strategies in a Semimartingale Set-up We consider trading within the time interval [0, TI for some finite horizon date T > 0. For the sake of expositional clarity, we restrict our attention to the case where only three primary assets are traded. The general case of k traded assets was examined by Bielecki et al. [5]. We first recall some general properties, which do not depend on the choice of specific dynamics of asset prices. In this section, we consider a fairly general set-up. In particular, processes Y', i = 1,2,3, are assumed to be nonnegative semi-martingales on 6,P)endowed with some filtration G. We assume a probability space (R, that they represent spot prices of traded assets in our model of the financial market. Neither the existence of a savings account, nor the market completeness are assumed, in general. Our goal is to characterize contingent claims which are hedgeable, in the sense that they can be replicated by continuously rebalanced portfolios consisting of primary assets. Here, by a contingent claim we mean an arbitrary &--measurable random variable. We work under the standard assumptions of a frictionless market. 3.1 Unconstrained strategies Let = (+I, +2, +3) be a trading strategy; in particular, each process $' is predictable with respect to the filtration G. The wealth of equals
+
+
and a trading strategy
+ is said to be seIf-financingif 3
Vt(#) = Vo(+)+
C +: '=I
dy:,
V f E LO. 7-1.
0
Let @ stand for the class of all self-financing trading strategies. We shall first prove that a self-financing strategy is determined by its initial wealth and the two components $2, +3. To this end, we postulate that the price of Y 1 follows a strictly positive process, and we choose Y' as a numCraire asset. We shall now analyze the relative values:
Lemma 3.1. (i) For any
+
E
@, we have
(ii) Conversely, let X be a GT-measurablerandom variable, and let us assume that there exists x E IR and 6-predictable processes I$, i = 2,3 such that
+
Then thereexistsa 6-predictableprocess 4' such that thestrategy = (+I, @2,@3) is self-financingandreplicates X. Moreover, the wealth process of 4 (i.e. the time-t price of X) satisfies Vt(@)= v:Y:, where
Proof. The proof of part (i) is given, for instance, in Protter [34]. In the case of continuous semimartingales, this is a well-known result; for discontinuous processes, the proof is not much different. We reproduce it here for the reader's convenience. Let us first introduce some notation. As usual, [X,Y] stands for the quadratic covariation of the two semi-martingalesX and Y, as defined by the integration by parts formula: XtYt = XoYo +
S
XU- dY, t
$
Y,- dX,
+ [X, Y]f.
For any cadlag (i.e., RCLL) process Y, we denote by AYt = Yt - Yt- the size of the jump at time t. Let V = V(+) be the value of a self-financing strategy, and let V1 = V1($) = V($)(Y1)-I be its value relative to the numeraire Y1. The integration by parts formula yields
c:=~
$fd ~ f Hence, . From the self-financing condition, we have dVt = using elementary rules to compute the quadratic covariation [X, Y] of the two semi-martingales X, Y, we obtain
We now observe that
Y:- d(y:)-l+ (Y:-)-' d ~ +: d[(yl)-l, yllt = ~(Y:(Y:)-~)= o and
yf-d(y;)-l
+ (y;-)-l
Consequently,
dv: =
dy; + d[(Y1)-l, Yi]t = d((y:)-'r,).
+: d~:'
+ 4;
d ~ y ,
as was claimed in part (i). We now proceed to the proof of part (ii). We assume that (27) holds for some constant x and processes +2, +3, and we define the process V' by setting (cf. (28))
Next, we define the process
as follows:
where Vt = V: Y:. Since dV: =
c ;+f~dYi1, ~ we obtain
From the equality
it follows that
and our aim is to prove that dVt =
c;=,+fd ~ fThe . last equality holds if
~ y = ~
i.e., if AV: = $~AY;~, which is the case from the definition (28) of V1. Note also that from the second equality in (29) it follows that the process $' is indeed G-predictable. Finally, the wealth process of $ satisfies Vt($) = VtY: for every t E [0, TI, and thus VT($) = X. We say that a self-financing strategy @I replicates a claim X E &-if
or equivalently,
Suppose that there exists an e.m.m. for some choice of a numeraire asset, and let us restrict our attention to the class of all admissible trading strategies, so that our model is arbitrage-free. Assume that a claim X can be replicated by some admissible trading strategy, so that it is attainable (or hedgeable). Then, by definition, the arbitrage price at time t of X, denoted as nt(X), equals Vt($) for any admissible trading strategy that replicates X. In the context of Lemma 3.1, it is natural to choose as an e.m.m. a probability measure Q1 equivalent to Pon (a,&-) and such that the prices Yill, i = 2,3, are G-martingales under Q1. If a contingent claim X is hedgeable, then its arbitrage price satisfies
+
nt(X) = Y:IEQI (X(Y;)-'
I GI).
We emphasize that even if an e.m.m. Q1 is not unique, the price of any hedgeable claim X is given by this conditional expectation. That is to say, in case of a hedgeable claim these conditional expectations under various equivalent martingale measures coincide. In the special case where Y: = B(t,T) is the price of a default-free zero-coupon bond with maturity T (abbreviated as ZC-bond in what follows), Q1 is called T-forward martingale measure, and it is denoted by QT. Since B(T,T) = 1, the price of any hedgeable claim X now equals nt(X) = B(t, T) IEQ,(X I Gt). 3.2 Constrained strategies In this section, we make an additional assumption that the price process Y3 is strictly positive. Let $ = (ql,q2,+3) be a self-financing trading strategy satisfying the following constraint:
for a predetermined, G-predictable process Z. In the financial interpretation, equality (30) means that a portfolio @ is rebalanced in such a way that the total wealth invested in assets Y1, Y2 matches a predetermined stochastic process Z. For this reason, the constraint given by (30) is referred to as the balance condition. Our first goal is to extend part (i)in Lemma 3.1 to the case of constrained strategies. Let @(Z)stand for the class of all (admissible) self-financing trading strategies satisfying the balance condition (30). They will be sometimes referred to as constrained strategies. Since any strategy E @(Z)is self-financing, from dVt(+) = d ~ fwe , obtain
+
c:=~ +f
By combining this equality with (30), we deduce that
Let us write yt3 = Yf(~;)-l,Z: = Zt(Y;)-l. The following result extends Lemma 1.7 in Bielecki et al. [4] from the case of continuous semimartingales to the general case (see also [5]). It is apparent from Proposition 3.1 that the wealth process V(@)of a strategy @ E @(Z)depends only on a single component of namely, +2.
+,
Proposition 3.1. The relative wealth VB(+) = Vt(+)(Y;)-l of any trading strategy E @(Z)satisfies
+
Proof. Let us consider discounted values of price processes Y', Y2,Y3, with Y3 taken as a numeraire asset. By virtue of part (i) in Lemma 3.1, we thus have
The balance condition (30) implies that
C $;Y;-' L
= z;,
and thus
By inserting (33) into (32), we arrive at the desired formula (31). The next result will prove particularly useful for deriving replicating strategies for defaultable claims.
Proposition 3.2. Let a GT-measurable random variable X represent a contingent claim that settles at time T. We set
where, by convention, Y; = 0. Assume that there exists a G-predictable process such that
+2,
Then there exist G-predictable processes and @3 such that the strategy @ = $) belongs to @(Z) and replicates X . The wealth process of@equals,for every f E [0,TI,
Proof. As expected, we first set (note that the process process)
is a G-predictable
and
Arguing along the same lines as in the proof of Proposition 3.1, we obtain
Now, we define
where Vt =
vBYB.
As in the proof of Lemma 3.1, we check that
+
and thus the process q3 is G-predictable. It is clear that the strategy = (+I, +2,+3) is self-financing and its wealth process satisfies Vt(+) = Vt for every t E [0, TI. In particular, VT(+) = X, SO that 4 replicates X. Finally, equality (37) implies (30), and thus belongs to the class @(Z). o
+
Note that equality (35)is a necessary (by Lemma 3.1) and sufficient (by Proposition 3.2) condition for the existence of a constrained strategy that replicates a given contingent claim X. 3.2.1 Synthetic asset Let us take Z = 0, so that E @(O). Then the balance condition becomes $ f ~ =f -0, and formula (31) reduces to
+
The process y2 = Y3Y', where Y' is defined in (34) is called a synthetic asset. It corresponds to a particular self-financing portfolio, with the long position in Y2 and the short position of Y? number of shares of Y1, and suitably re-balanced positions in the third asset so that the portfolio is self-financing, as in Lemma 3.1. It can be shown (see Bielecki et al. [5]) that trading in primary assets Y1, Y2,Y3 is formally equivalent to trading in assets Y1, Y2,Y3. This observation supports the name synthetic asset attributed to the process Y2. Note, however, that the synthetic asset process may take negative values. 3.2.2 Case of continuous asset prices In the case of continuous asset prices, the relative price Y' = Y2(Y3)-I of the synthetic asset can be given an alternative representation, as the following result shows. Recall that the predictable bracket of the two continuous semi-martingales X and Y, denoted as (X, y), coincides with their quadratic covariation [X, Y]. Proposition 3.3. Assume that the price processes Y1 and Y2 are continuous.
Then the relative price of the synthetic asset satisfies
-
where Yt := Y?'e-['f and at := (ln Y2,',1n Y3r1),=
(39)
In terms of the auxiliary process
J,.
(Y$')-'(Y;')-~ d(y2,',Y ~ , ' ) ~ .
7, formula (31)becomes
A
where +t = +:(Y?')-leaf. Proof. It suffices to give the proof for Z = 0. The proof relies on the integration by parts formula stating that for any two continuous semimartingales, say X and Y , we have
provided that Y is strictly positive. An application of this formula to processes X = Y2,' and Y = Y3,' leads to
The relative wealth V:(+) = Vt(@)(Y;)-lof a strategy
where we denote
+ E @(O) satisfies
+t = +:(Y:')-leal. A
Remark 3.1. The financial interpretation of the auxiliary process Y will be studied in Sections 4.1.6 and 4.1.8 below. Let us only observe here that if Y' is a local martingale under some probability Q then 7 is a
Q-local martingale (and vice versa, if 7 is a 6-local martingale under some probability 6 then Y' is a G-local martingale). Nevertheless, for the reader's convenience, we shall use two symbols Q and 6, since this equivalence holds for continuous processes only. It is thus worth stressing that we will apply Proposition 3.3 to predefault values of assets, rather than directly to asset prices, within the set-up of a semimartingale model with a common default, as described in Section 2.1. In this model, the asset prices may have discontinuities, but their pre-default values follow continuous processes. 4. Martingale Approach to Valuation and Hedging Our goal is to derive quasi-explicit conditions for replicating strategies for a defaultable claim in a fairly general set-up introduced in Section 2.1. In this section, we only deal with trading strategies based on the reference filtration IF, and the underlying price processes (that is, prices of defaultfree assets and pre-default values of defaultable assets) are assumed to be continuous. Hence, our arguments will hinge on Proposition 3.3, rather than on a more general Proposition 3.1. We shall also adapt Proposition 3.2 to our current purposes. To simplify the presentation, we make a standing assumption that all coefficient processes are such that the SDEs appearing below admit unique strong solutions, and all stochastic exponentials (used as Radon-Nikodym derivatives)are true martingales under respective probabilities. 4.1 Defaultable asset with total default In this section, we shall examine in some detail a particular model where the two assets, Y1 and Y2, are default-free and satisfy
where W is a one-dimensional Brownian motion. The third asset is a defaultable asset with total default, so that
Since we will be interested in replicating strategies in the sense of Definition 2.2, we may and do assume, without loss of generality, that the coefficients pi,tl aj,t, i = 1 , 2 are IF-predictable, rather than 6-predictable. Recall that, in general, there exist IF-predictable processes jT3 and F3such that (41)
-p3,taltsTl= p3,talt
We assume throughout that Y', > 0 for every i, so that the price processes Y', Y2 are strictly positive, and the process Y3 is nonnegative, and has strictly positive pre-default value.
4.1.1 Default-free market It is natural to postulate that the default-free market with the two traded assets, Y1 and Y2, is arbitrage-free. More precisely, we choose Y1 as a numkraire, and we require that there exists a probability measure E", equivalent to P on (a,fi),and such that the process Y211 is a IP1-martingale. The dynamics of processes (y1)-' and y2,' are
and dy? = yy((p2,t - p~,t+ ui,t(u~,t- 02,t))dt + (u2,t - ui,t)dwt), respectively. Hence, the necessary condition for the existence of an e.m.m. P'is the inclusion A c B, where A = ((f,o~)E [0, TI x CI : ( T ~ , ~ ( = o J(~z,t((l!)J ) and B = {(f,o) E [Of T ]x : , ~ ' , ~ ( w=) p 2 , t ( ~ )The ] . necessary and sufficient condition for the existence and uniqueness of an e.m.m. IF" reads
where the process 0 is given by the formula (by convention, 010 = 0)
Note that in the case of constant coefficients, if 01 = 0 2 then the model is arbitrage-freeonly in the trivial case when p2 = p1. Remark 4.1. Since the martingale measure P1 is unique, the default-free model (Y1,Y2) is complete. However, this is not a necessary assumption and thus it can be relaxed. As we shall see in what follows, it is typically more natural to assume that the driving Brownian motion W is multidimensional. 4.1.2 Arbitrage-free property Let us now consider also a defaultable asset Y3. Our goal is now to find a martingale measure Q1 (if it exists) for relative prices y2j1and Y3t1. Recall that we postulate that the hypothesis (H) holds under P for filtrations IF and G = IF v IH. The dynamics of Y3,' under P are
Let Q1 be any probability measure equivalent to P on (a,&), and let q be the associated Radon-Nikodfm density process, so that (45)
d ~16,l = qt dP I Q,?
where the process 7 satisfies
for some 6-predictable processes 8 and S, and 11 is a G-martingale under
n.
From Girsanov's theorem, the processes fi and 6, given by
are G-martingales under Q1. To ensure that Y211 is a Q1-martingale, we postulate that (43) and (44) are valid. Consequently, for the process y3r1to be a Q1-martingale,it is necessary and sufficient that satisfies
To ensure that Q1 is a probability measure equivalent to IP,we require that St 7 -1. The unique martingale measure QI is then given by the formula (45) where 11 solves (46), so that
We are in a position to formulate the following result. Proposition 4.1. Assume that the process 8 given by (44)satisfies (43), and
Then the model M = (Y1,Y2,Y3;cD) is arbitrage-free and complete. The dynamics of relative prices under the unique martingale measure Q1 are
Since the coefficients pi,t, ci,t, i = 1,2, are IF-adapted, the process W is an IF-martingale (hence, a Brownian motion) under Q1. Hence, by virtue of Proposition 1.1, the hypothesis (H) holds under Q1, and the IF-intensity of default under Q1 equals ~r = yt(1 + St) = yt +
r1.t - r2,t Cl,t
- 02.t
b3,t
- a,t)
Example 4.1. We present an example where the condition (48) does not hold, and thus arbitrage opportunities arise. Assume the coefficients are constant and satisfy: pl = p2 = ol = 0, p3 < -y for a constant default intensity y > 0. Then
where V($) represents the wealth of a self-financing strategy (+I, +2, 0) with +2 = Hence, the arbitrage strategy would be to sell the asset Y3, and to follow the strategy $.
2.
Remark 4.2. Let us stress once again, that the existence of an e.m.m. is a necessary condition for viability of a financial model, but the uniqueness of an e.m.m. is not always a convenient condition to impose on a model. In fact, when constructing a model, we should be mainly concerned with its flexibility and ability to reflect the pertinent risk factors, rather than with its mathematical completeness. In the present context, it is natural to postulate that the dimension of the underlying Brownian motion equals the number of tradeable risky assets. In addition, each particular model should be tailored to provide intuitive and handy solutions for a predetermined family of contingent claims, which will be priced and hedged within its framework. 4.1.3 Hedging a survival claim
We first focus on replication of a survival claim (X,O,.r), that is, a defaultable claim represented by the terminal payoff XI(T<71r where X is an TT-measurablerandom variable. For the moment, we maintain the simplifying assumption that W is one-dimensional. As we shall see in what follows, it may lead to certain pathological features of a model. If, on the contrary, the driving noise is multi-dimensional, most of the analysis remains valid, except that the model completeness is no longer ensured, in general. Recall that F3stands for the pre-default price of Y3, defined as (see (21))
with y: = Y:. This strictly positive, continuous, IF-adapted process enjoys the property that Y: = alt<71Y:. Let us denote the pre-default values in = $(y:)-l, i = 1,2, and let us introduce the the numeraire y3 by pre-default relative price of the synthetic asset P2by setting 3.3
c3Y'
-
-
and let us assume that ul,t -02,t # 0. It is also useful to note that the process Y, defined in Proposition 3.3, satisfies
In Sections 4.1.6 and 4.1.8, we shall show that in the case, where a given by (39) is deterministic, the process has a nice financial interpretation as a credit-risk adjusted forward price of Y2 relative to Y1. Therefore, it is more when dealing with the general convenient to work with the process case, but to use the process 7when analyzing a model with deterministic volatilities. Consider an IF-predictable self-financing strategy @ satisfying the balance condition q5:Y: + @:Yt = 0, and the corresponding wealth process
7
Y*
@:F:.
Since the process V($) is IF-adapted, we see that Let Vt(4) := this is the pre-default - price process of the portfolio q5, that is, we have n,,,tlVt(+) = l,,,tlVt(+); we shall call this process the pre-default wealth of 4. Consequently, the process := vt(@)(y;)-l = 4: is termed the relative pre-default wealth. Using Proposition 3.1, with suitably modified notation, we find that the IF-adapted process V3(@)satisfiesl for every t E [O, TI,
v:(+)
Define a new probability on (a,FT) by setting
where dl]; = q; 8; dWt, and
2,
The process t E [OrTI, is a (local)martingale under Q*.We shall require that this process is in fact a true martingale; a sufficient condition for this is that
From the predictable representation theorem, it follows that for any X E FT, such that x(?;)-' is square-integrable under Q , there exists a constant x and an IF-predictable process q52 such that
We now deduce from Proposition 3.2 that there exists a self-financing strategy q5 with the pre-default wealth Vt(q5) = ?;V; for every t E [0, TI, where we set
Moreover, it satisfies the balance condition q5:~: t E [0, TJ.Since clearly VT(+) = X, we have that
+ #Y:
= 0 for every
and thus this strategy replicates the survival claim (X, 0, T).In fact, we have that Vt(@)= 0 on the random interval [T, T I .
Definition - 4.1. We say that a survival claim (X,O, T) is attainable if the process V3 given by (52) is a martingale under Q*. The following result is an immediate consequence of (51) and (52).
Corollary 4.1. Let X E 7;.be such that x(Y;)-~ is square-integrable under QT. Then the survival claim (X, 0, T) is attainable. Moreover, the pre-default price nt(X,0 , ~of) the claim (X, 0, T) is given by the conditional expectation
-
(53)
-
-
nt(X, 0, T) = Y; IEQ.(x(?;)-~
I Ti),
Vt
6
[0, TI.
The process ?(X, 0, ,r)(y3)-l is an IF-martingale under Q*. Proof. Since x(Y;)-' is square-integrable under Q*, we know from the predictable representation theorem that q2 in (51) is such that
(%(+:)'d(~),)
EQ.
<
m,
so that the process
v3 given by (52) is a true
martingale under Q*. We conclude that (X, 0, T) is attainable. Now, let us denote by nt(X, 0, T) the time-t price of the claim (X, 0, T). Since is a hedging portfolio for (X, 0, T) we thus have Vt($J)= nt(X, 0 , ~ ) for each t E [0, TI. Consequently,
for each t E [0, TI. This proves equality (53). Inview of the last result, it is justified to refer to Q' as the pricing measure relative to y3 for attainable survival claims. Remark 4.3. It can be proved that there exists a unique absolutely continuous probability measure Q on (a,BT)such that we have
However, this probability measure is not equivalent to Q, since its RadonNikodym density vanishes after T (for a related result, see Collin-Dufresne ef al. [13]). Example 4.2. We provide here an explicit calculation of the pre-default price of a survival claim. For simplicity, we assume that X = 1,so that the claim represents a defaultable zero-coupon bond. Also, we set yt = y = const, pi,t = 0, and ai,t = ai, i = 1,2,3. Straightforward calculations yield the following pricing formula
We see that here the pre-default price Eo(l,0, T)depends explicitly on the intensity y, or rather, on the drift term in dynamics of pre-default value of defaultable asset. Indeed, from the practical viewpoint, the interpretation of the drift coefficient in dynamics of Y2 as the real-world default intensity is questionable, since within our set-up the default intensity never appears as an independent variable, but is merely a component of the drift term in dynamics of pre-default value of Y3. Note also that we deal here with a model with three tradeable assets driven by a one-dimensional Brownian motion. No wonder that the model enjoys completeness,but as a downside, it has an undesirable property that the pre-default values of all three assets are perfectly correlated. Consequently, the drift terms in dynamics of traded assets are closely linked to each other, in the sense, that their behavior under an equivalent change of a probability measure is quite specific. As we shall see later, if traded primary assets are judiciously chosen then, typically, the pre-default price (and hence the price) of a survival claim will not explicitly depend on the intensity process. Remark 4.4. Generally speaking, we believe that one can classify a financial model as 'realistic' if its implementation does not require estimation of drift parameters in (pre-default)prices, at least for the purpose of hedging
and valuation of a sufficiently large class of (defaultable)contingent claims of interest. It is worth recalling that the drift coefficients are not assumed to be market observables. Since the default intensity can formally interpreted as a component of the drift term in dynamics of pre-default prices, in a realistic model there is no need to estimate this quantity. From this perspective, the model considered in Example 4.2 may serve as an example of an 'unrealistic' model, since its implementation requires the knowledge of the drift parameter in the dynamics of Y3. We do not pretend here that it is always possible to hedge derivative assets without using the drift coefficients in dynamics of tradeable assets, but it seems to us that a good idea is to develop models in which this knowledge is not essential. Of course, a generic semimartingale model considered until now provides only a framework for a construction of realistic models for hedging of default risk. A choice of tradeable assets and specification of their dynamics should be examined on a case-by-case basis, rather than in a general semimartingaleset-up. We shall address this important issue in the foregoing sections, in which we shall deal with particular examples of practically interesting defaultable claims. 4.1.4 Hedging a recovery process Let us now briefly study the situation where the promised payoff equals zero, and the recovery payoff is paid at time T and equals Z, for some IFadapted process Z. Put another way we consider a defaultable claim of the form (0, Z, T). Once again, we make use of Propositions 3.1 and 3.2. In view of (35), we need to find a constant x and an IF-predictable process 42 such that
Similarly as in Section 4.1.3 we conclude that, under suitable integrability conditions on qT, there exists @* such that dqt = qb:dY;, where qt = 1 7;).We now set IEQ
v:
so that, in particular, = 0. Then it is possible to find processes 0' and $3 such that the strategy $ is self-financing and it satisfies: = and Vt($) = Zt + $:Yz for every t E [0, TI. It is thus clear that V,($) = Z , on the set (T I TI and VT($) = 0 on the set (T > TI.
vt(@) v;?;
4.1.5 Bond market For the sake of concreteness, we assume that Y: = B(t, T) is the price of a default-free ZC-bond with maturity T, and Y: = D(t, T) is the price of a defaultable ZC-bond with zero recovery, that is, an asset with the terminal payoff Y3,= I1l~
for some IF-predictable processes p(t, T ) and b(t, T). We choose the process Y: = B(t, T ) as a numeraire. Since the prices of the other two assets are not given a priori, we may choose any probability measure Q equivalent to P on (Q,GT)to play the role of Q1. In such a case, an e.m.m. Q1 is referred to as the forward martingale measure for the date T, and is denoted by QT. Hence, the Radon-Nikodym density of QT with respect to IF' is given by (46) for some IF-predictable processes 8 and 5, and the process
w:
=
wt -
t
e,,du, v t t LO, TI,
is a Brownian motion under QT. Under QT the default-free ZC-bond is governed by dB(t, T)) = B(t, T)(fit, T )dt + b(t, T)~w:) where p(t, T ) = p(t, T ) + 8tb(t, T). ~ e7tstand for the IF-hazard process of T under QT,SO thatTt = - ln(1 - Ft), where = QT(T5 t 1%). Assume that the hypothesis (H) holds under QT SO that, in particular, the process T is increasing. We define the price process of a defaultable ZC-bond with zero recovery by the formula
where the second equality follows from Lemma 1.3. It is then clear that Y': = D(t, T)(B(t,T))-' is a QT-martingale,and the pre-default price 6(t, T ) equals E(t, q = ~ ( tq,
1 @.
The next result examines the basic properties of the auxiliary process?(t, T ) given as, for every t E [0, TI,
The quantity?(t, T) can be interpreted as the conditional probability (under QT)that default will not occur prior to the maturity date T, given that we observe 7; and we know that the default has not yet happened. We will be more interested, however, in its volatility process p(t, T) as defined in the following result.
Lemma 4.1. Assume that the IF-hazard process f;of T under QT is continuous. Then the processF(t, T), t E [0,TI,is a continuous IF-submartingaleand
for some IF-predictable process p(t, 7'). The pvocess?(t, 7') is offinitevariation ifand only if the hazard process f:is deterministic. In this case, we haveF(t, T) = ec-F~. Proof. We have r(t, T) =
IEQ~(P-~T 1%) = eT!~,,
where we set Lt = iEQT(e-'~1 E). Hence, ?(t, 71 is equal to the product of a
-
strictly positive, increasing, right-continuous, IF-adapted process erl, and a strictly positive, continuous IF-martingale L. Furthermore, there exists an IF-predictable process B(tfT) such that L satisfies
). (56) now follows by an with the initial condition la = ~ ~ ( e - ' ~Formula
7'). To complete the application of ItBfsformula, by setting P(t, T ) = e-'#t, proof, it suffices to recall that a continuous martingale is never of finite variation, unless it is a constant process.
Remark 4.5. It can be checked that p(t, T ) is also the volatility of the process
7,
Assume that?t = J~ du for some IF-predictable, nonnegative process A
y. Then we have the following auxiliary result, which gives, in particular, the volatility of the defaultable ZC-bond.
Corollary 4.2 The dynamics under QTof the pre-default price E(t, T ) equals
Equivalently, the price D(t, T ) of the defaultable ZC-bond satisfies under QT dD(t, T) = D(t, ~ ) ( ( p ( T) t , + b(t, T)p(t,T ) ) dl + a t , T) d ~ : d ~ ~ ) . where we set d(t, T) = b(t, T) + p(t, T). Note that the process p(t, T) can be expressed in terms of market observable~,since it is simply the difference of volatilities d(t, T ) and b(t, T) of pre-default prices of tradeable assets. 4.1.6 Credit-risk-adjusted forward price Assume that the price Y2 satisfies under the statistical probability IP
with IF-predictable coefficients p and 0. Let Fr(t, T) = Y:(B(t, T))-' be the forward price of Y;. For an appropriate choice of 8 (see 50), we shall have that dFp(t, T) = Fp(t, ~ ) ( o-t b(t, T)) dW:. Therefore, the dynamics of the pre-default synthetic asset
under Q* are
-
and the process Yt = .Y?'e-"l satisfies
6
Let be an equivalent probability measure on (a,&) such that (or, equivalently, Y*) is a 6-martingale. By virtue of Girsanov's theorem, the process ? i given by the formula
is a Brownian motion under 6 . Thus, the forward price Fr (t, T) satisfies under
6
(58)
dFr(t, 7') = Fr(t, ~ ) ( o-t b(t, T))(I%
+ P(t, T) dt).
It appears that the valuation results are easier to interpret when they are expressed in terms of forward prices associated with vulnerable forward contracts, rather than in terms of spot prices of primary assets. For
this reason, we shall now examine credit-risk-adjusted forward prices of default-free and defaultable assets. Definition4.2. Let Y be a &-measurable claim. An %-measurable random variable K is called the credit-risk-adjustedforward price of Y if the pre-default value at time t of the vulnerable forward contract represented by the claim n(T<.rl(Y- K) equals 0. Lemma 4.2. The credit-risk-adjustedforward price &(t, T ) of an attainable survival claim (X, 0, T),represented by a GT-measurableclaim Y = Xn(T
G.
Proof. The forward price is defined as an 7;-measurable random variable K such that the claim
is worthless at time t on the set (t < 7). It is clear that the pre-default value at time t of this claim equals Zt(X, 0, T) - KD(~, T). Consequently, we obtain a FY(~ T ,) = z~(x,0, .r)(D(t,T))-'. Let us now focus on default-free assets. Manifestly, the credit-riskadjusted forward price of the bond B(t, T) equals 1. To find the credit-riskadjusted forward price of Y2, let us write
where a is given by (see (39))
Lemma 4.3. Assume that a given by (59) is a deterministicfunction. Then the credit-risk-adjustedforward price of y2 equals &(t, T)for every t E [0, TI.
Proof. According to Definition 4.2, the price &(t, T) is an %-measurable random variable K, which makes the forward contract represented by the claim D(T, T)(Y; - K) worthless on the set {t < T}. Assume that the claim Y: - K is attainab1e.l Since D(T, T) = 1, from equation (53) it follows that the pre-default value of this claim is given by the conditional expectation
'Attainability of this claim can be shown in a similarway as the attainabilityof a vulnerable call option considered in Section 4.1.7.
Consequently,
-Fr(t, T) = E ~ ( Y 1;5 )= E6(FP(T, T) 1 5 )= Fr (t, T )eaT"l,
as was claimed. It is worth noting that the p r o c e s s ~ ( tT) , is a (local)martingale under the pricing measure Q, since it satisfies A
A
Under the present assumptions, the auxiliary process Y introduced in Proposition 3.3 and the credit-risk-adjusted forward price Fv(t, T ) are closely related to each other. Indeed, we have Fp(t,7') = YtenT, so that the two processes are proportional. 4.1.7 Vulnerable option on a default-free asset We shall now analyze a vulnerable call option with the payoff
Our goal is to find a replicating strategy for this claim, interpreted as a survival claim (X,O, T) with the promised payoff X = CT = (Y; - K)+, where CTis the payoff of an equivalent non-vulnerableoption. The method presented below is quite general, however, so that it can be applied to any survival claim with the promised payoff X = G(G) for some function G : lR + IR satisfying the usual integrability assumptions. We assume that Y: = B(t, T), Y: = D(t, T ) and the price of a default-free asset Y2 is governed by (57). Then
We are going to apply Proposition 3.3. In the present set-up, we have Y? = Fr(t, T) and Yt = Fr(t, T)e-"1. Since a vulnerable option is an example of a survival claim, in view - of - Lemma 4.2, its credit-risk-adjusted forward price satisfies Fcd(t, 7') = C?(D(~, T))-'. A
Proposition 4.2. Suppose that the volatilities cr, b and fl are deterministicfuncfions. Then the credit-risk-adjusted forward price of a vulnerable call option written on a default-free asset y2 equals
where
and
v2(f, T ) =
S
(o,- b(u, T ) )du. ~
The replicating strategy @ in the spot market satisfies for every t E [0, TI, on the set { t < TI,
where d+(f,T ) = d+(&(t, T), t, T). Proof. In the first step, we establish the valuation formula. Assume for the moment that the option is attainable. Then the pre-default value of the option equals, for every t E [0, TI,
In view of (60),the conditional expectation above can be computed explicitly, yielding the valuation formula (61). To find the replicating strategy, and establish attainability of the option, we consider the It6 differential dFcd(t,7') and we identify terms in (52). It appears that (63)
d ~ ~ T~ ) = ( tN(d+(t, , T ) )d'Tyz(t,T ) = N(d+(t,T))eNT dFt = N(d+(t,T))?~enT-"'d e ,
so that the process r # ~in ~ (51)equals
+:c
-
-
Moreover, is such that + : ~ ( tT, ) + = 0 and 4; = c ; ( D ( ~T))-'. , It is easily seen that this proves also the attainability of the option. Let us examine the financial interpretation of the last result. First, equality (63) shows that it is easy to replicate the option using vulnerable forward contracts. Indeed, we have
A
Fcd(T,T) = X =
+
ST
7 Cd,
D(O, T )
N(d+(t,T))d ~ (t,pT )
-
and thus it is enough to invest the premium C; = C( in defaultable ZCbonds of maturity T , and take at any instant t prior to default N(d+(t,T ) ) positions in vulnerable forward contracts. It is understood that if default
occurs prior to T, all outstanding vulnerable forward contracts become void. Second, it is worth stressing that neither the arbitrage price, nor the replicating strategy for a vulnerable option, depend explicitly on the default intensity. This remarkable feature is due to the fact that the default risk of the writer of the option can be completely eliminated by trading in defaultable zero-coupon bond with the same exposure to credit risk as a vulnerable option. In fact, since the volatility 9, is invariant with respect to an equivalent change of a probability measure, and so are the volatilities 0 and b(t, T), the formulae of Proposition 4.2 are valid for any choice of a forward measure QT equivalent to P (and, of course, they are valid under P as well). The only way in which the choice of a forward measure QT makes an impact on these results is through the pre-default value of a defaultable ZC-bond. We conclude that we deal here with the volatility based relative pricing a defaultable claim. This should be contrasted with the more popular intensity-based risk-neutral pricing, which is commonly used to produce an arbitrage-free model of tradeable defaultable assets. Recall, however, that if tradeable assets are not chosen carefully for a given class of survival claims, then both hedging strategy and pre-default price may depend explicitly on values of drift parameters, which can be linked in our setup to the default intensity (see Example 4.2).
Remark 4.6. Assume that X = G(Y+)for some function G : lR -+ lR. Then the credit-risk-adjustedforward price of a survival claim satisfies &(t, T) = v(t, Fv (t, T)), where the pricing function v solves the PDE
with the terminal condition v(T,fi = in Section 5 below.
~ ( f iThe . PDE approach is studied
Remark 4.7. Proposition 4.2 is still valid if the driving Brownian motion is two-dimensional, rather than one-dimensional. In an extended model, the volatilities cst, b(t, T) and P(t, T) take values in lR2 and the respective products are interpreted as inner products in R2. Equivalently, one may prefer to deal with real-valued volatilities, but with correlated one-dimensional Brownian motions. 4.1.8 Vulnerable swaption
In this section, we relax the assumption that y1 is the price of a defaultfree bond. We now let y1 and Y2 to be arbitrary default-free assets, with
dynamics dl'; = ~
i ( ~ idt , t
+ +j,,
dwt), i = I, 2.
We still take D(t, T) to be the third asset, and we maintain the assumption that the model is arbitrage-free, but we no longer postulate its completeness. In other words, we postulate the existence an e.m.m. Q1, as defined in Section 4.1.2, but not the uniqueness of Q1. We take the first asset as a numeraire, so that all prices are expressed in units of Y1. In particular, Yitl = 1 for every t E lR+,and the relative prices Y211 and Y3,' satisfy under Q1 (cf. Proposition 4.1)
It is natural to postulate that the driving Brownian noise is twodimensional. Under this assumption, we may represent the joint dynamics of y2,1and Y3r1 under as follows
where W1, W2 are one-dimensional Brownian motions under Q1, such that d(W1,W2)t = pt dt for a deterministic instantaneous correlation coefficient p taking values in [-I, I]. We assume from now on that the volatilities oi, i = 1,2,3 are deterministic. Let us set
6
(a&-)
and let be an equivalent probability measure on such that the process Yf = YYe-"l is a 6-martingale. To clarlfy the financial interpretation of the auxiliary process F i n the present context, we introduce the concept of credit-risk-adjusted forward price relative to the numeraire Y1.
Definition 4.3. Let Y be a Gpmeasurable claim. An %-measurablerandom variable K is called the time-t credit-risk-adjusted Y1-forwardprice of Y if the pre-default value at time t of a vulnerable forward contract, represented by the claim
equals 0.
Gyl
The credit-risk-adjusted Y1-forward price of Y is denoted by (t,T), and it is also interpreted as an abstract defaultableswap rate. The following auxiliary results are easy to establish, along the same lines as Lemmas 4.2 and 4.3. Lemma 4.4. The credit-risk-adjusted Y1-forward price of a survival claim Y =
(X,0, T ) equals F
~(t,T~ ) =Izt(x1, 0, .r)(D(t,v1-l
where x1= x(Y;)-' is the price of X in the numtraire Y 1 ,and Zt(x1,0 , ~is) the pre-default value ofa survival claim with the promised payojFX1. Proof. It suffices to note that for Y = niT<,)X1we have
, to consider the pre-default values. where X1 = x ( Y ~ ) - ' and Lemma 4.5. The credit-risk-adjusted Y1-forward price of the asset Y 2 equals A
(65)
A
Fyzlyl(t,T) = Y:' ear-'! = YteNTl
where a is given by (64). Proof. It suffices to find an 7;-measurable random variable K for which
Consequently, K = Fyzlyl(t,T), where
where we have used the facts that yt = ~ ; ' e - " ~is a Q-martingale, and a is deterministic. We are in a position to examine a vulnerable option to exchange defaultfree assets with the payoff
The last expression shows that the option can be interpreted as a vulnerable swaption associated with the assets Y 1 and Y 2 . It is useful to observe that
so that, when expressed in the numeraire Y1, the payoff becomes
where c:,~ = c:(Y:)-' and D1(t,T) = D ( ~ , T ) ( Y : ) -stand ~ for the prices relative to Y 1 . It is clear that we deal here with a model analogous to the model examined in Sections 4.1.5 and 4.1.7 in which, however, all prices are now relative to the numeraire Y 1 . This observation allows us to directly derive the valuation formula below from Proposition 4.2. Proposition 4.3. The credit-risk-adjusted Y1-forward price of a vulnerable call option written with the payofgiven by (66)equals
-
where d*(Et,T) =
In f - In K
&
i v 2 ( t ,T )
v ( f ,T )
and
The replicating strategy @ in the spot market satisfies for every f E [0,TI, on the set { t < r } ,
where d+(t,T ) = d+(Fp(t,T ) ,1.7). Proof. The proof is analogous to that of Proposition 4.2, and thus it is omitted. 13 It is worth noting that the payoff (66)was judiciously chosen. Suppose instead that the option payoff is not specified by (66),but it is given by an apparently simpler expression
Since the payoff Cd, can be represented as follows
4
where G ( y l ,y2, y3) = y 3 ( ~ 2- Kyl)+, the option can be seen an option to exchange the second asset for K units of the first asset, but with the payoff
expressed in units of the defaultable asset. When expressed in relative prices, the payoff becomes
= D'(T, T ) Y ~ .It is thus rather clear that it is not longer possible where I(T<,~ to apply the same method as in the proof of Proposition 4.2. 4.2 Two defaultable assets with total default We shall now assume that we have only two tradeable assets, and both are defaultable assets with total default. This case is also examined by Carr [12], who studies the imperfect hedging of digital options. Note that here we present results for the perfect hedging. We shall briefly outline the analysis of hedging of a survival claim. Under the present assumptions, we have, for i = 1,2,
where W is a one-dimensional Brownian motion, so that
-
Y: = n1t<,,y:,
y: = n,t<,,y:,
with the pre-default prices governed by the SDEs
The wealth process V associated with the self-financing trading strategy ($I, $*) satisfies, for every t E [0, TI,
- -
= Y:/Y:. Since both primary traded assets are subject to total where default, it is clear that the present model is incomplete, in the sense, that not all defaultable claims can be replicated. We shall check in Section 4.2.1 that, under the assumption that the driving Brownian motion W is onedimensional, all survival claims satisfying natural technical conditions are hedgeable, however. In the more realistic case of a two-dimensional noise, we will still be able to hedge a large class of survival claims, including options on a defaultable asset (see Section 4.2.2) and options to exchange defaultable assets (see Section 4.2.3). 4.2.1 Hedging a survival claim For the sake of expositional simplicity, we assume in this section that the driving Brownian motion W is one-dimensional. This is definitely not
the right choice, since we deal here with two risky assets, and thus their prices will be perfectly correlated. However, this assumption.is convenient for the expositional purposes, since it ensures the model completenesswith respect to survival claims, and it will be later relaxed anyway. We shall argue that in a model with two defaultable assets governed by (68), replication of a survival claim (X,O,.r) is in fact equivalent to replication of the promised payoff X using the pre-default processes.
+;,
Lemma 4.6. Ifa strategy i = 1,2, based on pre-default values pi, i = 1,2, is a replicating strategy for an FT-measurable claim X, that is, if+ is such that the process Vt(+) = +:?: + +:?: satisfies, for every t E [0, TI,
then for the process Vt(+) = +:Y:
This means that a strategy
+ +:Y:
we have, for every t
E
[O,T],
+ replicates a survival claim (X, 0 , ~ ) .
Proof. It is clear that Vt(+) = n{t,,t Vt(+) =
-
Vt(q5). From
it follows that
that is,
+:dy: + +:dy:
= d(a,t
It is also obvious that VT(+)= XIL{T<TI Combining the last result with Lemma 3.1, we see that a strategy (+I,+2) replicates a survival claim (X, 0 , ~whenever ) we have
for some constant x and some IF-predictable process qb2,where, in view of (691,
G,
We introduce a probability measure equivalent to IP on (a,&-), and such that Y 2 r 1 is an IF-martingale under 6. It is easily seen that the RadonNikodym density 7 satisfies, for f E 10, TI,
with
Ot =
p2,t - p1,t + o1.tto1,t - o2,t) Dl,t
- 02,t
I
provided, of course, that the process 8 is well defined and satisfies suitable integrability conditions. We shall show that a survival claim is attainable if the random variable x(Y$)-' is 6-integrable. Indeed, the pre-default value Vt at time t of a survival claim equals
-
and from the predictable representation theorem, we deduce that there exists a process such that
The component of the self-financing trading strategy 4 = (+I, +2) is then chosen in such a way that
To conclude, by focusing on pre-default values, we have shown that the replication of survival claims can be reduced here to classic results on replication of (non-defaultable)contingent claims in a default-free market model. 4.2.2 Option on a defaultable asset In order to get a complete model with respect to survival claims, we postulated in the previous section that the driving Brownian motion in dynamics (68) is one-dimensional. This assumption is questionable, since it implies the perfect correlation of risky assets. However, we may relax this restriction, and work instead with the two correlated one-dimensional Brownian motions. The model will no longer be complete, but options on a defaultable assets will be still attainable. The payoff of a (non-vulnerable) call option written on the defaultable asset Y2 equals
so that it is natural to interpret this contract as a survival claim with the promised payoff X = (?; - K)+. To deal with this option in an efficient way, we consider a model in which
where w1 and w2are two one-dimensional correlated Brownian motions with the instantaneous correlation coefficient pt. More specifically, we assume that Y: = D(t, T) = 1lt,Tt6(t,T) represents a defaultable ZC-bond with zero recovery, and Y: = ?ll,,Tt?~is a generic defaultable asset with total default. Within the present set-up, the payoff can also be represented as follows cT= G(Y;, Y;) = (Y; - KY;)+, where g(y1, y2) = (y2 - Kyl)+,and thus it can also be seen as an option to exchange the second asset for K units of the first asset. The requirement that the process ytl = follows an IFmartingale under implies that
y(Y:)-'
-
-
where fi-= (W1, W2) follows a two-dimensional Brownian motion under Q. Since Y; = 1, replication of the option reduces to finding a constant x and an IF-predictable process cp2 satisfying
To obtain closed-form expressionsfor the option price and replicating strategy, we postulate that the volatilities olt, o2,t and the correlation coefficient pt are deterministic. Let Fv(t, T ) = ?:(E(t, T))-l (Fc(t,T ) = Ct(D(t,T))-l, respectively)stand for the credit-risk-adjusted forward price of the second asset (the option, respectively). The proof of the following valuation result is fairly standard, and thus it is omitted.
--
Proposition 4.4. The credit-risk-adjustedforward price of the option written on Y2 equals A
~ ~ T) ( t= ,Fp (t, ~ ) n ( d + ( F ~T), ( tt,, T)) - ~ ~ ( d - ( F ~77, ; l (f ttT)). ,
Equivalently, the pre-default price of the option equals
where d * ( f , t , ~=)
ln
f-In K f v2(t,T ) v(tl T )
and (u:,$ + G ; , ~ 2pu~l,u02,u)d~. Moreover the replicating strategy @ in the spot market satisfiesfor every t on the set ( t < T),
E
[0,TI,
4.2.3 Option to exchange defaultable assets
We work here with the two correlated one-dimensional Brownian motions, so that
where d(wl, W2)t = pt dt for some function p with values in 1-1, I]. The model is no longer complete, but it is still not difficult to establish a direct counterpart of Proposition 4.4 for the exchange option with the payoff KYk)+. In fact, the next result shows that the pricing formula expressed in terms of pre-default prices has the same shape as the standard formula for the option to exchange non-defaultable assets with dynamics (68). It is notable that we do not need to make any assumption about the behavior of the default intensity. We only assume that the coefficients in (73) are such that there exist an e.m.m. for the process ?2r1, where
(q-
so that we implicitly impose mild technical conditions on drift coefficients. Proposition 4.5. Assume that the volatilities ul,u2 and the instantaneous cou-
relation coeficient p are deterministic. Then the pre-default price of the exchange option equals
where
and
Moreover the replicating strategy q5 in the spot market satisfiesfor every f E [0, TI, on the set ( t < T),
The pricing formula for the option on a defaultable asset (see Proposition 4.4) can be seen as a special case of the formula established in Proposition 4.5. Similarly as in Sections 4.1.7 and 4.1.8, we conclude that the pricing and hedging - - of any attainable survival claim with the promised payoff X = g ( Y i , Y;) depends on the choice of a default intensity only through the pre-default prices 7; and yf. This property shows that we have correctly specified the hedging instruments for a claim at hand. Of course, the model considered in this section is not complete, even if the concept of completeness is reduced to survival claims. Basically, a survival claim can be hedged if its promised payoff can be represents as X = Fkh(~y). 5. PDE Approach to Valuation and Hedging In the remaining part of the paper, we take a different perspective, and we assume that trading occurs on the time interval [0, T]and our goal is to replicate a contingent claim of the form
which settles at time T. We do not need to assume here that the coefficients in dynamics of primary assets are P-predictable. Since our goal is to develop the PDE approach, it will be essential, however, to postulate a Markovian character of a model. For the sake of simplicity, we assume that the coefficients are constant, so that
The assumption of constancy of coefficients is rarely, if ever, satisfied in practically relevant models of credit risk. It is thus important to note that it was postulated here mainly for the sake of notational convenience, and the general results established in this section can be easilv extended to a non-homogeneous Markov case in which pilt = pi(f,Y:-, Y:-, Y:-, Hi-), oijt= oi(t, Y:-, Y:-, Y:-,Ht-), etc.
5.1 Defaultable asset with total default We first assume that Y1 and Y2are default-free, so that ~1 = ~2 = 0, and the third asset is subject to total default, i.e. ~3 = -1,
We work throughout under the assumptionsof Proposition 4.1. This means ;: HT)is attainable, that any Q1-integrablecontingent claim Y = G(Yk,Y;, Y and its arbitrage price equals (75)
nt(Y) = Y!
(Y(Y$)-'1 &),
V t E [0, TI.
The following auxiliary result is thus rather obvious.
Lemma 5.1. The process (Y1,Y2,Y3, H) has the Markov property with respect to thefiltration G under the martingale measure Q1. For any attainable claim Y = G(Yk, Y?,Y;; HT)there exists afunction v : [0, TI x JR3 x {O,1)+ lR such that nt(Y) = v(t, Y:, Y:, Y;: Ht). We find it convenient to introduce the pre-default pricing function v(.;0) = v(t, yl, y2, y3; 0) and the post-default pricing function v(.; 1) = ~ ( tyl, , y2, y ~1). ; In fact, since Y: = 0 if Ht = 1, it suffices to study the post-default function v(t, yl, y2; 1) = v(t, yl, y2,O; 1). Also, we write
Let y > 0 be the constant default intensity under P,and let C > -1 be given by formula (48).
Proposition 5.1. Assume that thefunctions v(. ;0) and v(. ;1)belong to the class C1p2([0,TI x IR:, JR). Then v(t, yl, y2, y3; 0) satisfies the PDE
subject to the terminal condition v(T, yl, y2, y3; 0) = G(y1, y2, y3; O), and v(t, yl, y2; 1)satisfies the PDE
subject to the terminal condition v(T, yl, y2; 1) = G(y1, y2,O; 1).
Proof. For simplicity, we write Ct = nt(Y). Let us define
Then the jump A c t = Ct - Ct- can be represented as follows:
We write d; to denote the partial derivative with respect to the variable yi, and we typically omit the variables (t, Y:-, Yt-, Y:-, Ht-) in expressions dtv, djv, Av, etc. We shall also make use of the fact that for any Bore1 measurable function g we have
since Y; and Y:- differ only for at most one value of u (for each w ) . Let tt = n(t
and this in turn implies that
We now use the integration by parts formula together with (42) to derive dynamics of the relative price = Ct(Y:)-'. We find that
Hence, using (47), we obtain
This means that the process ?admits the following decomposition under Q1
From (75), it follows that the process c i s a martingale under Q1. Therefore, the continuous finite variation part in the above decomposition necessarily
vanishes, and thus we get
Consequently, we have that
Finally, we conclude that
Recall that tt = Illtory. It is thus clear that the pricing functions v(., 0) and v(.;1)satisfy the PDEs given in the statement of the proposition. The next result deals with a replicating strategy for Y . Proposition 5.2. The replicating strategy $for the claim Y is given byformulae
$;Y; = v-+:y: -+;3y;.
Proof. As a by-product of our computations, we obtain
The self-financing strategy that replicates Y is determined by two components $2, $3 and the following relationship:
By identification, we obtain $I;Y;~ = (Y:)-lAv and
This yields the claimed formulae. Corollary 5.1. In the case of a total default claim, the hedging strategy satisfies the balance condition. Proof. A total default corresponds to the assumption that G(y1, y2, ~3~ 1) = 0. We now have v(t, yl, y2; 1) = 0, and thus $;Y3 = v(t, Y:, Y:, Yjl-; 0) for every 1 E [0,7]. Hence, the equality $:Y: + &Y: = 0 holds for every t E [0, TI. The last equality is the balance condition for Z = 0. Recall that it ensures that the wealth of a replicating portfolio jumps to zero at default time. 5.1.1 Hedging with the savings account
Let us now study the particular case where Y1 is the savings account, i.e., d ~ =: r ~ dt, : Y: = 1, which corresponds to pl = r and 01 = 0. Let us writeF= r + T;; where
stands for the intensity of default under Q1. The quantityrhas a natural interpretation as the risk-neutral credit-risk adjusted short-term interest rate. Straightforward calculations yield the following corollary to Proposition 5.1. Corollary 5.2. Assume that 0 2 # 0 and d ~ =: r ~ dt, : d ~ =; Y;(pidt d ~ = : ~
+ 0 2 dwt),
; - ( ~dt 3
+ 0 3 dWt - d ~ t ) .
Then the function v(. ;0) satisfies atu(t, y2, y3; 0) + ry~d~u(t, y ~y3; , 0) +@3a3u(f,y2, y3; 0) -W, y2, y3; 0)
with u(T, y2, y ~0); = G(y2, y3; O), and the function u(. ;1)satisfies 1 dtv(t, y2; 1)+ ry232u(t,y2; 1)+ 5a:y:d~~(t,y2; 1)- ~ ( ty2;, 1)= 0 with u(T, y2; 1) = G(y2,0; 1). In the special case of a survival claim, the function u(.;l) vanishes identically, and thus the following result can be easily established.
Corollary 5.3. The pre-default pricingfunction v(.;O) ofa survival claim Y = lIT,TIG(Y+, Y): is a solution of the following PDE:
with the terminal condition v(T, y2, y3; 0) = G(y2, y3). The components @2 and +3 of the replicating strategy satisfy
Example 5.1. Consider a survival claim Y = l(~,,~g(Y:), that is, a vulnerable claim with default-free underlying asset. Its predefault pricing function u(.; 0) does not depend on ys, and satisfies the PDE (y stands here for y2 and a for 02)
with the terminal condition u(T, y;O) = lllt,Tlg(y).The solution to (76) is v(t, Y) = e F-r)(t-T)ur,g,2(t,Y) = ej;(t-T) Ur,g,2(t,Y), where the function ur*s,2 is the Black-Scholes price of g(YT)in a Black-Scholes model for Yt with interest rate r and volatility 02.
5.2 Defaultable asset with non-zero recovery We now assume that
with ~3 > -1 and ~3 # 0. We assume that Yo3 > 0, SO that Y: > 0 for every t E IR,. We shall briefly describe the same steps as in the case of a defaultable asset with total default. 5.2.1 Arbitrage-free property As usual, we need first to impose specific constraints on model coefficients, so that the model is arbitrage-free. Indeed, an e.m.m. Q1 exists if there exists a pair (8, <) such that
To ensure the existence of a solution (9,c) on the set T < t, we impose the condition Pl - P2 01--=01-01 - 02
P 1 - P3
01
- a3 '
that is, pl(03 - (52) + ~ 2 ( 0 1 - 03)
+ ~ 3 ( 0 2- 01)
= 0.
Now, on the set T 2 t, we have to solve the two equations
If, in addition, (a2- 01)lc3f 0, we obtain the unique solution
so that the martingale measure Q1 exists and is unique. 5.2.2 Pricing PDE and replicating strategy We are in a position to derive the pricing PDEs. For the sake of simplicity, we assume that Y1 is the savings account, so that Proposition 5.3 is a counterpart of Corollary 5.2. For the proof of Proposition 5.3, the interested reader is referred to Bielecki et al. [7]. Proposition 5.3. Let n2 f 0 and let Y1, Y2, y3satis&
dy; = ry: dt,
d ~= : Y ? ( P ~dt + 02 d ~ t ) ,
Assume, in addition, that 02(r- p3) = 03(r- p2) and K~ # 0, ~3 > -1. Then the price of a contingent claim Y = G(Y$,Y;, HT)can be represented as nt(Y)= v(t,Y:, Y:, Ht), where the pricingfunctions v(.;0)and v(.;1)satisfy thefollowing PDEs
and
subject to the terminal conditions v(T,y2, y3; 0 ) = G(yz,y3; O),
v(T,y2, ~ 3 1); = G(y2,y3; 1).
The replicating strategy @ equals
$7 and
@:
=
1 =(v(t,
Y?,Y:-(l + a); 1)- v(t. Y:, Y:-; o)),
is given by @:Y:+ @:Y: + @:Y; = Ct.
5.2.3 Hedging of a survival claim
We shall illustrate Proposition 5.3by means of examples. First, consider a survival claim of the form
Then the post-default pricing function v*(.;1) vanishes identically, and the pre-default pricing function vg(.;0) solves the PDE
with the terminal condition vg(T, y2, y3; 0) = g(y3). Denote a = r - ~ 3 and y p = y(1 + li3). It is not difficult to check that vg(t, y2, y3; 0) = ep(T-t)un,g3(t,y3) is a soy) is the lution of the above equation, where the function w(f, y) = v"~g*~(f, solution of the standard Black-Scholes PDE equation
with the terminal condition w(T, y) = g(y), that is, the price of the contingent claim g(YT) in the Black-Scholes framework with the interest rate a and the volatility parameter equal to 03. Let Ct be the current value of the contingent claim Y, so that
The hedging strategy of the survival claim is, on the event (f < T},
5.2.4 Hedging of a recovery payoff As another illustration of Proposition 5.3, we shall now consider the contingent claim G(Y$ Y;, H T ) = l l , ~ ~ , ~ g ( Ythat + ) , is, we assume that recovery is paid at maturity and equals g(Y+). Let ug be the pricing function of this claim. The post-default pricing function vg(.; 1) does not depend on y3. Indeed, the equation (we write here y2 = y)
with ug(T, y; 1) = g(y), admits a unique solution vr,gt2,which is the price of g(YT)in the Black-Scholes model with interest rate r and volatility 02. Prior to default, the price of the claim can be found by solving the following PDE
with vg(T, y2, y3; 0) = 0. It is not difficult to check that
vg(t, y2, y ~0); = (1 - eJ'(t-T))vr,g.2(4 ~ 2 ) . The reader can compare this result with the one of Example 5.1.
Two defaultable assets with total default We shall now assume that we have only two assets, and both are defaultable assets with total default. We shall briefly outline the analysis of this case, leaving the details and the study of other relevant cases to the reader. We postulate that 5.3
so that with the pre-default prices governed by the SDEs
In the case where the promised payoff X is path-independent, so that for some function G, it is possible to use the PDE approach in order to value and replicate survival claims prior to default (needless to say that the valuation and hedging after default are trivial here). We know already from the martingale approach that hedging of a survival claim X I is formally equivalentto replicating the promised payoff X using the pre-default values of tradeable assets
We need not to worry here about the balance condition, since in case of default the wealth of the portfolio will drop to zero, as it should in view of the equality Z = 0. We shall find the pre-default pricing function v(t, yl, y2), which is required to satisfy the terminal condition v(T, yl, yz) = G(y1, yz), as well as the hedging strategy (+I, @). The replicating strategy - -is such that for the pre-default value C of our claim we have Ct := v(t, Y:, Y:) = +:Ti + @:?;, and
+
-
(78)
dCI, = 4; dT;
Proposition 5.4. Assume that a1 f satisfies the PDE
02.
+ +; dy;.
Then the pre-default pricingfunction v
with the terminal condition v(T, yl, yz) = G(y1, yz).
-
Proof. - We shall merely sketch the proof. By applying Ita's formula to v(t, Y:, Y:), and-comparing the diffusion terms in (78) and in the It8 differential dv(t, Y:, Y:), we find that
where @' = @'(t,yl, y2). Since @'yl = v(t, yl, y2) - @2y2,we deduce from (79) that yigia~v+ ~202d2u= TI + q2y2((52- ol), and thus @2y2=
ylaldlv + y2~2d2v- vo1 02
- 01
On the other hand, by identification of drift terms in (79), we obtain dtv + yi(pi + y)div + y2(p2 + Y ) ~ Z V
Upon elimination of @' and @2, we arrive at the stated PDE. Recall that the historically observed drift terms are pi = pi than pi. The pricing PDE can thus be simplified as follows:
+ y, rather
The pre-default pricing function v depends on the market observables (drift coefficients, volatilities, and pre-default prices), but not on the (deterministic) default intensity. To make one more simplifying step, we make an additional assumption about the payoff function. Suppose, in addition, that the payoff function is such that G(yl,y2) = ylg(y2/yl) for some function g : IR, + IR (or equivalently, G(yl,y2)= y~h(y1Jy2) for some function - h : lR+ -,IR). Then we may focus on relative pre-default prices ?t = Ct(Y:)-' and y2,' = Y:(Y;)-'. The = q t , Y?) corresponding pre-default pricing function q t , z), such that will satisfy the PDE
-
--
with terminal condition q T , z ) = g(z). If the price processes Y 1 a n d Y 2 in (68) are driven by the correlated Brownian motions W and with the constant instantaneous correlation coefficient p, then the PDE becomes
-
Consequently, the pre-default price Ct = Y j q t , ??) will not depend diand thus, in principle, w e should rectly on the drift coefficients jl'l and be able to derive an expression the price of the claim in terms of market observables: the prices of the underlying assets, their volatilities and the correlation coefficient. Put another way, neither the default intensity nor the drift coefficients of the underlying assets appear as independent parameters in the pre-default pricing function. Before we conclude this work, let u s stress once again that the martingale approach can be used in a fairly general set-up. By contrast, the PDE methodology is only suitable when dealing with a Markovian framework. In a forthcoming paper [8], w e analyze a more general situation where a traded defaultable asset is a credit default swap, so that its dynamics involve also a continuous dividend stream.
12,
Acknowledgments. Some results of this work were presented by Monique Jeanblanc at the "International Workshop on Stochastic Processes and Applications to Mathematical Finance" held at Ritsumeikan University on March 3-6, 2005. She deeply thanks the participants for questions and comments. The first version of this paper was written during her stay at Nagoya City University on the invitation by Professor Yoshio Miyahara, whose the warm hospitality is gratefully acknowledged. The work was completed during our visit to the Isaac Newton Institute for Mathematical Sciences in Cambridge. We thank the organizers of the programme Developments in Quantitative Finance for the kind invitation.
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A Large Trader-Insider Model Arturo Kohatsu-Higa and Agnes Sulem INRIA-Rocquencourt Domaine de Voluceau, Rocquencourt, B.P. 105, F-78153 Le Chesnay Cedex, France We give some remarks on the anticipatingapproach to insider modelling introduced by the authors recently. In particular, we define forward integrals by using limits of Riemmann sums. This definition is well adapted to financial applications. As an application, we consider a portfolio maximization problem of a large trader with insider information. We show that the forward integral is a natural tool to handle such problems and we compute the optimal portfolios for an insider and a small trader. Key words: Anticipating Calculus, Information asymmetry, large traders.
1. Introduction In this article, we would like to explain the anticipating approach to insider information. The section on the forward integrals properties relies on Chapter 3 of Nualart (1995). Nevertheless, as we have not found a standard reference for this material in the form of the forward integral we will do it here in detail. For this we need to introduce the basic tools of differentiation on the Wiener space. 7,P) Consider the interval [0, T ] and a complete probability space (a, on which a standard one dimensional Brownian motion W is defined; {7t]tGI0,Tl denotes the filtration generated by W, augmented with the P-null sets and made right continuous. Since all the results in the paper rely heavily on Malliavin calculuslwe introduce some of its terminology briefly. We denote by CT(IRn)the set of C" bounded functions f from JRn to lRl with bounded derivatives of all orders. If S is the class of real random variables F that can be represented as f(Wt,, . . . ,Wt,,) for some n E N, tl, . . .,t, E [0, TI and f E Cr(IRn),we can complete this space under the
Sobolev norm 11.lllrP given by
2 ( w t , , .. .,Wt,,)lmtil(s), obtaining a Ba-
where D is defined as D,F = i=l
nach space, usually indicated with D1,p. Analogously, we can construct the space Dkrpby completing S under the Sobolev norm
where D ~ ~ , =, ~D,,F. . .D,,F. Finally, we denote Dm =
nnd"
prl krl We denote the adjoint of the.closableunbounded operator
by 6;. This operator is called the Skorohod integral. The domain of 6: is the set of all processes u in L2([0,TI x Cl) such that
for some constant C possibly depending on u and llF112= E(IFI2)ll2. If u E Dom(6;), then 6;(~)is the square integrable random variable determined by the duality relation
Note that the above construction can be carried through for any fixed time interval [s, S], in the space L2([s,S] x a).We will also use the notation
For a stochastic process @, we say that 4 E L ~if Jthe~ following norm is finite: 11@1/:2
=E
[ST
o ~@(S)rds] +E
[STST o
o I D ~ @ (dsdu] s)~.
2. The Forward Integral Consider an insider, that is an agent that has sensible information about the future values of a stock, who may also have an influence on the evolution of the stock price. This is called a large trader-insider. In general one would like to study models of the type
Here n represents the insider's strategy which is adapted to a filtration 6, which may be bigger (or just different) than the filtration generated by the Wiener process W with natural filtration 7. Therefore S is also adapted to 6 (if 7 C G)and the above stochastic integral will be an anticipating integral commonly known as the forward integral of Russo-Vallois. Next, we define the forward integral. For this, given any partition 0 = to < ... < t , = T such that r n a ~ ( t ;+ti; ~ i = 0, ..., n - 1)-+ 0 as n + m, let q(s) := max(ti;ti
s s).
Then we can define the forward integral as follows: Definition 2.1. Let @ : [0, TI x R -+ be a measurable continuous process. Theforward integral of $ with respect to W(.) is defined by
if the limit exists in probability and is independent of the partition sequence taken. This definition does not coincide exactly with the original definition of Russo-Vallois, unless we put some additional assumptions. Note that the above definition is local. That is, let @ be forward integrable such that for a measurable set A c C2 we have that $ 1 = ~ 0, Then l'$(t)lAd-W(f)= 0. In that sense, as in Nualart, (1995). page 45 we will use the local defintion of all the spaces to appear below. First let us start proving that the expectation of this integral is not zero and therefore the usual rules of calculus do not apply. In particular, usual martingale properties are not true. For parallel martingale properties of anticipating integrals, see the interesting articles of Tudor (2004) and Pecatti-Theieullen-Tudor (2005). Definition 2.2. Let @ : [0, TI x R + IR be a measurable process such that
@(t)E IL112. We say that @ E IL:~ if the following stability property is satisfied: for any sequence of partitions 0 = to < ... < t , = T such that its norm tends to zero as n + w, there exists the trace process D,+@E L2([0,T] x a)such that
In such a case we say that @
E
I L $ ~and we define
This norm will serve to control the variance of the forward integral as it is shown in the next Theorem. Theorem 2.1. Suppose that @ E 1~:'~. Then theforward integral of @ exists, the limit in the definition 2.1 being satisfied in L1(C2) andfurthermore
where 6 denotes the Skorohod integral. Furthermore,
Proof. In order to prove that the integral exists we use the following formula (see formula (1.12) in page 130 in Nualart (1995a))
Then the existence of the forward integral follows from Definition 2.2. Furthermore we have that each element in this expression belongs to L2(R) and therefore we have that
The last estimate is obtained similarly. We have
Therefore,
Then the Riemmann sum sequence is bounded in L2(51) and therefore converges in L2(fJ)as it converges in L1(51). Then taking limits in the above inequality we obtained the desired result. Next we prove that the integral process is a continuous process. Theorem 2.2. Suppose that
+ t IL?,
m for some p > 2 then the process version.
such that E
[I' ( I D , + ( u ) ~ JT
d s y du] <
{$+(s)d-W(s);t E [O, TI] has a continuous
Proof. Use Proposition 5.1.1 in Nualart (1995a).
Now we give the formula for the quadratic variation.
Theorem 2.3. Given any sequence of partitions of the interval \Oft], n, : 0 = to < ... < tn = t such that max{ti+i- ti;i = 0, ..., M - 1 ) + 0 as n + m, we have that
for
+
g(f+'
2
w(~))
-1
1 ~ ( s )ds 1 ~a s .
E E:~.
Proof. First suppose the simple case that there exists a fixed partition 0 = so < ... < S m = f S U C that ~
where F; E lD1r2. In such a case we obviously have that @ is forward integrable and furthermore
We then also have that for the sequence of partitions n; = (ti;i = 0, ..., n ]u {sj;j = 0, ..., rn) then
as n + w because the partition {si;j = 0, ..., rn} is fixed and the forward integrals are L2- continuous in the time variable. Therefore without loss of generality we will suppose that (s,;j = 0, ..., rn} c n,.Then we have that
As the partition {sj;j = 0, ..., rn] is fixed we have that
as n
+ w.
Therefore
Finally the result follows from the following density argument:
Now we give the It8 formula that is necessary for our calculations. Before we need a preliminary Lemma.
Lemma 2.1. Suppose that @ E IL:2 nIL2l4 with D,+@ E IL1,2 and b is a stochastic Define the process process with b E
Then f (., X)Q
E
IL? for any f
E
c ~ ( [TIo x, IR).
L~
Proof. First, note that b(s)ds E L.:' Furthermore, one clearly has that
In fact, D ,
The other properties beingdear, the assertion consider
ltb(s)ds
b(s)ds =
E
J~DUb(s)ds.
IL? follows. Next,
Therefore we have that
Finally by the chain rule and product rule, we have that
and Jf
Ds+ (f (-,x)Q)= d * ( . r X)Ds+X@+ f (s, X(s))D,+@. Theorem 2.4. Suppose that @ E I L : ~ ~ I L with ~ , ~ D,+Q E JL1r2 and b is a stochastic
process with b E IL112 then for any f E c ~ ( [ oTI, x IR)we have that
Proof. In order to prove that the integral exists we find first a smooth approximation of the process cjj of the type
where F; = cjj(si) E ID1t2and 0 = so < ... < sn = T is a fixed partition and
as n + oo. Note that in this case one has that
NOWdefine ql(s) = inf(si;si > S) and p(s) = similarly the approximation process
<
S U ~ ( S ~ ; S ~s).
We define
Consider any partition 0 = to < ... < t, = t such that it contains all the points sj, j = 0, ..., n. Using the Taylor expansion we have
f (2Xn(f))= f (0. x) +
(ax
f (ti, x"(ti))(xn(ti+~) - xn(fi))+ atf (fi, xn(ti))(ti+l- ti))
Here TIn(tj)denotes a value between Xn(ti)and X"(ti+l)and fi a value between ti and ti+l. Obviously, f(t, Xn(t))converges a.s. to f (t, X(t)) as n + 00. The last term above, as in the previous Theorem 2.3 converges to
In fact, one can easily reduce the problem to the calculation of the limit of
The first term converges to zero as n + w and the second converges first as rn + w to
and to
Lt dxxf (s,x(s))r)(s)'ds
as n -+m. The other terms converge clearly
So we only have to consider the last $"(s)d-W(s). First, as rn + m this term which is Z% d , f(ti, Xn(ti)) term converges a s . as all the othe; terms converge. Therefore this limit is the forward integral Jtd , f (s, Xn(s))+"(s)d- W ( s ) . The forward integral
rt'-'
d xf (s, X(s))$(s)d-W(s) exists due to Lemma 2.1. The rest of the argument follows using a subsequence that converges at a fast speed. That is, take a uniform partition of the interval [0, TI, say si = Tiln, then consider the sequence $Y such that SUPtsTIX(t) - Xn(t)l I n-€ for e > 112. We will then have that for the same sequence ti = Tilrn
m-1
= m+m lim
C i=O
( d x f ( t i . ~ ( t i ) ) $ ( f i )- axfc4f.it( xn(ti))+"(ti)) (W(ti+,) - w ( t i ) ) .
Now we consider the subsequence for which n = rn to obtain that the above limit converges to zero. Then the result follows. 0 Remark 2.1. 1.The previous proof also gives a sense to the integral
L~J x f 6,XO))d-X(s).
2. In fact the original definition of the forward integral by Russo-Vallois is somewhat different to the one given here. In general, their definition is more general. Nevertheless, once one wants that this integral becomes the limit of Riemman sums then one is forced to the above framework. Still, we remark that the above conditions can be somewhat relaxed but the general idea remains. 3. For example, the above proof is also satisfied in local form. That is, the result is also satisfied if C#I E IL>;oc n ILE with D,++ E IL$f and b is a
stochastic process with b E I , : ; : and f E C112([0, T ] x IR). For the definition of these spaces see Nualart [251. 4. The fact that the above It6 formula demands an extra condition (D,+~#JE in comparison with its counterpart in Skorohod integral form is well documented in the literature. In particular, in the case of the Stratonovich-Skorohod integral. Nevertheless as our restriction comes from the financial interpretation of the models to be used we accept them as natural. 3. A First Toy Example
Rather than following the general theory exposed in Kohatsu-Sulem (2006),we will give some examples in order to illustrate the theory. In this section, we consider a first toy model where the dynamics of the prices are given by
(2)
dS(t) = S(t)(p+ bW(T))dt+ oS(t)d-W ( t )
where p and b are real numbers, o > 0. We suppose moreover that p(t) = p = constant. The interpretation of this model when b 2 0 is that the insider introduces a higher appreciation rate in the stock price if W ( T )> 0. Given the linearity of the equation of S this indicates that the higher the final stock price the bigger the value of the drift in the equation driving S. Some cases of negative values for b can also be studied but the practical interpretation of such a study is dubious. Furthermore we remark that usually in this model we assume that the trades of the insider are not revealed to the public. This is also an interesting modelling issue which is also assumed by Kyle and Back. They assume that the cumulative trades of the insider plus a Wiener process in the insider's filtration are public information. The Wiener process is interpreted as the effect of the so-called noise traders. This interpretation can also be applied in any of the cases studied with the enlargement of filtration approach and as we will see it can also be applied here. The difference here is that we will introduce large trader-insider models with finite utility where there can also be small traders that act rationally.
In order to compare with the theory given in our previous article, we decide to first give an approach which is easier to introduce at this stage but that later will not be possible to apply. This is the set-up of enlargement of filtration. For this, consider the filtration Gt = 5 7t o(W(T)). In this filtrationit is well known that W is a semimartingaleand its semimartingale decomposition is given by
where w is a Wiener process in 6. Therefore in this case, as the forward integral becomes a semimartingale integral we have that the model for S is
Therefore the optimization of the logarithmic utility for this model is done through classical methods. Briefly, one has that the wealth process associated with this price process is given by
Then the discounted wealth, P(f) = e-"V(t) can be written as
The solution to the above equation is
~ ( t =) V(O) exp
(1
(,u - r + ~ w ( T )+
T --s W(s) W(T)
1
Therefore if we consider the optimization of the logarithmicutility we have the following problem max I(n) ne3I3(
where
and for any filtration 3-1 C 6 satisfying the usual conditions we define
x is 'H adapted;
I'
(n(s)12ds < oo
We then have the following theorem
Theorem 3.1. Assume that 3-1 is anyfiltration included in 6. Then the optimal portfolio for the above problem is given by
and the optimal value is given by
In particular, lim JG(t,2 ) = m,
(3)
t+T
while lim JVH(t, f c ) < co t+T
for 5% = a(S(s);s I t). Furtlzermore the functions JG(t,6 ) and J f l ( t ,f i ) are increasing in b. A far more general theorem was given in Kohatsu-Sulem (2006). Proof. In order to obtain the result first note that given that x have that
E Next the function
[I
O T C ( S ) ~ K ( S )= ]
0.
E A f l ( t ) , we
is a strictly convex function adapted to the filtration f i . Therefore the maximal value is obtained for the value h given in the statement of the theorem. The limit wealth for the full insider is infinite because
The last result follows by noting that
By using a formula for conditional expectations of Gaussian random variables (see Kohatsu-Sulem (2006))one obtains that
Therefore the result follows because
To finish one only needs to note that
-
(p;ayt
+
x202 $ , ( ( b T(b2T + ~+ )2bo)s + ~+ ( a2 ~ - sds. ))~
Finally differentiating with respect to b it follows that JH(t,h)is increasing.
There are various other interesting remarks that are made in KohatsuSulem (2006) with respect to the interpretation of this result. This result says that in various situations the insider which acts as a large trader may have effects in the market and the small trader only uses a projection of this market in order to optimize its utility. This projection does not transfer the information from the insider to the small investor. This example also reflects the fact that there is not only one insider but various insiders that may act depending on the filtration that
one takes between Tft = o(S(s);s 5 t)and Gt = 7t v o(W(T)).Finding examples where the calculations can be done explicitely will be an interesting subject of future research. This toy example, which can be solved using the simple technique showed here was solved in Kohatsu-Sulem (2006) using a powerful technique consisting on optimization in an anticipating framework. We will show in the next section an example which can be considered as a nontrivial application which cannot be solved using the previous technique. Before that we will discuss another issue related with (3). In fact with a small modification we can obtain that the optimal logarithmic utility of the insider is finite.
Theorem 3.2. Consider thefiltration G;= 'fi V n (w(T) + W'((T- s)'); s 2 t ) where W' is another Wiener process independent of W and 8 E (0'1). Then we have that limJ@(t,fi)< 03. t-+T
Proof. First note that the first part of Theorem 3.1 can be applied to the filtration Gt = 5 V o(W(T))V o(Wf(s);s_< To). Therefore we only need to compute
T-s
W ( T )- W ( S )+ W'((T - s)') T-s+(T-s)'
From here it follows that the logarithmic utility is finite if 8 < 1 To finish we prove a theorem that can be interpreted as the non-existence of arbitrage or the issue of non-conspicuous insider trader.
Theorem 3.3. For any filtration 7-f included in Q such that S is ?(-adapted, suppose fhat there exists an 54-opfimalporffolio f i E IL:~ which leads to afi'nife logarithmic utility. Then there exists an 54-Wiener process W w such that
Proof. Just to avoid explicit notation let p,(o) = p+bW(T). If there exists an optimal portfolio f i then it minimizes the logarithmic utility of this trader which is
Applying variational calculus to the above expression we obtain that
Furthermore note that
Then
Therefore by Lkvy's characterization of the Wiener process we have the result.
y.
Note that in the classical Merton model fi(s) = Therefore the previous theorem states that the small trader will not find any anomaly in his trading of the stock even if this is influenced by an insider. This result also says that if we interpret Ww as the effect of 54-noise traders then the market maker will only see the information in the stock price itself. 4.
Continuous Stream of Information In this section, we consider for 6 > T fixed
In this model, the insider has an effect on the drift of the diffusion through information that is 6 units of time in the future. This continuous deformation of information may be used to model streams of information rather than one single piece of information. In this case, it is difficult to see what is the information held by the insider but hisher effect on the market is known. One first important remark is the following proposition.
. Proposition 4.1. W is not a semimartingaleon thefiltration (%+6)tE10,Tl Proof. Consider the definition of semimartingale as given in Protter's book page 52. If W is a (7;+h)-semimartingale,then for any partition whose norm tends to zero and always smaller than 6, consider the process
This process is then ('f;+a)-adaptedand converges uniformly to zero but its stochastic integral converges to the quadratic variation of W leading to a contradiction. This shows that the insider filtration does not even correspond to (Ft+b)tEIO,T1. The definition for the insider's filtration in the particular case that 6 2 T is
6t = 7; V o(W(T)) V ~ ( W ( +S 6) - W(T);s 5 t). Then the calculations can be carried out as in the previous section. Nevertheless, we need to be more precise here in the general case. We do this here. In such a situation, we have to clearly use the anticipative set-up given in the first section. Therefore we have to find the solution for the equation of the prices. Proposition 4.2.
S(t) = S(O)exp ((p - $2)
t +b
la +
~(sldso~(t))
is the unique solution of equation (4) in the space 1~:;~~. The proof of this result follows directly from the It6 formula given in theorem 2.4. We are interested in computing the optimal policy of the small investor with filtration ' H t = o(S,; s 5 t). From the previous proposition, we have that f i t = o (Y(s); s l f),
L'+'
where Y(s) = b W(r)dr + oW(s). Now we study the wealth process associated with this price process. The wealth process is defined as the solution of
where the interpretation of d-S(t) is as in Definition 2.1. Note that in order that this equation among others has a sensible financial interpretation we introduced in Section 2 the forward integral as a limit of Riemmann sums. Then the discounted wealth, v(t) = e-rt V(t) can be written as
As before the solution to the above equation is
We will later show that the optimal portfolios proposed satisfy the conditions stated in Section 3. With these assumptions, we have that the limit of the logarithmic wealth process can be written as
The class of admissible portfolios is given by 9 7 = { nis ';H adapted; n E IL?'}.
Theorem 4.1. Define the following portfolio
where a(s) =
- lim E h-0
lfir E 1 ~ : ' ~then fi is the optimal portfolio for the above problem for anyfiltration 'H and the optimal value is given by
A more general theorem was proved in Kohatsu-Sulem (2006). Proof. In order to obtain the result we have to prove first that the functional J is strictly convex. For this, let no and nl E 3.Then we have that for any a E (O,1) ](an0 + (1- a)nl) < aJ(no) + (1- a)J(nl).
This property clearly comes from the factor - $ - ~ ( sin ) ~the expression for J. Next, we find the first directional derivative of J. Consider for n, v E 3,then DnJ(n): = lirn J(n + ev) - J(n) OE'
E.
If we set the above equation equal to zero for all v E 54 and in particular for v = Xlbo,t,lfor X E D1t2we have by a density argument that
Now note that f i satisfies the above equation. In fact, replacing ir in the above equation, we have E
[s
=
[
I I
-a lim E W ( S+ h, - W ( S ) 54 / ,]ds + o ( w ( t o ) w ( s o ) )fiS0
so
-, lim E h+O
h W ( s+ h) - W ( s ) ds + (W(t0)- W(s0)) fis, h
[L
h+O
"
I
I
= 0,
by continuity of the paths of the Wiener process. Therefore f i has to be optimal. In fact, for all /3 E 54 and E E (0,I), we have
Now, with
1 = 1 + 7 we have 1- E 1
lim -(J(-) L'OE
A 1 + 17 - J(ir))= lim - ( J ( f i 1-E 0 11
+ ip?) - J(ir))= D*J(fi).
Then we get 1 DgJ ( 2 )= lim -(I(? EO'
E
+ ep) - J(ir))2 DfiJ(ir)+ J(p) - I(??).
We conclude that
l(B)- 1(fi)5 DpJ(fi)- DfiJ(fi);fi,B In particular, using that DpJ(ng)= 0, we get
J(B> - I(n*)5 0, which proves that n' is optimal.
E 54.
To find the optimal expression for the utility it is enough to note that
therefore the optimal utility is
From here the result follows. A very useful property is that the optimal portfolios in a smaller filtration is just a projection.
\
Proposition 4.3. Let 'H' c 'Hz c 6 be two filtrations satihing the usual conditions such that there is an optimal portfolio 112 in 7 i 2 within a class of protfolios If3iw~c then there is an optimal portfolio f i l in 3-[' which satisfies
Therefore in order to prove the existence of the optimal portfolio it is essential to compute a or at least obtain its existence and some regularity properties. We do this, first in the case that 6 2 T . This is done in the next proposition. Proposition 4.4. Suppose that 6 2 T. The optimal logarithmic utility portfolio
in the filtration
3-[
c 6 is given by
The optimal value is given by
In particular, lim JG(tl6 ) = cm, t-T
while limJ,H(t,it) < m t+T
for 5% = o(S(s);s I t).Furthermore the functions JG(t,it) and creasing in b.
J H ( f ,fi)
are in-
Proof. Define Y ( t ) = b J ~ W(r)dr * ~ + oW(t). Then for 6 2 T
-
E [ W ( t + b)/7ft]= (b(t + 6) + o)M
1
g(t, u)dY(u)
where M Mi = o-l ((b6 + 20) (e? - 1 ) + o ( e f + l)rland g(t, u ) = e $ ( 2 t - ~ )+ e : ~ In fact, note that Y is a Gaussian process. Therefore E [ W(s)/%t] = J~h(s, f,u)dY(u)for a deterministic function h. To compute h we compute the covariances between W ( s )and the stochastic integral and Y ( v )for some v I t I s I T . First E [W(s)Y(v)]= bsv + o(s A v).
Also
Therefore the above two expressions have to be equal. After differentiation of the equality with respect to v I t three times, we obtain
Solving this differential equation gives
Next one verifies that for the following constants, the covariances coincide.
21
Cl(s,t ) = eTC*(s,t).
Therefore, we have that
s-t
dY(u).
Then the result follows. Next, using Theorem 4.1, we have that the possible optimal portfolio n* defined by
. fact, all the properties are obtained through the process Y. We is in 1 ~ : ~ In do not give the details of this verification. Then the optimal utility is finite as it is given by
If
I(!, n*)= l0g(v0)+ 2 E
[I
H * ( s ) ~ ~ s. ]
Remark 4.1. When s I T, we have that
where
This shows that even the information on all the prices of the interval [0, does not reveal the information held by the insider to the small trader. As before we can also show that the insider's utility is finite if we use s 6 ) + W1((T- t)@); s < t ) for 0 < 1. Similarly the filtration Gi = V v ( ~ ( + we can also obtain a representation theorem such as Theorem 3.3. Instead we will take a look at the case 6 < T. We use a different shortcut through the anticipating Girsanov's theorem. For details and notation we refer to Chapter 4 in [25]. Theorem 4.2. Consider the case 6 < T. Then there is noarbitrage for thefiltration f i t = o (S(s);s< t ) and the logarithmic utility for the optimal portfolio value for
this investor isfinite.
Proof. We apply Theorem 4.1.2 in [25]in the interval [0,T + 61 with the transformation T ( w ) = w + b l ( . I T ) w(s+6)ds,
I
defined in C[O,T + 61. Then we have that if T(a1) = 0 then w ( t ) = 0 for all t E [T,T + 61. Therefore
That is, by finite induction we have that T is an injection. To prove that it is sujective one follows a similar pattern. Next we have that detz ( I + Du) > 0 for u,(w) = bl(s I T)w(s+ 6 ) and that under the change of measure
9 = det2( I + Du) exp (dP
T
S
T
bW(s + 6)dW(s)-
W ( s + 6)'ds)
then w = T ( W )has the law of a Wiener process under Q. Therefore there exists an equivalent martingale measure for this problem. In order to compute the optimal portfolio one uses the dual method. p and define That is, denote m = ( ~ - ~ -( r) dQ' = detz (I + Du) exp dP
Then the optimal portfolio value is
The optimal portfolio value is finite because E [log
(%)I
< m.
Off course an interesting problem is to compute explicitely the optimal portfolio for the case 6 < T. Although one may consider that the large trader effect is somewhat hidden in this paper through the process appearing in the drift. We remark that this may be considered as a first learning step towards more complex models. Some of these models were presented in Kohatsu-Sulem (2006)or Kohatsu (2005).
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Lectures on Probability Theory and Statistics. Ecole d'etg de Probabilitis de Saint-Flour X X V . Lect. Notes in Maths. 1690. Springer-Verlag. 27. Dksendal, B. and Sulem, A.: Partial observation in an anticipative environment. Preprint University Oslo 31/2003. 28. Peccati, G., Thieullen, M., and Tudor, C., 2005. Martingale structure for Skorohod integral processes. to appear in The Annals of Probability. 29. Protter, P., 2004. Stochastic Integration and Diflerential Equations. A New Approach. Springer-Verlag. New York. 30. Russo, F. and Vallois, P., 1993. Forward, backward and symmetric stochastic integration. Probab. Th. Rel. Fields 97,403-421. 31. Russo, F. and Vallois, P., 2000. Stochastic calculus with respect to continuous finite quadratic variation processes. Stochastics and Stochastics Reports 70,140. 32. Russo, F. and Vallois, P., 1995. The generalized covariation process and It6 formula. Stochastic Process. Appl. 59,81-104. 33. Skminaire de Calcul Stochastique 1982/83, Universite Paris VI, 1985. Grossisements de Fitrations: Exemples et Applications. T. Jeulin and M. Yor (eds.)Lect. Notes in Maths. 1118. Springer-Velag. Berlin. 34. Tudor, C., 2004. Martingale type stochastic calculus for anticipating integrals, Bernoulli 10(2), 313-325. 35. Yor, M., 1985. Grossissement de filtrations et absolue continuite de noyaux. In Grossissements de Filtrations: Exemples et Applications. T. Jeulin and M. Yor (eds.) Lect. Notes in Maths. 1118. Springer-Verlag. Berlin.
[GLP & MEMM] Pricing Models and Related Problems Yoshio Miyahara Graduate School of Economics, Nagoya City University Mizuhochou Mizuhoku, Nagoya, 467-8501, Japan The [GLP & MEMM] pricing model (= [Geometric Levy Process & Minimal Entropy Martingale Measure] pricing model) has been introduced as a pricing model for the incomplete financial market. This model has many good properties and is applicable to very wide classes of underlying asset price processes including the geometric stable processes. We explain those good properties and see several examples of this model. After that we investigate the calibration problems of [GLP & MEMM] model. Key words: Geometric L6vy Process, Relative entropy, Minimal entropy martingale measure, Stable process, Calibration 1. Introduction The [Geometric Lbvy Process & MEMM] pricing model was first introduced in [36]. This model is one of the incomplete markets, and is based on the geometric Levy process and the minimal entropy martingale measure (= MEMM). We assume that the value process of bond is given by Bt = exp(rf1, where r is a positive constant. The price process of the underlying asset is denoted by St. 1.1 Black-Scholes model The explicit form of Black-Scholes model (Geometric Brownian motion model) is given by
and the stochastic differential equation (SDE) form is given by
where Wt is a standard Wiener process.
The risk neutral measure Q is uniquely determined by the Girsanov's lemma. Under the Q the process I?r, = Wt + (p - r ) c l t is a Wiener process and the price process St is expressed in the form of (3)
St = ~ ~ e ( ~ - i . ' ) ~ + or~ ' ~ d& = St (rdt + od wt).
The price of an option X is given by e-rTEQ[X].The theoretical 8-S price of the European call option, C(So, K, T), with the strike price K and the fixed maturity T is given by the following formula
where @(d)is the normal distribution function and
1.2 Properties of B-S models 1.2.1 Distribution of log returns The log return is the increment of the logarithm of St, (6)
A log St =
1 log St+,t - log St = (p - -02)at + oa Wt, 2
and the log return process is ( p - io2)t+ oWt. The distribution of the log return (or the log return process) of the B-S model is normal. This is convenient for the calculation of the option prices. For example we have obtained the explicit formula of the price of European call options. But it is said that the distributions of the log returns in the real market usually have the fat tail and the asymmetry. These facts suggest us the necessity to consider another models. 1.2.2 Historical volatility and implied volatility Under the setting of the B-S model, the historical volatility of the process is defined as the estimated value of (T based on the sequential data of the price process St. We denote it by;;. On the other hand the implied volatility is defined as what follows. Suppose that the market price of the European ) : d were given. Then the value of o call option with the strike K, say , which satisfies the following equation
is the implied volatility, and this value is denoted by o r ) . We remark here that the implied volatility o r ) depends on the strike value K, and that on the contrary the historical volatility Tdoes not depend on K.
We first consider the case where the market value of options obey to the Black-Scholes model, and so the market price' :c is equal to the theoretical B-S price CK . In this case the solution of the equation for the implied volatility is equal to the original o and it holds true that o r ) = o = constant. This means that if the market obeys exactly to the Black-Scholes model, then the implied volatility o?) should be equal to the historical volatility o. But in the real world this is not true. It is well-known that the implied volatility is not equal to the historical volatility, and the implied volatility is sometimes a convex function of K, and sometimes the combination of convex part and concave part. These properties are so-called volatility smile or smirk properties. 1.3 Generalization of B-S model 1.3.1 Geometric LCvy Process models We start from the explicit form of Geometric Brownian motion: St = ~ ~ e ( ~ - ~It "may ~ )be~ a+natural ~ ~ ~idea . to replace the Wiener process with the more general Levy processes Z t and set
oy)
This type processes called the Geometric Levy Processes (GLP). The [GLP & MEMM] pricing model is one of this type of generalisation of B-S model. The class of Levy processes are very wide and the distributions of St may have the fat tail property and may be asymmetric. 1.3.2 Stochastic volatility models We start from the SDE form dSt = St ( p d t + o d W t ) . When we replace the Brownian motion with a Levy process, we obtain the equation described in the previous subsection (see §2). When we randomize the volatility o as follows (9)
dSf = St ( p d t + i i t d W t ) ,
where 3t is a stochastic process, then we obtain the so-called stochastic volatility models. 1.4 Our Goal The purposes of this lecture are, 1)we introduce the [GLP & MEMM] pricing model and see that this model has many good properties, and next 2) we review some relating problems of this model, in theoretical sense and (or) in practical fence (for example, the fitness analysis and calibration analysis).
2. Geometric LCvy Process Pricing Models We assume that the value process of bond is given by
where r is a positive constant. A pricing model consists of the following two parts: (A) The price process St of the underlying asset. (B) The rule to compute the prices of options. For the part (A)we adopt the geometric Levy processes, so the part (A) is reduced to the selecting problem of a suitable class of the geometric Levy processes. For the part (B) we adopt the martingale measure method, so the part (B) is reduced to the selecting problem of a suitable martingale measure Q, and then the price of an option X is given by e-rTEQ[X]. 2.1 Geometric Levy processes The price process St of a stock is assumed to be defined as what follows. We suppose that a probability space (0,F,P) and a filtration ( 5 , O It ITI are given. We also suppose that a Levy process Zt is defined on this probability space and that the price process St of a stock is given in the form
Throughout this paper we assume that 7; = o(S,, 0 5 s I t ) = o(Z,, 0 I s I t ) and 7 = FT. We give here the definition of Levy process and the characterization of it (see [45]). Definition 2.1. A stochastic process (Zt]on R~ is a Levy process if the following conditions are satisfied. 1) For any choice of n 2 1 and 0 I to < t l . . . < t,, random variables Zto,Zt, - Z,, Zt, - Zt,,. . . ,Zt,,- Zto,-l, are independent (independent increments property). 2) Zo = 0 a.s. 3) The distribution of Z,+t - Z, does not depend on s (temporal homogeneity or stationary increments property). 4) It is stochastically continuous. 5) There is no E 7 with P(Ro) = 1 such that for every o~ E n o , Zt(o1) is right-continuous in t 2 0 and has left limits in f > 0.
In this lecture we discuss the case of d = 1. The Levy process Zt is characterized by the generating triplet (a2,v(dx),b), where u2 is a non-
negative constant, v(dx) is a measure such that
and b is a constant. By the use of this generating triplet, the characteristic function of Zt is
Using Ito formula, we know that St satisfies the following stochastic differential equation
where itis another Levy process given by
And the price process St has the following expression
where &(i)tis the DolCans-Dade exponential (or stochastic exponential) of
Zt. (18)
~ ( i )=, eit-iO-+ < ~ ,n i( ~ l +> AZ.)P-"~~ ~ s
The generating triplet of 2tl say (c2,?(dx), 6), is
Remark 2.1. (i) It holds that supp o c (-1, co). (ii) If v(dx) has the density n(x), then ?(dx) has the density ii(x) and ii(x) is given by
(22)
1
A(x) = -n(log(1 + x)). l+x
(iii) The relations between Zt and 2t are more precisely discussed in [30] , where the stochastic logarithm of Xt, L(X)t, is defined and the following relations are obtained.
Many candidates for the suitable Levy process have been proposed. We give some examples below. 1)Stable process (Mandelbrot and Fama (1963)) 2) Jump diffusion process (Merton (1973)) 3) Variance Gamma process (Madan (1990)) 4) Generalized Hyperbolic process (Eberlein (1995)) 5) CGMY process (Carr-Geman-Madam-Yor (2002)) 6) Normal inverse Gaussian process (Barndorff-Nielsen(1995,1977)) 2.2 Equivalent martingale measures A probability measure Q on (0,F ) is called an equivalent martingale measure of St if Q P and e-"St is (5,Q)-martingale. Since the geometric Levy process model is incomplete in general, there are many equivalent martingale measures. For the part of (B) of the pricing model we have to select a special martingale measure. Many candidates for the equivalent martingale measure have been proposed as follows. 1)Minimal Martingale Measure (MMM) (Follmer-Schweizer (1991)) 2) Variance Optimal Martingale Measure (VOMM) (Schweizer (1995)) 3) Esscher Martingale Measure (ESMM) (Gerber-Shiu (1994), B-D-E-S (1996)) 4) Minimal Entropy Martingale Measure (MEMM) (Miyahara (1996), Frittelli (2000)) 5) Utility Martingale Measure (Utility-MM) 6) Mean Correcting Martingale Measure (MCMM)
-
3. Esscher Transformed Martingale Measures 3.1 Esscher transforms
The Esscher transform is very popular and thought to be very important method in the actuary theory (see [24] ). Esscher has introduced the risk function and the transformed risk function for the calculation of collective risk. His idea has been developed in his work [IS] and by many authors, and played very important roles in the option pricing theory. 3.1.1 Esscher transforms and risk processes We give several definitions. Definition 3.1. Let R be a random variable and h be a constant. Then the
probability measure P f F ' defined by
is called the Esscher transformed measure of P by the risk variable R and the index h, and this measure transformation is called the Esscher transform by the risk variable R and the index h. Definition 3.2. Let Rt,O 5 t 5 T, be a stochastic process. Then the Esscher transformed measure of P by the risk process Rt and the index process h, is the probability measure p(f:;Lh defined by
This measure transformation is called the Esscher transform by the risk process Rt and the index process h,. Definition 3.3. In the above definitions,if the index index process is chosen , measure of St, then P[::~~~,~, is called so that the ~ ( f ? ~ ~ is~ a, ~martingale the Esscher transformed martingale measure of St by the risk process Rt, and it is denoted by PgoS:) or simply
~kfss).
3.1.2 Simple return process and compound return process When we give a certain risk process Rt, we obtain a corresponding Esscher transformed martingale measure if it exists. As we have seen in the previous section, the GLP has two kinds of representation such that St = so&= So&(2)t.So the processes Zt and Zt are both candidates for the risk process. We shall see the economical meaning of them. For this purpose, we will review the discrete time approximation of geometric Levy processes. Set
According to the above two kinds of expression of St, we obtain two kinds of approximation formula. First one is
Second approximation is
where E ( Y ( ~is ) )the ~ discrete time Dol6ans-Dade exponential of Y;), and )Y; is defined from the following relations
So we obtain
From this we obtain
and we know that A$) is the simple return process of s;). On the other hand, we obtain from the definition of
) the increment of log-return and it is called the and we know that A Z ~ is compound return process of s;).
Remark 3.1. The terms "simple return" and "compound return" were introduced in [I] ,and well known in economics (See also [47] ).
(F,8)
For t E we define z?) = z;), Yj") = $). Then it is easy to see that the process z?) converges to the process Zt when n goes to m. On the other hand the process converges to the process Tt as follows. The process St satisfies the stochastic differential equation dSt = St-dzt. From this it follows that dZt = (For the justification of this formula, see [30] .) Comparing this formG1a with (31), we know that the process Yj") is the approximation process in the procedure of solving the stochastic
e) F.
differential equation for Zt. This fact means that the process ~j") converges to the process Zt. Based on the above observation, it is natural for us to give the following definition.
Definition 3.4. The process Zt is called the "simple return process" of St, and the process Zt is called the "compound return process" of St. 3.2 Esscher transformed martingale measures for geometric Levy processes We next study the existence and uniqueness of Esscher transformed martingale measures. 3.2.1 Esscher Martingale measure (ESMM) Suppose that Zt is adopted as the risk process. Then, if the corresponding Esscher transformed martingale measure P::) is well defined, then it should be called the 'compound return Esscher transformed martingale measure'. This is the Gerber-Shiu's Esscher martingale measure introduced in [24],and the term 'Esscher martingale measure' is usually suggesting this compound return Esscher transformed martingale measure We suppose that the expectations which appear in what follows exist. Then the martingale condition for an Esscher transformed probability (ESS) measure Q = PZ,o,TI,h is
This condition is equal to the following condition
and this is also equivalent to the following expression,
where @(u)is the characteristic function of Z1 (@(ti)= Ep[eiUZ1]). To formulate the existence theorem, we set
Then we obtain
Theorem 3.1. I f the equation
(37)
f(EsMM)(h) = r,
has a solution h', then the ESMM of St, P ( ~ ~exists ~ ~and) ,
The process Zt is also a L&vyprocess under P ( ~and~ the~generating ~ ) triplet of Zt under P ( ~ say ~ ~( ~ (~E s )M M, )dESMM)(dx), ~, b(ESMM)), is
(Proof) The equation (37) is equivalent to the condition (3.16). Therefore P ( ~ is ~ a~martingale ) measure of St. Z~O.TI,~' The characteristic function of Zt under P by definition
(
~ = P~Z,o,T,,h" ( ~~ ~+t (~E )S M) M )(4,is
And this is equal to
By simple calculation we obtain (44)
@YMM)(u) = exp ( t (-to2u2 + i(b + h*o2+ ~ , x , ~x(ewX l i - l)v(dx))u
+
sw(eiux- 1 - i -Do
~ ~ l ~ ~ ~ ~ ~ ~ ~ ) ~ ~ v ( d ~
Using this formula, we can see that the martingale condition (3.16) is reduced to the equation (3.18). (Q.E.D.) 3.2.2 Minimal Entropy Martingale Measure (MEMM) Next we consider the case where Zt is adopted as the risk process. If exists, the corresponding Esscher transformed martingale measure P::: then it should be called the "simple return Esscher transformed martingale measure". We will see here the relation between the Esscher transform and the minimal entropy martingale measure (MEMM). We first give the definition of the MEMM.
Definition 3.5. If an equivalent martingale measure P* satisfies
(45)
H(P81P)I H(Q1P)
VQ :
equivalent martingale measure,
then P' is called the "minimal entropy martingale measure (MEMM)" of St. Where H(Q(P)is the relative entropy of Q with respect to P log[$$ld~,i f Q << P, otherwise, From the proof of [23] it follows that Proposition 3.1. The simple return Esscher transformed martingale measure PLESS) of St is the minimal entropy martingale measure (MEMM) of St. Z[O.TI
Based on the above results, we give the following definition. Definition 3.6. (i) The compound return Esscher transformed martingale measure P:::) is called the "Esscher martingale measure (ESMM)" and denoted by P(ESMM). (ii) The simple return Esscher transformed martingale measure PLESS)is ZIO.TI called the "minimal entropy martingale measure (MEMM)" and denoted ). by P' (or P(MEMM)
As we have mentioned above, the MEMM, P', is the simple return ) . existence Esscher transformed martingale measure. (P' = P ! ~ ~ ~ )The ZIO.TI theorem of the MEMM is obtained in [23]. Set
Then the following result is obtained ([23] ). Theorem 3.2. If the equation
has a solution 0', then the MEMM of St, P', exists and
The process Zt is also a U v y process under P* and the generating triplet of Zt under P', say (D'~, v*(dx),b*), is
(Proof)The results of this theorem follows directly from the proof of [23] . (Q.E.D.)
Remark 3.2. (i) The above result is improved to the multi dimensional cases (see [22] or [17]). (ii)Thefunction f(MEMM)(0) is also expressed as follows
And the process 2t is also a Levy process under P' and the generating triplet of zt under P' is
(iii)Thefunction f(MEMM)(€J) is a non-decreasing function of 8. (iv) If St is integrable, then it holds that (57)
E(St) = SOexp(tf
(MEMM)
(0)).
(vi) If the condition that f ( M E M M ) ( ~ )> r is satisfied, then the solution 8' of (48) is negative (8' < 0) if it exists. Such cases occur very often. 3.2.3 Mean Correcting Martingale Measure (MCMM) For the jump-diffusion type models, the Brownian motion can be adopted as the risk process. In that case the corresponding Esscher transformed martingale measure is the mean correcting martingale measure (MCMM) (see [46] or [8] ). We suppose that the expectations which appear in what follows exist. Then the martingale condition for an Esscher transformed probability
(ESS) measure Q = PW,O,Tlrh is
This condition is equal to the following condition (59)
~
~
[= erEp[ehW1], e ~ ~
~
~
~
and this is also equivalent to the following expression,
So the martingale condition is
To formulate the existence theorem, we set
Then we obtain
Theorem 3.3. I f the equation
has a solution h*, then the MCMM of St, I='(MCMM), exists and
The process Zt is also a Lhy process under I='(MCMM) and the generating triplet of Zt under P ( ~ ~say ~( (~ 3 )( ~, V~ ( M ~ C~ ~)~ 2) ( b(MCMM)), dX), is
~
]
3.3 Comparison of ESMM and MEMM
The ESMM, MEMM and MCMM are all obtained by Esscher transforms, but they have slightly different properties. The MCMM is supported in only the case of a > 0. So we restrict our attention on the ESMM and MEMM, and we will survey the properties and the differences of them. 3.3.1 Corresponding risk process The risk process corresponding to the ESMM is the compound return process, and the risk process corresponding to the MEMM is the simple return process. The simple return process seems to be more essential in the relation to the original process rather than the compound return process. In this sense we can say that the MEMM is more reasonable martingale measure than the ESMM. 3.3.2 Existence condition As we have seen in the previous section, for the existence of ESMM, P ( ~ ~ the~following ~ ) , condition (68) is necessary. On the other hand, for the existence of MEMM, P', the corresponding condition is (69)
S
~(e' -
v(dx) < m.
Ilxl>lJ
This condition is satisfied for wide class of Levy measures, if 8' < 0. Namely, the former condition is strictly stronger than the latter condition. This means that the MEMMmay be applied to the wider class of models than the ESMM. This difference does work in the stable process cases. In fact the MEMM method can be applied to the geometric stable model but the ESMM method can not be applied to this same model. 3.3.3 Corresponding utility function The ESMM is corresponding to power utility function or logarithm utility function (see [24] or [25] ). On the other hand the MEMM is corresponding to the exponential utility function (see [21] or [25] ). We remark here that, in the case of ESMM, the power parameter of the utility function depends on the parameter value h' of the Esscher transform. 3.3.4 Properties special to MEMM a) Minimal distance to the original probability: The relative entropy is very popular in the field of information theory, and it is called Kullback-Leibler Information Number(see [27] ) or Kullback-Leibler distance (see [12] ). Therefore we can state that the MEMM is the nearest equivalent martingale measure to the original probability P in the sense of Kullback-Leibler distance. Recently the idea of
minimal distance martingale measure is studied. In [25] it is mentioned that the relative entropy is the typical example of the distance in their theory.
b) Large deviation property: The large deviation theory is closely related to the minimum relative entropy analysis, and the Sanov's theorem or Sanov property is well-known (see, e.g. [14] ). Applying this theorem on the paths spaces of the price process, we can conclude that that the MEMM is the most possible empirical probability measure of paths of price process in the class of the all equivalent martingale measures. In this sense the MEMM should be considered to be the exceptional measure in the class of all equivalent martingale measures. The Sanov's theorem is Theorem (Sanov) [14] Let p be a probability measure on the Polish space C and let p,, E Ml(Ml(C)) be the distribution under pn of the C7=,a,,. Then H(.lp) is a good, convex rate function on MI(C) and (p, : n 2 1) satisfies the full large deviation principle with rate function H(.Ip). c) Indifference Price: The MEMM is related to the indifference prices with the exponential utility in the following sense. Let p,.(X) be the indifference price of X, where y is the parameter for the absolute risk aversion. Then it holds that
lim pl,(X)= e - r T ~[XI p Y-0
We also remark that, in the case of MEMM, the relation of the MEMM to the utility indifference price is known (see [44] ,[23] and [48] ). As the result of the above discussions, we can say that the MEMM has many better properties than the ESMM in the theoretical sense. 4. [GLP & MEMM] Pricing Model In this section we explain the [GLP & MEMM] pricing model and see examples of the model. 4.1 Model Now we give the definition of the [GLP & MEMM] Pricing Model. The [GLP & MEMM] Pricing Model is such a model (see [36] ):
(A) The price process St is a geometric Levy process (GLP). (8)The price of an option X is defined to be e-rTEp.[X], where P' is the MEMM. Of course this model can be considered for the cases where the MEMM exists. The existence condition is given in Theorem 2 of Section 3.
4.2 Examples of [GLP & MEMMI Pricing Model In this section we see several examples of [GLP & MEMM] Pricing Models. To do this, we have to check the existence condition of the MEMM, i.e. we have to examine that the given geometric Lkvy process St = So exp Zt satisfies the conditions of Theorem 2. We denote the function f(MEMM)(8) by f (0) for simplicity, namely
Then the condition is that the following equation has a solution 19'.
4.2.1 Geometric Variance Gamma Model The Levy measure of Variance Gamma process is of the following form (see [31J).
where C, CI,c2 are positive constants. The following results are obtained (see [23] ). Proposition 4.1. 1)Ifcz 5 1, then the equation f(0) = r has a unique solution 8',and the solution is negative. 2) i'fcz > 1and f(0) 2 r, then the equation f(8) = r has a unique solution 8*,
and the solution is non-positive. 3) Ifc2 > 1and f (0) < r, then the equation f (8) = r has no solution. 4.2.2 Geometric CGMY Model The Levy measure of the CGMY process is
whereC > 0,G 2 0,M 2 0,Y < 2(see[3]). If Y I0, thenG > OandM > 0 are assumed. We mention here that the case Y = 0 is the VG process case, and the case G = M = 0 and 0 < Y < 2 is the symmetric stable process case. In the sequel we assume that GI M > 0. For this model the following results are obtained (see [41] ). Proposition 4.2. 1)IfM I 1, then the equation f(0) = r has a unique solution 8',and the solution is negative. 2) IfM > 1and f(0) 2 r, then the equation f(0) = r has a unique solution O*,
and the solution is non-posifive. 3) IfM > 1and f(0) < r, then the equation f(8) = r has no solution.
4.2.3 Geometric Stable Model We consider the stable model. Suppose that Zt is a stable process and let (0,v(dx),b) be its generating triplet. The Levy measure is
where 0 < a < 2 and we assume that
The following results are obtained (see [23]). Proposition 4.3. Under the assumption cl, c2 > 0, the equation f (0) = r has a unique solution 8', and the solution 8' is negative. Remark 4.1. Consider the case where cl, c2 > 0. Under the original measure P, St, t > 0 is not integrable. But under the MEMM P*, any moments Ep.[l~tlk], k = 1,2,. . .,of St are finite. This fact follows easily from the result that 8' is negative, and this property is very useful for the study of option pricing of this model. 4.3 Option Pricing and Volatility Smile/Smirk Properties In order to apply the [GLP & MEMM] Pricing Models to the financial problems, we have to establish the methods to compute the option prices. Namely we have to compute the expectations Ep[F(w)],where F(w) is a functional of Levy process. 4.3.1 European Type Options If a contingent claim C is depending only on the terminal value of the stock price ST = SoeZ', then we can compute the price of C as what follows. Let C = ST) = f(sOeZT) = F(ZT), (F(z) = f ( S 0 8 ) ) , and set C(t,y) = EP'[e-r(T-t)f(ST)ISt = y] and C(t,z) = ~ p [ e - ' ( ~ - ' ) ~ ( z T=) Z( ]~ = t Ep. [e-r(T-t) f (sT)1st= SoeZ]. (Remark that C(t,z ) = C(t,SoeZ).) Since the process Zt is a Levy process with the generating triplet (u2,v*(dx),b'), C(t,z) satisfies the following equation under the assumption of the smoothness of C(t,2).
Solving this equation, we obtain the option price C(0,So) = C(0,O).
FFT Method for European Call Options The fast Fourier transform method (FFT method) is very useful for the computation of option prices. We need to compute the such an expectation Ep[F(w)],and in the case of European type options such type of expectations EP[G(ST)].If we know the distribution function p*,(z) of ZT under PI, then E p [ G ( S ~ )=] G(z)p;(z)dz. L6vy process is characterized by the generating triplet, and the generating triplet is given explicitly in the characteristic function. So we can assume that the characteristic function FT(u)of ZT under PI is given and the density function p;(z) is obtained as the inverse Fourier transform of @;(u). For the computer simulation of the theoretical prices of European call options, the FFT method is very useful. Carr and Madan have introduced their idea in [4], and their method has been improved by Cont and Tankov in [lo]. We rearrange their ideas in such a form that we can easily apply the formula to our [GLP & MEMM] pricing models. The characteristic function $;(u) of Zt under the MEMM P* is 4.3.2
where q * ( u )= $;(u). Let p;(dz) be the distribution of Zt under the MEMM P',and assume that p;(dz) = p;(z)dz. Then
The price of European call option is
T). Then Set K/So = ek, and define c(k;So, T ) = C(So, so@,
We introduce the so-called time value of option
and let [(v;So,T ) be the Fourier transform of C(k;So, T )
Using (4.12)
and e-rT+;(v - i) - eiurT C(v; So. T) = So
iv(l
+
iv)
The characteristic function @;(u) is computed directly from the generating triplet (02,b',v8(dx)),so S(v; So,T )is obtained from the above formula. Next, by (4.14), Z(k; SO,T) is obtained by the inverse Fourier transform
and (87)
~ ( kSo, ; T) = C(k; So, T) + (So- e - r T ~ ) f , K = sock.
Finally we obtain the price of the European call option C(S0,K, T) as (88) C(So, K, T) = c(log(KISo);So, T) = C(log(K1So);So, T) + (So- e - r T ~ ) + 4.3.3 Volatility SmileISmirk Properties
The volatility smilelsmirk properties are reported for many market prices of options. This fact tells us that the Black-Moles model is not necessarily best model, and that we should study other models which may have the volatility simile/smirk properties. It is known that the [GLP & MEMM] models have those properties (see [39] ). 5. Physical World and MEMM World
The behavior of the price process St is governed by the original probability P, and the movement of St is observable. This is the real world (=Physical world). On the other hand the price of an option X is computed as the expectation e-rTEp[~], namely the process St is supposed to obey the MEMM P". This world is differ from the real world, and this world should be called the imaginary world (=MEMM world). 5.1 From Physical World to MEMM World Suppose that the price process St = SoeZtis given and the generating triplet of Zt is (a2,v, b). Let 8' is the solution off (0) = r, where the function f ( 8 ) is defined by (4.6). Then, by Theorem 3 in 54.2, the generating triplet ( d 2 ,v*, b*)of Zt under P" is
This triplet determines the prices of options in the framework of [GLP & MEMM] pricing model. Remark 5.1. The existence condition for 8' (i.e. B' is the solution off (8)= r ) is equivalent to the following martingale condition (M*)
We should notice that the 8' does not appear explicitly in this formula, and that this formula is just the same condition that P is a martingale measure of the price process St. Concerning to the martingale condition for more general cases of semimartingales, see [47] . 5.2 From MEMM World to Physical World We study the inverse roblem of the previous subsection. Suppose that the generating triplet (o*', v*,b.)of Zt under P' is given. Since we assume that P' is martingale measure, the condition (M')is satisfied. We try to construct a probability P such that under P the price process St = SoeZfis geometric Levy process and the MEMM of St = So$' with P is
P'.
Let 8' be any real number (it is usually supposed that 8' < 0) and set
where we assume that all integrals are converge. Then suppose that we could construct the probability measure Pe. such that under Pg' the process Zt is a Levy process with the generating triplet (a;.,oe., be.). It is easy to see that P' is the MEMM of St = So$' with P P . We remark here that there are many geometric Levy processes whose MEMM is just the same P*.
5.3 Example: Geometric Stable Process Case Parameters in the physical world: (a, cl, cz, b), cl+c2>O,-m
0 < a < 2, cl, c2 2 0,
Parameters in the MEMM world: (8', a', c;, c;, b*), + c; > 0, - w < b* < a,where
8' < 0,O < a' < 2,
c;, c; 2 0, c;
l,x
ot (x) $(dx) = et)'(8-l)c* 1 IXl(a'+l) dx.
and the following martingale condition
must be satisfied. So, if we have given the values of (ON,a*,c;, c;, ), then the value of 'b is determined by the above condition ( M ) . 5.4 Diagram of Physical World and MEMM World Physical World MEMM World
(a2,v, b) under P
Zt
( d 2 ,v', b*) under P'
(52,G, b) under P
zt
(5", F,6') under P'
Zt: log-return process, Zt: simple return process
Estimation of Levy Processes in the Physical World Usually this procedure is carried on under the restriction of the class of Levy processes, for example the stable process class, VG process class, etc. Therefore the estimation problem of the process is reduced to the parameter estimation problems. There are many papers on this subject. (see [37] for example). If the MLE is possible and easy, then this method may be good. But this method is not easy to apply our cases. 6.
6.1 Characteristic Function Method of Moments 1) Characteristic Function and Moment Generating Function: The characteristic function (in the sense of distribution) $(u) of X is defined by (98) $(u) = +(u;X) = = exp {+(u)), i = The sample characteristic function &(u) is given by
fi.
Note that &(u) is a consistent estimator of $(u): (100)
1
n-+m
( u )=( u ) ,
-m
< u < m.
The moment generating function M(u) of X is defined, if it exists, by M(u) = M(u; X) = ~[e"], -m < u < m, (101) and the sample moment generating function fi(u) is given by
The sample moment generating function fi(u) is a consistent estimator of M(u). 2) Moment Equations for Characteristic Function:
When we take the function eiuXas the function fu(X)for the generalized method of moments, then the generalized moment equations are (103) +(u) = $J"(u), -03 < U < 00. If the moment E[xk] exists, then it is well-known that
and the classical moment equations are
3) Moment Equations for Moment Generating Function: Suppose that the moment generating function M(u) exists. In such cases we can take the function euXas the function fu(X)for the generalized method of moments, and then the generalized moment equations are
(106)
M(u) = M,(u),
-m
< u < m.
6.2
Estimation of LCvy Processes Set Z = Z1. The corresponding characteristic function +(u) is
(107)
+(u) = ~ [ e ~ = ~ exp ' ] {+(u)}
What we have to do is to estimate the generating triplet (u2,v(dx),b) of the distribution of Z. These parameters explicitly contained in the characteristic functions. So it is natural for us to apply the characteristic function method to those estimation problems. Set
then (li,j = 1,2,. . .) is i.i.d. with the same distribution as Z1, since Levy process has temporally homogeneous independent increment. So, if we are given a sequential data of a Levy process Zt, then we can apply the method described above to estimate the distribution of Z = ZI, namely we can apply the generalized moment equation or the classical moment equation when it exists (see [37] or [2] ). Fitness Analysis of the Models Suppose that the sequential data of the price process St of underlying asset and the data of market prices of options. From these data, we have to select a model which is most fitting to the given data. This is the fitness analysis of the models. 7.1 Procedure of fitness analysis Collecting data: the sequential data of the price process St, and the data of market prices of options. Selection of the most fitting model to the obtained data.
7.
To solve this problem, we first fix a type of model (for example, [Gstable? & MEMM]), and we take the following steps. 1)Estimation the price process of the underlying asset in the physical world from the sequential data of it. 2) Calculation of the MEMM from the estimated parameter, and computation of the theoretical prices of options in the estimated MEMM world. 3) Analyzing the fitness of the theoretical prices to the market prices. We carry on the above procedures for several types of models, and the final step is 4) Determination of the most reasonable model.
7.2 Fitting error of the model
Denote the estimated probability by F, (or equivalently, the estimated be the corresponding MEMM. generating triplet by (?-,~b)), and let Then the theoretical price of option C is ~~,[ce-'~]. We denote this value by C'. Suppose that the data, rli, I = 1,2,. . .,L, of market prices of options Ci are given. Then we define the fitting error of the model by
Procedure to obtain the fitting error: Physical World
MEMM World
Data: ( E i } (time series data) Estimcted: F Transforged: @,Zb) 8* @*,P,V ) (European Call Options) Theoretical prices: Data: ( q r ](European Call Options, in the Market)
Remark 7.1. the second candidate for the fitting error is
7.3 Example: Geometric Stable Process Case Parameter in the physical world: (a,cl, c2, b)
Estimators: (Z,c,~,b)
c*
81 is determined by
The process which determines the theoretical option prices is
7.4
Fitness analysis of the estimated model As the results of the above procedure, if the value E*
is small, then the fitness of the model to the data is good. The value E* depends on the model, namely on the selected class of the process (for example, class of stable processes, class of CGMY processes, etc.) We can conclude that the class whose fitness error is the smallest is the best class (see [40] .) 8. Calibration The calibration is similar to the fitness analysis, but usually the calibration is done based on only the data of option prices in the market (see [ll] ). Namely the calibration solves the following minimization problem
where y* is the parameter of amodel in the risk neutral world. For example, in the geometric stable process case, the calibration means the estimation of y* = (O*, a*,c;, c;, b*)in the MEMM world. Suppose that the above minimization were attained at y(*)= f a ' ) , then the calibration error is
The second candidate for the calibration error is the root mean-square error (RMSE) given by
Calculation of the Option Prices: Expectation of Functionals of Levy Processes In order to apply the [GLP & MEMM] Pricing Models to the financial problems, we have to establish the methods to compute the option prices. Namely we have to compute the expectations Ep[F(w)], where F(w) is a functional of Levy process. 9.1 Asian Option Let A(S, K) be the Asian option, namely 9.
(119)
A(S,K ) = max(
f
T
~ t d-t K,01.
Then the price of A(S., K) is e-lTEp.[A(S.,K)]. For the computation of the above value, in [49] the following interest results have been obtained. Set qt = $ (1 - e-Cf)), and let X t be a stochastic process determined by the following equation
Then it holds that
and so the calculation problem of the price A'(&, K) = e-'TEp.[A(S(o~), K)] is reduced to the calculation of C - ' ~ E ~ [ XNext ~ ] . we set
and define a new probability measure Q' by
then the process Yt is a Markov process under Q* and solves the following equation (125) m 1 d W t + (b.- -02)dt 2 xF,(dudx) xiV;(dudx) -m -, 1+ x
+S
-Sm
Next we define V(t, y) as the solution of the following equation
where V*(dx)is the Levy measure of Zt under the probability P*, and Zt is the LCvy process which appears when St is expresses in the form of S = SoE(Z)t (DolCans-Dadeexponential). Then V(O,2) = V(0, qo - e-rT"so ) is the price of the Asian option A(S, K). The points of this result is that the problem to obtain the price of Asian option is reduced to the problem to obtain the price of European type option of Markov process. 10. Utility Indifference Prices and Risk Measure
We consider the exponential utility function
and set (129)
Jn(c,B) := SUP EP[U,(C+ G(@)T- B)1 Be0
= sup Ep[l - exp{-a(c + G ( 0 )- B ] ] tl€@
where O is a suitable set of strategies, G ( 0 )is the gain of a strategy 8, B is a contingent claim. We give the definition of the utility indifference price p,(c, B) (see [13] 94.2, or [26] ). Definition 10.1. The value p,,(c, B) which satisfies the following equation
is called the utility indifference price of B.
It is easy to see that the value p,(c, B) does not depend on c, so we use the notation p,(B). The quantity J,(c, B) is related to the relative entropy by the following duality relation (131)
L(c, 8)= 1- exp{- inf (H(MP)+ ac - E ~ [ ~ B ] ) } QEM
= 1 - e-,'
expla sup QEM
where M is a convex subset of local martingale measures corresponding to O (see [22]). And the utility indifference price p,(B) has the following property (132)
lim pn(B) = Ep.[B].
(See [44] ,[23] and [48] .) This result suggests us that -EP. [B] may be an example of the reasonable coherent risk measure. 11. Generalization of the [GLP & MEMM] Pricing Model 11.1 Multi dimensional cases
The multidimensional cases are very important in the practical sense, because of the fact that many options are based on the index, for example Nikkei 225, and the index is the combination of the multi dimensional price processes of underlying assets. Suppose that the price processes are given by
where Zf = (z:, . . .,z:)~is a d-dimensional L6vy process. This process is equivalent to
where Zt = (Z:, . . . ,Z;')Tis the corresponding d-dimensional Lkvy process, and the price processes S; have the following expression
~ Dolkans-Dade exponential of 2;. where ~ ( 2 1is)the The results described in the previous sections for 1-dimensional case are generalized to the multi dimensional cases (See [22] and [17] ).
The points are the following two. 1) The MEMM P' is obtained by the Esscher transform by 2. 2) The processes Zt and Zt are also Levy processes under P'. Remark 11.1. The Esscher transform by Zt is unique if it exists, but the Esscher transform by Zt is not necessary unique in the multi dimensional cases (See [30] ). 11.2
[GLP & MEMMI models with defaultable risk We have started from the following type models St = SoeZt, 0 5 t 5 T, or the following SDE dSt = st-dZt, where Zt is a Levy process such that supp C c (-1, w).In this case the model is without defaultable risk. If we permit the case supp V c [-I, m), then the defaultable risk is in mind. To do this we introduce a new Levy process 2j") (A > 0), defined on a new probability space (a(",7(", P(~)), whose Levy measure ~("(dx)is
~", ~ j " =) So&(Z(")t. And let s:" be the solution of d~:")= S ~ ~ ) d Z namely We can assume naturally that the original processes St, Zt and Zt are P(@),where the Levy defined on the same probability space (f2(A),7(*, measure of Zt is ij(dx). Next we define the following stopping time
is independent of St, Zt and zt, and that It is easy to see that dA)
It is obvious that ~ j " = ) StliT(~~),tl. Suppose that the MEMM of s?),P(")*,exists. Then Z?) is Levy process ) ' the Levy measure of 2j")under P ( ~ )is* under ~ ( 4and
where 8(" is the solution of the following equation for 8
We can see that Zt is also a Levy process under ~ ( 3and ) ' the Levy measure Therefore Zt is also a Levy process under P(~)*. of it is eH'A"x~(dx).
Remark 11.2. The Levy measure of Zt under P ( ~ )(=* ev"'xB(dx)) is different from the Levy measure of Zt under P',which is eWxB(dx). From these results we know that T(" is independent of St and Zt under P(")*,and
The theoretical prices of options can be computed as the expectation with respect to P("*. In particular, The prices of European type options are easily computed, using the above properties of d").
Remark 11.3. The arguments of this subsection are possible in the MEMM setting, but not possible in the ESMM setting because the process z:")such = S~$Y' is not well-defined. that sj'\) = So&(Z("))t 11.3 semimartingale process model
The martingale theory is established in the framework of semimartingale process. So, in the theoretical or mathematical sense, it is natural to study the semimartingale process models. In fact many articles are studied under the semimartingale setting. Among them the generalization of the [GLP & MEMM] Pricing Model is discussed in [9] , where the entropy-Hellinger martingale measure is introduced. Acknowledgments. The author likes to thank professor Albert Shiryaev for valuable comments.
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Topics Related to Gamma Processes Makoto Yamazato Department of Mathematical Sciences University of the Ryukyus, Senbaru 1, Nishihara-cho, Okinawa, 903-0213Japan The aim of this paper is to explain important but not popular properties related to gamma processes and show the applicability of these properties. We define subclasses (CME and its subclasses) of the class of infinitely divisible distributions, which are generated by mixtures and convolutions from gamma distributions, and study their properties. Then we apply the obtained results to the unimodality of the distributions in the above classes, the boundedness in space-time parameters of transition densities of subordinators generated by CME distributions and the determination of the class of hitting time distributions of 1-dimensionalgeneralized diffusion processes. Finally, we remark that some subclasses of the class CME and the class of selfdecomposable distributions are often used in mathematical finance. Key words: gamma process, convolution, mixture
1. Introduction
Gamma process appears in various fields both theoretical and practical. Its 1-dimensional distribution - gamma distribution has various remarkable properties (refer [23]). The aim of this paper is to introduce and explain important but not popular (in Japan) properties related to gamma processes and show the applicability of these properties. We restrict ourselves to 1-dimensionalprocesses. We define subclasses ME, CE, CG, CEME and CME of the class of infinitely divisible distributions, which are generated by mixtures and convolutions from gamma distributions, and study their properties. For this purpose, we mainly use the fact that the distributions in these classes can be decomposed into exponential distributions and gamma distributions (Sections 3-6). Then we apply the results obtained in Sections 3-6 to unimodality of the distributions in the above classes (Section 7), boundedness in space-time
parameters of transition densities of CME, processes (Levy processes generated by CME distributions on [0, m)) in Section 8 and determination of the class of hitting time distributions of 1-dimensional generalized diffusion processes (Section9). In the final section, we remark that distributions of some subclasses of the class CME and, class L (class of selfdecomposable distributions) and related stochastic processes are often used in mathematical finance. Almost all facts described in this paper are already known. However, some results are new (Theorems 3.3,3.4,6.3,8.2,8.3, Example 8.1) and the proof of Theorem 3.5 is simplified. 2. Infinitely divisible distributions and LCvy processes
Definition 2.1. A probability measure p on IR is said to be infinitely divis, I R such that ible if for every n 2 1, there is a probability measure ~ i on ~1 = (pn)"' where ( ~ n ) ~ is' the n-fold convolution of y,. Theorem 2.1. A probability measure p on IR is said to be infinitely divisible i f and only i f its characteristicfunction is represented as follows:
where, y
E
lR,o 2 0 and v is a measure, called the Ltvy measure of p, satisfying
J' 1
A
1xl2v(dx)< m.
We denote by I the class of infinitely divisible distributions on IR.
Definition 2.2. A probablity measure p on IR is said to be selfdecomposable if for any 0 < c < 1, there is a probablity measure p, on lR such that
We denote by L the class of selfdecomposabledistributions. The following characterization of the class L is well known.
Theorem 2.2. A probability measure p on IR is selfcecomposable ifand only if it is infinitely divisible and its characteristicfunction is represented as
where, y E IR,o 2 0 and k(u) is a nonnegative measurablefunction nondecreasing on (-m,0) and nonincreasing on (0, m) satisjijing S(lxl-l A (xl)k(x)dx< m.
Characteristic function of an exponential distribution p with density ~ is represented as
e
Hence the exponential distribution is selfdecomposablewith L6vy measure u-'e-". The class of exponential distributions is regarded as a subclass of the class of gamma distributions. A probability measure p is said to be a gamma distribution with parameter (a,j3) if it is absolutely continuous with density B" xn-le-PX for x > o otherwise. The parameter a is called the shape parameter and B , is called the scale parameter. The gamma distribution is selfdecomposablewith Levy measure au-'e-pU . Definition 2.3. A stochastic process (Xt : t 2 0 ) on IR is a Levy process if the following conditions are satisfied: (1) For any choice of n 2 1 and 0 to < tl < . . . < t,, random variables Xt,, Xt, - Xt,, . . .,Xt,, - Xt ,,-, are independent (2)XO = 0 a.s. (3) The distribution of X,+t - X, does not depend on s. (4) limbloP(IXt1 > E ) = 0 for E > 0. (5) Xt is right-continuous in t 2 0 and has left limits in t > 0. Theorem 2.3. Let ( X t )be a Levy process on IR. Then the distribution of Xt is infinitely divisiblefor every t 2 0. Ifthe characteristicfunction of X I is represented as
then the characteristicfunction of Xt is represented as
Let ( X t }be a Levy process. If its distribution at t = to > 0 is an exponential distribution with density j3e-px, (j3 > 0), then the distribution at t > 0 is a gamma distribution with density -x(t~h-l)e-px.
-
~
3. Mixtures o f exponential distributions
We say that a probability measure p is an ME, distribution if there is a probability measure G on (0, w ]such that
p([O,x ] )=
(2)
S
( 1 - e-")G(du)
for x > 0.
(0,mI
The measure p may have a point mass G ( { m ] at ) the origin and the distribution function of p is infinitely differentiable on (0,w ) . Similarly, we say that a probability measure p is an ME- distribution if there is a probability measure G on [-w, 0 ) such that
p([x,O])=
S
( 1 - e-"')G(du)
for x < 0.
[-4)
Theorem 3.1. ([21])A probability measure p on [0,w) is an M E , distribution if and only if there is a nonnegative and absolutely continuous measure Q on (0,w ) with density bounded by 1 a.e. satisfying udlQ(du)< m such that,for s E IR,
%
where L p stands for the Laplace transform of p. We have
We call measures G appearing in (2)and Q appearing in (3) G-measure and Q-measure of p E ME,, respectively. We denote ME = ME, * ME-.
Theorem 3.2. ([29]) Let p+ E ME, and p- E ME- and let p = p, * p- E ME. Let G , and G- be mixing distributions of p, and p-, respectively. Denote
and d, = inf(v > 0 : G+((O,v])> 0 ) .
lfd- < d,, then the Laplace transform f p(s) of p exists for -d+ < s < -d- and is represented as
Theorem 3.3. Let p E ME+ and let q(u)du be its Q-measure. Assume that Jw tq(u)du = m, equivalently, p is absolutely continuous w.r. t. Lebesgue measure. Let f be the density of p(dx). Then f is bounded i f and only i f ? ( I - q(u))du < m . Proof. Let G be the G-measure of p. Since f ( x )is bounded iff m and since
sLp(s) =
1
%G(~U)
Jm
uG(du) <
+
as s + m, we have that f (x)is bounded if and only if lim,,, The quantity s f p ( s ) is written as
sLp(s) < m.
The first and the second terms tend to -
;q(u)du > -m and
I*:
- ( I - q(u))du5 m,
respectively, by monotone convergence theorem. The third term minus fourth term is equal to
which is bounded in s. If J* h ( l - q(u))du < m, then since lim,,, exists,
sLp(s) <
exists and finite. Hence, lim,,,
sLp(s) < w. Converse is obvious.
0
Theorem 3.4. Let p E ME, and let q(u)du be its Q-measure. Assume that 1 iq(u)du)at the ;q(u)du < m, equivalently, p has a point mass exp(origin and is absolutely continuous w.r.t. Lebesgue measure on (0, w ) . Let f be the density of the absolutely continuous part of p(dx). Then lirnXlof ( x ) = iq(u)du)and hence f is bounded ifand only q(u)du < q(u)du exp(-
ifi*
00.
Proof. Let G be the G-measure of p. ue-"G(du)dx,
&_,
Since p(dx) = G ( ( W } ) ~ ~+( ~ X )
as s -+ w. Hence we have
The left hand side of the above equality is written as
By lfH8spital's rule,
ass -+
w.
In order to solve some problems, it is necessary to extend the domain of the Laplace transform of an ME, distribution to the complex number plane 43. We denote by 43, the open upper half plane in 43. A function analytic on 43, with positive imaginary part is called a Pick function ([8]). By P(a, b), we denote the subclass of Pick functions which admit an analytic continuation by reflection across the interval (a,b) into the lower half-plane. Pick functions are related to moment generating functions. In order to apply to Laplace transforms, we consider other classes of functions dual to Pick functions : Let
Proposition 3.1. ([S], [8]) In order that a function h belongs to F(0, w), it is necessary and sujicient that there are n E IR,/3 5 0 and a measure o on [0, m) satisfying ho,C,,(l+ u2)-lo(du) < m such that
The measure CJ is called the spectral measure of h. Since the imaginary part of the principal value Log h(s) (s E 43,) for h E P(0, m) belongs to [-n,O], we have another representation (exponential representation). Proposition 3.2. ([5], [ 8 ] ) In order that a nonnegativefunction h on (0, w) not identically 0 belongs to P(0, w), it is necessary and suflcient that there are a y E IR and a measurablefunction q(u) satisfiing 0 I q(u) I 1a.s. such that
for s E @+. This proposition shows that the Laplace transform of an ME+-distribution belongs to F(0, m) with y = - $r -q(u)du and o(du) = uG(du). Remark 3.1. For s E C+ U (0, w)
s+a=
explexp(-
r(& g(&
i)du + loga) if a > 0, if a = 0. - &)du) -
The imaginary part of the quantity in the braces belongs to [O, n]. Lemma 3.1. Let p
c+u
E
ME, with exponential representation (3). Then, for s
E
%
where y = - q ( ~ ) d ~ . The imaginary part of the quantity in the braces belongs to [-n, n]. It belongs to [0, n],in particular, ifa = 0.
&
Remark 3.2. Although the reciprocal does not belong to the class ME+, it has a representation in Proposition 3.2 and hence belongs to
-
P(0,w).
&
Lemma 3.2. Let p s = -a + ib,
E
ME+ and let G be its G-measure. Let a > 0. Then for
ibf p(s) -+ aG({a))as b J. 0. Proof. Write Lp(s) as
Thus.
The second term in the right hand side of the above equality tends to 0 as b J. 0, since b2u{(u- a)2+ b2}-I is bounded in b E (0,1], b2{(u-a)2 + b2}-I J. 0 as b J. 0 if u # a and Ibu(u - a)((u - a)2+ b2)-'1 Ibu(u - a)-'l A iu. The following result is obtained by Aronszajn and Donoghue Jr. ([I])under more general setting and hence the proof is complicated. The proof in our restricted setting is simplified in [28]. Theorem 3.5. ( [ I ] , [28]) Let p E ME+ and let G and q(u)du be its G-measure and Q-measure, respectively. Let a > 0. Then G((a])> 0 ifand only ifthere is an E > 0 such that
Moreover,
Proof. By Lemma 3.1, we have
l
m
ibLp(s) = exp (log a +
a(u -a) - b2 - ibu U{(U - a)2 + b2}
for s = -a + ib (a, b > 0). By Lemma 3.2, ibLp(s) approaches to aG({a])as b J 0. Assume that G((a])> 0. Then, Arg(ibLp(s))
-+
0, log libLp(s)l -+ log(aG({a)))
as b -,0. Here, Arg denotes the principal value of the argument. Thus,
and
as b J 0. By the monotone covergence theorem, we have lim
JW
~ J O
a(u - a )
U{(U
-
+ b2t (q(u) - l[a,w)(u))u
Also, we have
Thus 02
a Jw-(q(u) u(u 0
- l [ , w ) ( u ) ) l u= - l o g G ( { a ) )> -m.
-0)
Hence for 0 < E < a
Conversely, assume ( 4 ) for some E > 0 . Then, by the same argument as above, we have
4,
Since b l ~ - a l ( l u - a l ~ + b ~_<) - ~ we have (5)by the assumption (4). Therefore,
Remark 3.3. Lemma 3.2 and Theorem 3.5 are valid for F(0, oo) function with u(du) = uG(du).
4. Convolutions of exponential distributions Let
form 2 1andal, ..., am> 0 ) and let CE~_ be the mirror image of CE:. We denote by CE,, CE- and CE the closures in the weak convergence sense of CE:, CE! and CE: * CE!, respectively (CE = Convolutions of Exponential distributions) . Theorem 4.1. Let p p E I and
E
P([Ofm)). Then, p is a CE+ distribution if and only if
for u > 0 and n(x) = (x-l Cke-akx) for x > 0 where, y' 2 0, q(u) = CklCak,,)(u) with (0, m)-valuedfiniteor infinite sequence (ak}satisfying Ckakl < m. Theorem 4.2. Let p E P(R).Then, p is a CE distribution ifand only i f p E I and there is an R\{O)-valuedfiniteor infinite non-decreasing sequence {ak}such that . . I a k < a k + l< . . . I a - l < O < a l < . . . < a [ < a e c lI . . . ,
and the Uvy measure v of p is represented as v(dx) =
x Ck,,, e-akx)d~for x > 0, (1x1-I Ck
5. P6lya frequency functions Definition 5.1. A function f(x) defined on (-m, m) is said to be a PF,function (a Polya frequency function of order r ) or an r-times positivefunctionifforn =2,3, ..., randforxl < x 2 . . . < x n r y l< y2 < ... < ynr
Definition 5.2. A function f ( x ) is said to be a PF (or totally positive ) function if it is r-times positive for any r 2 2.
PF (or PF,) function is said to be a PF (or PF,) density if it is a probability density. The concept PF-function is considered by P6lya et al. and PF,function is introduced by Schoenberg ([19]). PF and PF, densities have sign diminishing property. This fact is used in hypothesis test of number of modes using kernel density estimate ([14]).For an increasing sequence XI < x2 < . . . < x,, v(xl, . . .,x,) denotes the number of sign changes of the sequence. For a function f on R, we define
v ( f ) = sup ~ (( ~ f1 1 ., . .,f (xn)) for all n and all choices of the sequence { x ~ ] : = ~ . Theorem 5.1. Let f be a PF density and let g be a probability density on IR. Let
h(x) =
S f ( x - y)g(y)dy. Then
Theorem 5.2. Let f be a PF, density and let g be a probability density on IR. Let h(x) = S f ( x - y)g(y)dy. I f v ( g ) s r, then
Theorem 5.3. A necessary and sufficient condition that a probability density on IR be PF is that its Laplace transform is represented as
where o2 2 0, y E R, aj are real and 0 < 0212 + CF, aj2 < m. This theorem and Theorem 4.1 show that the class of PF densities coincides with the class of nondegenerate CE distributions. It is known that a density function is a PF2 density if and only if its logarithm is concave. Example 5.1. Normal densities and exponential densities are PF densities. Rather a nontrivial example is $( [ I l l ) . A gamma density with parameter (a,/?), a 1 1, is a PF2 density.
Total positivity is related to Riemann hypothesis. Let <(s)be Riemann's zeta function. Let 5(s) = i s ( s - 1)n-fSr($)<(s)and let E(z) = 5 ( $ + iz). The function E is written as
S ( z )= 2
$
@(u)cos luzdu,
where
m
O(u) = 4 E(2n4n2e9'- 3n2n$U)e-dnp, n=l
%
(refer 1221). Riemann hypothesis is equivalent to that is the Laplace transform of a PF density (Thorin, private communication with Bondesson ([5] p.124), and [16] et al.). 6. Classes CME, CEME and CG
Definition 6.1. We say that a probability measure p on IR is a CME distribution if p E I and its Levy measure v is absolutely continuous with density l represented as
'
Y 0. e-yuQ(du)for y < 0,
= {~I;-yu~(du) for
where Q is a measure on IR \ (0)satisfying
We denoted this class by B in [29]. This class on [0, m) is called gcmed (= generalized convolutions of mixtures of exponential distributions)in [5] et al. The class CME (= Convolutions of Mixtures of Exponential distributions) is closed under convolution and weak convergence ([29]). We define CEME by the class of convolutions of CE distributions and ME distributions. We denoted this class by CME in [29]. It seems that the name CME is more suitable to the above class in Definition 6.1. We say that a distribution p belongs to class CG (Convolutions of gamma distributions) if it belongs to the class CME and the Q-measure Q is absolutely continuous and the density is nonincreasingon (-m, 0) and nondecreasing on (0,m). This class is called ggc in [5] et al. The class CG coincides with the weak closure of finite convolutions of gamma distributions ([5]). By the nondecreasingness of the density of the Q-measure of CG distribution, we see that every CG distribution is selfdecomposable. It is easy to see the inclusions. L U
CEcCGcCEMEcCMEcI. U
ME
Example 6.1. Every 1-dimensionalstable distribution belongs to CG. Levy measure of an a-stable (0 < a < 2) distribution is written as follows: c+x-"-l
&(x)=
rlxl-"-l
+ I)-' LW~ " e - * ~ d ufor x > 0, = c-T(a + 1)-l l ~ l ~ e - * ~for d ux < 0 - c+T(a -
where c+,c- 2 0 and c+ + c- > 0. The C G distribution on [0, co) (= CG+ distribution ) has a remarkable representation resemble to (3).
Theorem 6.1. ([S]p.49) Let p
E
CG+ with Laplace transform
where q 2 0 is nondecreasing, 0 < lim,,, q(u) = a < uV1q(u)du< m, then the density of p is represented as
%
m
and satisfies
f ( x ) = xL'-' h(x)
where h is completely monotone. Proof. For the proof of this theorem, the following fact is essential: Let X and Y be random variables with gamma distributions with parameters (al,/?)and (az,/?), then (&, &) and X + Y are independent. If a CG-distributed random variable X is a sum of n independent gamma distributed random variables, then it can be represented as X = clXl + ~ 2 x + 2 . . + cnXn, where X I , X2,. . .,Xn are independent and scale parameter 1 gamma distributed random variables and cl, c2, . . .,c,, > 0. Let Y = ELl Xk. Then Y is ( a , 1)-gamma distributed and Y,X1 / Y,X2/ Y, .. .,Xn / Y are independent. Hence, Y,clXl / Y,c2X2/Y,. . . ,cnXn/ Y are also independent. This shows that X / Y and Y are independent. Hence X is a mixture of gamma distributions with shape parameter a . Taking a limit, we obtain the theorem. For CME on [0, w) (= CME+)distribution, we have a similar but slightly weaker result. Lemma 6.1. Let p1 and p2 be mixtures ofgamma distribution with shape parameters a and /3, respectively. Then the convolution pl * p2 is a mixture of gamma distributions with shape parameter a + p.
Proof. It is enough to notice that, for x, a, /3, y, 6 > 0,
Theorem 6.2. ([5])Let p
E
CME+ with Laplace transform
where q 2 0,0 < ess q(u) = a < oo and satisfies the density f of p is represented as
1
u-lq(u)du < oo, then
where n is the integer satisfying n - 1 < a I n and G is a measure on (0, co) satisfying
Moreover,
Proof. The probability measure p is represented as a convolution of n ME+ distributions. Hence by Lemma 6.1, we have (7). Since
we have (8). Since A'ze-Ax 5 x-"nne-" for x > 0 and A > 0, we have (9)by (8).
In the above theorem, n can not be replaced by a. A counter example is seen in [5]. A fact similar to Theorem 3.3 holds for CME+ distributions as follows.
Theorem 6.3. Let p E CME, with Laplace transform
q(u) = a , and where 0 I q(u) I a for a 2 1 and for all large u > 0, lirn,,, satisfies u-lq(u)du < m. Let f be the density of p. Then xl-" f ( x ) is bounded ifand only if
Jm
:(a - q(u))du < m.
Ifq(u) I a u-a.e., then (10)
lim xl-" f( x ) = r(a)-' exp[
nu - ( u + l)q(u) du].
110
Especially, i f p E GC,, then the right-hand side of the above equality is represented as
Proof. Let ql(u) = q(u)Aa and let p1 be a CME distribution with Q-measure ql(u)du. Since a > 0, pl has a density g. The quantity saLpl(s)is written as
The first and the second terms tend to -
l1
i 4 1 ( u ) d u> -m and
Lm:
-(a - q ~ ( u ) ) d5u m,
respectively, by monotone convergence theorem. The third term minus fourth term is written as
which converges to
as s + m by the dominated convergence theorem. By Theorem 6.2, g(x) = O(x-l) as x -, m. Hence xl-"g(x) is bounded for large x. Note that lim,,,s"~pl(s) exists allowing infinity. Hence, :(a - ql(u))du < w if and only if lim saLpl(s)= exp(
s'm
Am
w.
u(u + 1)
Am -
Assume that (a- q ( u ) ) d< ~ m, equivalently, b ( a-ql (u))du< oo. By Tauberian theorem, we have p1([0,x]) ix" as x J, 0 where C is the right hand side of (10).We have, by L'hospital's rule, g(x) Cxa-l as x J 0. This shows (10).We also have that xl-"g(x) is bounded. The CME distribution ,~12with Q-measure q(u) - ql(u)is written as pho(dx) + (1 - p)h(x)x. Since a 2 1, we have
-
I sup x ~ - " ~ ( < x )w . X
Conversely, assume that xl-" f(x) is bounded. Since p 2 has a point mass at the origin, xl-"g(x) is bounded near the origin. Hence xl-"g(x) is :(a - ql(u))du.c bounded. Then sflLpl(s)is bounded. Hence we have m, equivalently, Jm :(a - q(u))du< m. (11)is straightforward.
Am
7. Unimodality
Definition 7.1. A probability measure p on IR is said to be unimodal if there is a E IR such that the distribution function of p is convex on (-w, a ) and concave on (a, w). Definition 7.2. A probability measure /i on IR is said to be strongly unimodal if for every unimodal distribution p, the convolution p * p is again unimodal. Theorem 7.1. ( [ l o ]A) probability measure p on IR is strongly unimodal Yand only ifit is absolutely continuous with logarithmic concave density (PF2density). Theorem 7.2. Every CEME distribution is unimodal.
Proof. ME-distributions are unimodal with mode 0. CE-distributions are strongly unimodal. Hence CEME-distributions are unimodal.
It is easy to see that every stable distribution is selfdecomposable. Every selfdecomposabledistribution is unimodal ([25]). Hence every stable distribution is unimodal. The proof of unimodality of selfdecomposable distributions is not simple. But, since we know that stable distributions are CG-distributions, the proof of unimodality of stable distributions is quite simple as follows. Theorem 7.3. ([26])Every 1-dimensional stable distribution is unimodal. Proof. 1-dimensional stable distributions belongs to CG and CG is a subclass of CEME. Hence 1-dimensional stable distributions are unimodal by Theorem 7.2. Boundedness of transition densities of CME. processes Let (X(t)}be a Levy process. If the distribution of X(l) is a CME, distribution, then we call (X(t)}a CME, process. Assume that, in this section, the distribution of X(1) is absolutely continuous. Namely, Jmt ~ ( d u=) w for the Q-measure of the distribution of X(1). We consider in this section, under what condition the transition density of {X(t)}is bounded in the space variable x and the supremum in x tends to 0 as time variable t tends to oo. The following result is shown in [24].
8.
Theorem 8.1. Let (X(t)}be a CME, process with transition density p(t, x). Then for any 0 < to < tl and xo > 0,
sup
p(t, x) .= m.
te[to,tl Ixe[~~,m)
1. Let 0 < a < band let 1 0
for x E [a, b], for x E [a, bIC.
Let p E CME+ with Q-measure q(u)du. Then we have p([O, x]) = p + q(1e-ax)for x > 0 where p = and q = 1- p. Hence
where
Let fk(x) =
m.Then, by Stirling formula r ( z ) - fizZ-112e-z, (z -+ nkX*-le-"x
m),
The quantity sup, p,(x) can be regarded as an integral of sup, fk(x) with respect to a binomial distribution {(;f)p"-kqk}&o. The binomial distribution tends to 6,(dx) as n + m. Hence lim sup p, ( x ) = 0.
n-rm
2. Let Q(dx)= 6,(dx) (a > 0) and let p be a CME+-distribution corresponding to Q. Then pf*(dx)= e-""60(dx) + pt(x)dx
where
In this case, we also have
sup pt(x) -t 0 as t
-t
m
X
Theorem 8.2. Let p E CME,. Let 1 for x E [a,b], 0 for x E [a,bJc a s . m
$q2(x)dx= mand ?(l-q2(x))dx < m where0 < a < b. 0 5 q ~ ( xL) 1 a.s., Iffhe Q-measure Q of p is given by Q(dx) = (ql(x)+ q2(x))dx+ Q J ( ~ xthen ), dpf* lim sup -( x ) = 0. ~ < x < m dx
f+m
Proof. Let 1 I n 5 t < n + 1. Then Q-measure of pt* is writen as
nql(x)dx+ q2(x)dx + Qddx) where
Ql(dx) = (t - n)ql(x)dx+ ( t - l)qz(x)dx+ tQs(dx). Probability measure corresponding to 92 is absolutely continuous and the density h is bounded by Theorem3.3. Hence the probability measure corresponding to q2(x)dx+ Qa(dx)has a density g(t, x) and it satisfies that r
where p2 is a probability measure corresponding to Q4(dx). Let p y be a probability measure corresponding to nql(x)dx. Then for p = %,
and suppn(x) + 0 as n + co, X
where pn(x) = C;=,( k ) p q -. Density f( t l x )of pf' satisfies n
n-k knkxk-'e-""
f
(f,X )
I png(t,x) + sup pn(x) X
5 pn sup h(x) + sup pn(x) X
X
Theorem 8.3. Let ,u E CME,. Let 0 I q2(x) I 1 as.,
I f the Q-measure of p is given by Q(dx) = 6,(dx) + q2(x)dx+ Q3(dx)where a > 0, then
dpf' lim sup -( x ) = 0. t+cao5x<m dx
Proof. The proof is the same as the proof of above theorem using 2 instead of 1. Remark 8.1. If q2 does not satisfy A(1 - q2(u))duim, then there is an example for which the density of pt*(dx)is not bounded for all t > 0. The following Example 8.1 exhibits such an example. Example 8.1. Let p E ME, with G-measure G. Assume that G((co))= 0 and $ uG(du) ~ ( l o ~ xas) x- + ~ m. Then p is absolutely continuous with respect to Lebesgue measure and p has a density
-
f ( x ) := Hence that
lm ue-""G(du)
1 X-'(log -)-2 as x J, 0. X
Lmi(l- q(u))du < co by Theorem 3.3. Integration by parts yields rx
By Abelian-Tauberian theorem for Laplace transform, this asymptotic relation is equivalent to that 1 - (- log -)-I as s s
S
e-"F(dx)
Then,
+ m.
(lm t
(12)
e - ~ ~ ( d x ) )(logs)-' as s + m.
This is equivalent to that (13)
-
F'*(x) (- log x)-t as x 0,
by Abelian-Tauberian theorem. Hence f "(x) is unbounded in x > 0 for each t > 0. Iff '*(x)is monotone for all small x, then ft'(x) it(- logx)-'-'
-
be the Laplace transform of p. Then
" --du1 q(u) U+S
U
- tloglogsass +
m
by (12). By Abelian-Tauberian theorem for Stieltjes transform ([4]), this asymptotic relation is equivalent to that
-&
9
If is monotone for all large u, then q(u) as u + m. Conversely, assume that T d u log log x as x + m.
1
-
Then we have (13) by Abelian-Tauberian theorems for Stieltjes transform and Laplace transform. 9. l-dimensional generalized diffusion processes
Let ( B ( t ) }be a one-dimensional Brownian motion and let t(t,x) be its local time. We denote by M the class of right continuous nondecreasing function m on [-oo, m] to [-m, m] with m(+m) = f m and m(0-) = 0. For m E M, we define tj = tj(m) by
for j = 1,2 and we define a measure m(dx) on [-w, w ]by
Here [ C l , &Ic is the complement of
[ E l , C,].
Let
Define a stochastic process ( X ( t ) ,51 by X(t) = B(+-'(t)) and the life time
5 = i n f ( t > O : X ( t ) = t l o r t 2 ] i f (]#0, =w otherwise. This process is a strong Markov process with state space Em = (supp m)l~t,,ez) and is called the generalized diffusion process corresponding to the function m (see [12]). The measure restricted to ( t l , E 2 ) is called the speed measure of the process {X(t)].For y E Em, we define the hitting time of y by T, = inf(t > 0 : X(t) = y] if ( 1 #0, = 00 otherwise. If ltjl< w and C j E Em, where Em is the closure of Em in R,then we define .rej by y by limy,,,, T~ for j = 1,2, respectively We denote by Em the set with t j ( j = 1,2) adjoined to Em whenever Itjl < w and ej E Em. If P x ( ~<, 00) > 0 for x in Em and y in Em, we define
We denote by
the class of conditionnal hitting time distributions of generalized diffusion processes. In [27](Theorem I), the following characterization is obtained.
Proposition 9.1. In order that a probability measure p on IR, belongs to Hgd, it is necessary and suficient that there are a CE+ distribution p1 with Laplace transform f pl(s) = & and an ME+ distribution p2 with p2((0})= 0 such that p = p1*p2 and (ai)is either empty or a strictly increasing Fnite or infinite) sequence and the spectral measure (T of (sLp2(s))-' has a positive point mass at ai for each i.
n
This proposition shows that Hgd is a proper subclass of CEME,. We can restate this result in terms of Q-measure using Theorem 3.5 in Section 3. Theorem 9.1. ([28])In order that a probability measure p on R+belongs to Hgd, it is necessary and suficient that its Laplace transform is represented as
where ql and q 2 satisfy thefollowing conditions: I. (a) ql = 0 or (b) ql is a non-decreasing stepfinction with step size 1, ql(0) = 0 and jump points (aj]of q1 satisfy C j ayl < w (hence, 0 is not a jump point).
SO
1
2. 0 I q2(u) 2 1, :qp(u)du < m, 3. In case (b) in 1, q 2 satisfies
for
( E ~ with }
Am $q~(u)du=
m.
aj > ej > 0.
Proof. Apply Theorem 3.5 to the reciprocal of the expression of sLp(s) in Lemma 3.1. This theorem shows that every stable distribution p with Laplace transform 1
L ~ ( s=) exp Jm(L - -)cuffdu 0
u+s
U
belongs to Hgn Here, c > 0 and 0 < a < 1. ;-stable (a = ;) distribution is the hitting time distribution of Brownian motion. It is not known what kind of generalized diffusions correspond to other stable distributions. Gamma distribution does not belong to Hgdif and only if the shape parameter a is greater than 1. 10. Levy processes appearing in mathematical finance
Recently, various types of Levy processes, namely VG (Variance Gamma), NIG (Normal Inverse Gaussian), GIG (Generalized Inverse Gaussian), GH (Generalized Hyperbolic) processes often appear in Mathematical Finance literature as a model of stock price. Also, stationary processes of Ornstein Uhlenbeck type are used as volatility processes in stochastic volatility models (refer [3] and references therein). The class of stationary distributions of the statinary one-dimensional processes of Ornstein Uhlenbeck type coincides with the class of selfdecomposable distributions on
IR ([MI). We show that the above classes (VG, NIG, GIG, GH) of processes (or distributions) belongs to the class CG or the class of selfdecomposable distributions. Let p(x; p, 6) be the density of the positive p/2-stable distribution with Laplace transform
Boyarchenko and Levendorskii ([7]) called a probability measure on [0, co) with density p(x; p, 6, y) = eb'"'p(x;p, 6)e-iJ"", y > 0 a Tempered Stable distribution and denoted TS(p,6, y) ([7]). Its Laplace transform is written as eQ('), where
and the Levy density is given by
They ([6]) called an infinitely divisible distribution p on IR a KoBoL distribution of order p < 2 if it is infinitely divisible with the Levy measure
where c, > 0 and A, > 0. This shows that the Tempered Stable distribution is a one sided KoBoL distribution of order p/2. KoBoL distribution is a CG distribution with the density of its Q-measure
KoBoL distribution with p = 0 is called VG distribution. The use of VG distribution in finance is proposed by Madan and Seneta [13]. In [15], KoBoL distribution is called tilted stable distribution and the name "tempered stable" is used for an infinitely divisible distribution with Levy measure
where u E (0,2) and the measure Q(du) := q(u)du on IR\{O) satisfies (6). The meaning of "tilted" is explained in [3].
We denote by KA the modified Bessel function of the third kind with index A. Let Yt = fit + Bt where (Bt} is a Brownian motion. Let (Zt} be a subordinator generated by TS(p, 6, y)-distribution independent of {Yt}. Then the characteristic function of the subordination Xt = Yz, is of the form et@(~), where +(z) = 0(;z2 - ipz). It is rewritten as
dmi.
Since the 1-dimensional distributions of a suborwhere LY = dinated process of a Brownian motion with drift by a selfdecomposable subordinator is selfdecomposable ([lq), the distribution of Xt is selfdecomposable for each t > 0. Adding ipz to (14) and then letting p = 1, we get the NIG distribution with characteristic function exp ipz + 6[(n2- p2)'R - (a2- (p + i,z)2)l/fl).
(
The transition density is given by
where (y) = (1 + A probability measure on (0, w) is said to be a GIG distribution if it has a density
where the parameters satisfy
Halgreen [9] and, Shanbhag and Sreehari [20] showed that GIG distribution is a CG distribution. A probability measure with characteristic function
is called GH distribution. It is absolutely continuous with respect to Lebesgue measure and the density is represented by KA-l12. GH distributions are obtained by the subordination of Brownian motion with drift
by GIG subordinator. They (191, 1201) proved that GH distributions are selfdecomposableby showing that the one dimensional distributions of a subordinated process of Brownian motion with drift by CG subordinator are selfdecomposable. Sato's result ([17]) is its extension. We remark that NIG distribution is a GH distribution with A =
-;.
References 1. N. Aronszajn and W. F. Donoghue Jr., On exponential representations of ana2.
3. 4. 5.
6. 7.
8.
9. 10. 11. 12.
13. 14.
15. 16.
lytic functions in the upper half-plane with positive imaginary part, J. Analyse Math., 5 (1956), 321-388. N. Aronszajn and W. F. Donoghue Jr.,A supplement to the paper on exponential representations of analytic functions in the upper half-plane with positive imaginary part, J. Analyse Math., 12 (1964), 113-127. 0. E. Barndorff-Nielsen and N. Shephard, Modelling by Levy processes for financial econometrics, in Uvy Processes Theory and Applications, Birkhauser (2001), 283-318. N. H. Bingham, C. M. Goldie and J. L. Teugels, "Regular Variation", Cambridge University Press (1987), Cambridge. L. Bondesson, Generalized gamma convolutions and related classes of distributions and densities, Lecture Notes in Statistics, 76 (1992) Springer-Verlag, New York. S. Boyarchenko and S. Levendorskii, Perpetual American options under Levy processes, SIAM J. Cotrol Optim. 40 (2002), 1663-1696. S. Boyarchenko and S. Levendorskii, "Non-Gaussian Merton-Black-Scholes Theory", Advanced series of statistical science & applied probability Vol. 9, World Scientific (2002), New Jersey-London-Singapore-HongKong. W. F. Donoghue, Jr., "Monotone matrix functions and analytic continuation", Springer 1974, Berlin Heidelberg New York. C. Halgreen, Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions, Z. Wahrsch. Verw. Gebiete 47 (1979) 13-17. I. A. Ibragimov, On the composition of unimodal distributions, Theor. Probability Appl. 1 (1956) 255-260. S. Karlin, "Total Positivity", Vol. 1, Stanford Univ. (1968), Stanford. S. Kotani and S. Watanabe, Krein's spectral theory of strings and generalized diffusion processes, Functional Analysis in Markov Processes (M. Fukushima, ed.), Lecture Notes in Mathematics, 923 (1982), 235-259, Springer, Berlin Heidelberg New York. D. 8. Madan and E. Seneta, The VG model for share market returns, J. Business 63 (1990),511-524. E. Mammen, J. S. Marron and N. I. Fisher, Some asymptotics for multimodality tests based on kernel density estimates. Probab. Theory Relat. Fields 91 (1992), 115-132. J. Roshski, Tempered stable processes, Mini-proceedings : 2nd MaPhysto Conference on Levy Processes Theory and Applications (2002), Aarhus University. B. Roynette et M. Yor, Couples de Wald indkfiniment divisibles Examples lies
17. 18. 19. 20. 21. 22. 23.
24. 25. 26. 27. 28.
29.
h la fonction gamma d'Euler et h la fonction zeta de Riemann, To appear in Ann. Inst. Fourier. K. Sato, Subordination and selfdecomposability,Statistics &Probability Letters 54 (2001) 317-324. K. Sato and M. Yamazato, Stationary processes of Ornstein-Uhlenbeck type, Lecture Notes in Math., 1021 Springer (1983) 541-551. I. J. Schoenberg, On P6lya frequency functions I. The totally positive functions and their Laplace transforms, Journal d'Analyse Math., 1 (1951), 331-374. D. N. Shanbhag and M. Sreehari, An extension of Goldie's result and further results in infinite divisibility, Z. Wahrsch. Verw. Gebiete 47 (1979) 19-25. F. W. Steutel, "Presetvation of infinite divisibility under mixing and related topics", Mathematical Center Tracts 33 (1970), Matematisch Centrum, Amsterdam. E. C. Titchmarsh, "The zeta-function of Riemann", Hafner (1972), New York. N. Tsilevich, A. Vershik and M. Yor, Distinguished properties of the gamma process, and related topics, Prepublication du Laboratoire de Probabilites et Modeles AlCatoires No. 575 (2000). S. Watanabe, K. Yano and K. Yano, A density formula for the law of time spent on the positive side of one-dimensional diffusion processes, preprint. M. Yamazato, Unimodality of infinitely divisible distribution functions of class L. Ann. Probability 6 (1978), 523-531. M. Yamazato, On strongly unimodal infinitely divisible distributions, Ann. Probability 10 (1982),589-601. M. Yamazato, Hitting time distributions of single points for 1-dimensional generalized diffusion processes, Nagoya Math. J. 119 (1990), 143-172. M. Yamazato, Characterization of the class of hitting time distributions of 1-dimensional generalized diffusion processes, Proc. 6th Japan-USSR Symposium on Probability Theory and Mathematical Statistics, (1992)422-428, World Scientific, Singapore-New Jersey-London-HongKong. M. Yamazato, On subclasses of infinitely divisible distributions on R related to hitting time distributions of 1-dimensionalgeneralized diffusion processes, Nagoya Math. J. 127 (1992), 175-200.
On Stochastic Differential Equations Driven by Symmetric Stable Processes of Index a Hiroya ~ashimotol,Takahiro Tsuchiya2and Toshio Yamada3 'Sanwa Kagaku Kenkyusho Co.,LTD, Department of Mathematical Sciences,Ritsumeikan University 1. Introduction In the first part of the present paper, famous Tanaka's equation is discussed in the case of symmetric stable processes. Then some important uniqueness results in one-dimensional case will be reviewed. The second part is devoted to obtain some results concerning comparison problems using Lamperti's method. Marcus integral plays an essential role to formulate comparison theorems. In the last part, a sufficient condition which guarantees the pathwise uniqueness in d-dimensional case, will be proposed. 2. On uniqueness problems: One dimensional case
We consider followingstochastic differentialequations driven by a symmetric stable process of index a.
where Zt is a symmetric stable process of index a of which characteristic function is given by
As is well known, famous Tanaka's example shows that the weak uniqueness does not imply the pathwise uniqueness in the case of SDE driven by a Brownian motion. We will mention that Tanaka type argument is still applicable to the case of the equation with respect to a symmetric stable process of index a . Theorem 2.1. Consider the equation
where
Then, the weak uniqueness holdsfor solutions to (31, but the pathwise uniqueness fails. To prove Theorem 2.1, the following theorem by Rosinski and Woyczynski [ll]plays essential roles. Theorem 2.2. Let F be an 7; := o(Zs;s 5 t ) -adapted process such that
and the random time ~ ( u:=) $ I Ft I" dt satisfies that u + 00. Consider the inverse of T and Bt:
T(U)
-+ M,P.s., when
Then the time changed stochastic integral T-I
(t)
Fs dZ,
zt =
is a Gtadapted symmetric stable process of index a. x We have also,
[Proof of the Theorem 2.11 Let Xt be a solution to (3). Then, we observe that T ( U ) := I sgn(Xt-) la dt = u + m,as u -t oo.
L"
Then by the Theorem 2.2, Xt is a symmetric stable process of index a with = 7; . So, any solution to (3) has the same law. Then the respect to 7,-1(~) solution to the equation (3) is unique in the weak sense. We will observe that the weak existence of a solution to (3). Let X, be a symmetric stable process of index a with Xo = 0 . Then, t
Zt =
sgn(Xs-1 dXs
is also a symmetric stable of index a and Xt satisfies
This means the existence of a solution to (3)in the weak sense. Now, let Xt be a solution to the equation (3):
Then (-Xt) satisfies t
-Xt =
s p ( - X s - ) dZs
So, (-Xt) is also a solution to (3). The pathwise uniqueness fails for the solution to (3). When is the solution to (1)or (2)is pathwise unique? If the coefficients are Lipschitz continuous, the Picard iteration method works very well and it proves the pathwise uniqueness for the solution to (1)or (2). In one dimensional case much weaker conditions suffice for uniqueness for SDE's with respect to one dimensional Brownian motion. For example (see [12]),a sufficient condition for pathwise uniqueness is that P - ~ ( udu ) = clro where p is the modulus of continuity:
So+
I 4 4 - 4 y ) I5 p(l x - y I). In view of the above result, one would hope that analogous weaker conditions would suffice for the pathwise uniqueness to the solution to (1)or (2). The following condition is due to Komatsu [7]. (See also Bass [2]).
Theorem 2.3. Suppose that
I
- o ( y ) I"<
p(l x - y
I),
Vx,Vy E R1,
where the modulus of continuity p is a continuous increasing function with p(0) = 0 such that
S
p-'(x)dx=oo,
Then,for all xo
E R1 the solution
is pathwise unique.
VE>O.
to the equation
Concerningto the solution to the equation (2) of which coefficient is time inhomogeneous, following Nagumo type modification to the condition in the above theorem is proposed in [6] .
Theorem 2.4. Suppose that
where thefunction p is continuous increasing with p(0) = 0 such that
and where thefunction h(t)is non-negative continuous such that
Thenfor all xo
E
R1, the solution to the equation
is pathwise unique. Remark 2.1. The conditions given in the Theorem 2.2 and Theorem 2.3 are sharp and best possible in some sense. 3.
Lamperti's method Given b(x)satisfying the Lipschitz condition:
I b(x) - b(y) I5 M I x - y I
Vx, V y E R1,
the equation driven by an one dimensional Brownian motion:
can be solved very simply using the following inequalities ~ . ( t ):= max I x?+')- x?) 15 sst
6 t
I b(x?') - MX?-')) I ds
6
t
where X? = xo and xi")= xo + Bf + ~(xP-") ds. Now we make a change of scale x + y = f (x) with f be Yf = f (Xt) . Then Ito's formula implies
E
c~(R'). Let Yt
where,
(b)
b(f)
= f'b+ y / 2 .
Lamperti's idea is to construct the solution to the equation (7) by solving (a) and (b) for f and b . Standard Picard's iteration procedure applies to a wider class of coefficients, but Lamperti's method is simpler, because it uses neither the martingale inequality nor Borel-Cantelli lemma. (See Lamperti [9] and also McKean [lo].) Just as in the same way as in one dimensional Brownian motion case, the following equation concerning a symmetric stable process of index can be solved:
Make a change of scale Yt = f (Xt) with increasing f E C3(R1). Then in the language of Marcus integral Yt satisfies
where 0 means the Marcus integral. Remark 3.1. For the precise definition of Marcus integral, see KurtzPardoux-Protter [8] and also Applebaum [I]. In the case where
Marcus integral:
stands for
where @' = q. 4. Some comparison results In the present section Zt stands for a symmetric stable process of index
a. In view of comparison results for solutions to SDE's concerning a one dimensional Brownian motion, one would hope that that solutions to SDE's driven by a symmetric stable process of index a: d ~ r=) a(Xt-)dZi + bi(Xt-) dt,
)!x
= xi, i = 1,2,
enjoy comparison properties under the conditions such that
and a satisfies the same condition as in Theorem 2.3. But, unfortunately the next example would suggest that in our case, comparison problems for solutions would take much more complicated aspects. Example 4.1. Consider the followingtwo equations:
Then the solutions to (11)are given by Dolean-Dade as follows;
Consider the following series of random times T,, n = 0,1,2, . . .,such that
Under the condition xl < x;!,we observe that
In the followings of the section, we will apply Lamperti's method to obtain some comparison results. Consider the following SDE's:
where bisatisfies the Lipschitz condition such that
As we have seen in the section 3, solutions to (13)can be constructed by simple way. We have two comparison lemmas. Lemma 4.1. (Weak comparison lemma) Suppose that for the equations (13)
and (ii) bl(x)I b;!(x), V X E R1.
Then, holds. Lemma 4.2. (Strong comparison lemma) Suppose that for the equations (13)
and (ii) bl(x)< b*(x), V X E R1.
Then, holds.
p(xj1) < xj2), t 2 0) = 1
Let f E C 3 ( ~ 'with ) f > 0. Make a change of scale Yj') = f ( ~ ? ) )Then . Y f ) satisfies following Marcus equation:
with
Suppose that the coefficients 13and b;. are given such that the equations (a) and (b)can be solved for f E C3(R1)f > 0 and bi . Then we obtain following comparison results for solutions to Marcus equations (14).
Theorem 4.1. (Comparison theoremfor Marcus equafions) Suppose that
i l ( x ) I 62(x), V XE R1, and Y (1) o I Y (2) o a.s., Then
P ( Y ~ '5) Y?);t E [0,m ) )= l holds. Theorem 4.2. (Strong Comparison theoremfor Marcus equations) Suppose that
&(x) < 62(x), V X E R', and Y:' < Yf)a.s., Then P ( Y< ~~
f )t ;E [0, m))= 1
holds. Remark 4.1. Lamperti's method is applicable to a wider class of Marcus equations driven by semimartingales.
5. Pathwise uniqueness: d-dimensional case Let Zt be a d-dimensional symmetric stable process of index a, (1 < a < 2): E[exp((T,zt))l= exp(-t 1 5 I") Consider the following stochastic differential equation driven by Zt :
Assumption 5.1. Assume that the coefficient matrix u = [oik] satisfies oik(x)= dikO(x), X
E
R~
Under the Assumption 5.1, we have the following theorem.
Theorem 5.1. Suppose that the continuousfunction p defined on [0,cu) with p(0) = 0 is increasing and such that:
is concave and
Let p be the modulus of continuity:
I ~ ( x-) o(y) 1 p(l x - y I), Vx, Vy E R ~ . Then for every yo E \R
the solution to
is pathwise unique. Remark 5.1. For examples, the functions such that
p ( ~ =) U ,
p(u) = u((a - 1)log l ) i , u
satisfy the conditions imposed on pin the theorem 5.2. Concerning the equation driven by a d-dimensional Brownian motion:
the functions such that p(u) = u,
1
1
p(u) = u(l0g -) 3 , U 11 11 p(u) = u(l0g -) 7 (loglog -) 3, . . ., u U imply the pathwise uniqueness. (See [13].)
References 1. D. Applebaum. Levy Processes and Stochastic Calculus. Cambridge Univ. Press (2004). 2. R. F. Bass. Stochastic differential equations driven by symmetric stable processes. Seminuire de Probabilitis XXXVI (2003), 302-313. 3. R. F. Bass, K. Burdzy and Z. Q. Chen. Stochastic differential equations driven by stable processes for which pathwise uniqueness fails, Stoch. Proc, their Appl. VO~ 111 . (2004), 1-15. 4. R. F. Bass. Stochastic differential equations with jumps. Probability Surveys. vol. 1 (2004), 1-19. 5. C. Doleans-Dade.Quelques applicationsde la formule de changement de variable pour les semimartingales. Z. Wahr. vol. 16 (1970), 181-194. 6. H. Hashimoto. On stochastic differential equations driven by symmetric stable processes-uniqueness and comparison problems (in Japanese),Master thesis (2004), Ritsumeikan University. 7. T. Komatsu. On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations of jump type. Proc. Japan Acad., vol. 58, Ser. A, no. 8 (1982), 353-356. 8. T. G . Kurtz, ~ . ~ a r d o uand x P. Protter. Storatonovich stochastic differential equations driven by general semimartingales. Ann. Inst. H. Poicark. vol. 31, no. 2 (1995), 351-377. 9. J. Lamperti. A simple construction of certain diffusion processes. lour. Math. Kyoto Univ., vol. 4 (1964),161-170. 10. H. P. McKean, Jr. Stochastic Integrals. Academic Press (1969). 11. J. Rosiliski and W. A. Woyczyhki. On Ito stochastic integration with respect to p-stable motion: Inner clock, integrability of sample paths,double and multiple integrals. Ann. Prob., vol. 14 (1986),271-286.
12. T. Tsuchiya, On the pathwise uniqueness of solutions of stochastic differential equations driven by multi-dimensional symmetric n stable class. submitted. 13. T. Yamada and S. Watanabe. On the uniqueness of solutions of stochastic differentialequations. lour. Math. Kyoto Univ.,vol. 11 (1971), 155-167. 14. S. Watanabe and T. Yamada. On the uniqueness of solutions of stochastic differential equations 11. ibid., vol. 11 (1971), 553-563.
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Martingale Representation Theorem and Chaos Expansion Shinzo Watanabe Ritsumeikan University 1. Introduction Given a filtration F = (51(i.e. an increasing family of sub a-fields of events), a martingale representation theorem is concerned with a representation of F-martingales as stochastic integrals by basic martingales. In the case of Brownian filtration, of which we shall discuss in Section 2, a result is well-known as ltdS representation theorem which states that every square-integrablemartingale can be represented as a stochastic integral by the path of Brownian motion. This theorem was first found by It6 ([I]) as a corollary of his theory on Wiener chaos expansion of L2-Wienerfunctionals and plays an important role in the problem of financial markets. In this respect, we would quote the following remark by Daniel Stroock in page 180 of [S]: "ln fact, it (1tOS representation theorem) shares with 1tOS formula
responsibility for the widespread misconception in thefinancial community that ItB is an economist." In this introduction, we will review such a theorem and discuss its idea in the simplest case of a random walk. Let (Ek)k=1,2,... be a coin tossing sequence, i.e., i.i.d. sequence with P(Ek = 1) = P(Ek = -1) = It is also called an i.i.d. sequence of random signs. If we set
1.
n=O x,,= { O , +...+t,,, n = 1,2,... X = (X,),,=o,l,... is a simple random walk on Z starting from the origin. Since tk= Xk - Xk-l, k = 1,2,. . ., X = ( X n )and (Sk)generate the same filtration F = { E l n = ~ , l , . . . , where
In the following, we take and fix N E Z++(:= ( n E Z I n > 0 1) and consider the time up to N; N is called the maturity in financial problems. As usual, a family Y = (Yn)OsnsNof random variables is called an F-adapted
process if Y n is ~l-measurablefor every 0 I n I N, and a family @ = (@k)l
and call it the martingale transform of X, more precisely, [Mo,@]-martingale transform of X. We denote this as M = Mo + 0 .X, for short. It is irnmediately seen that M is an F-martingale. The martingale representation theorem for the filtration F states, conversely, that every F-martingale can be represented by a martingale transform of X. Namely, Theorem 1.1. Every F-martingale M = (M,) can be represented in theform
by the constant Mo and an F-predictable process @ = (@k)lsksN.Here, Qk is determined uniquely by
Note that Mo = E(MN)by the martingale property, Here is a simple proof: Since Mk - Mk-1 is %-measurable, there exists a function dk(il,. . . ,ik)on the product space (-1, lIksuch that
By the martingale property,
and hence, we have d k ( t l r .. . ,&-I, 1) = -dk(tl,.. . , t k - 1 , -1). This implies, by setting @k = dk(
Obviously @ = (@k)lsksnis F-predictable and ( M k- Mk-i). Ek = Q k eE: = Ok. This completes the proof.
We will give still two different proofs; it may look too much exaggerated, as the proof given above is so simple. However, the ideas of the following proofs can be applied to the case of continuous time: indeed, they provide us with two different prototypes of the proof in continuous time, as we shall see in Section 2. Proof A. A real FN-measurablerandom variable Y has a representation as Y = f(El, . . . ,EN) by a function f (il, . . . ,i ~on) the product space (-1, 1tN, and the totality of such Y forms a 2N-dimensional (real) Hilbert space (n R ~ with ~ ) the in inner product (Y,Z) = E(YZ). A orthonormal basis (ONB) (Es}is given, indexed by a subset S c (1,2,. . . ,N), as Ss :=
n
ti, S c (l,2,. . . ,N), with the convention S1 = 1.
~GS
Take any Y and consider the orthogonal expansion of Y - E(Y) by the . Ck where k is the orthonormal system (Es;S f 8). Writing each 9 s as Ss\(kl largest element in S, we immediately have the following expansion N
Y - E(Y) =
@k
& where
is
- measurable.
Define an F-martingaleZ by a martingale transform Z = E(Y) + 0. X. Then we have Y = ZN. Hence, E(YIF,) = Z,. Noting that every F-martingale (M,,) is given in the form M, = E(Y)F,,)for some Y, we can now complete the proof. Proof B. Given an F-martingale M = (M,), define an F-predictable process @ = (Qk) by setting Define an F-martingale M X. Then we have
=
(M,,) by the [Mo, @]-martingaletransform of
Therefore, for an F-martingale L = M - fi,we see by (1.1) and (1.2) that E[(Lk - Lk-1)
'
CkIFk-l] = E[(Lk - Lk-1) . (Xk - Xk-l)lFk-l] = 0, k = 1,. . .,N.
By applying the next lemma, we can conclude that L = 0, implying M = k, which completes the proof.
Lemma 1.1. Ifan F-martingale L = (L,) satisfies that (1.3) Lo = 0 and E[(Lk - Lk-1) . (Xk - Xk-1)1Fk-l] = 0, k = 1,. . .,N, then, L = 0.
Proof. Every FN-measurab1erandom variable is bounded and hence, we can take a constant c > 0 such that max lLNl 5 1/(2c). Then, noting E(LN)= E(Lo)= 0,1+ CLN > and E(1+ CLN) = 1. Define a new probability Q on (62, FN) by setting
5
Then X = (X,) is an F-martingale under Q. Indeed, we have
because E[(L, - L,,-l)(Xn- XIl-1)1Fn-1]= 0 by (1.3), and E[(Lk - Lk-l)(Xn Xn-1)17n-l]= (Lk - Lk-l)E[(X,, - Xn-1)17;,-1] = 0 when k < n. Since X, - X,,-1 takes values 1or -1 only, this implies that
that is, X = (X,) is also a symmetric random walk under Q. We have which implies that LN = 0 and hence thus deduced that P = Q on (0,FN), L, = E[LN1%] = 0 for all 0 I n I N.
Remark 1.1. Generally for F-martingales L and M, we define an Fpredictable process (L, M) = ((L, M),) by
We say that two martingales are orthogonal if (L, M) = 0. Now, Lemma 1.1 can be stated as follows: An F-martingales L which is orthogonal to the basic martingale X = (X,) must be a constant.
Remark 1.2. The notion of martingale transforms has been introduced by D. L. Burkholder ([B]). In continuous time, a corresponding notion is that of stochastic integrals.
Example 1.1. We consider the case where above random signs ISk} are replaced by more complicated i.i.d. random variables. ~ For a positive integer L 2 3, let E = (al,. - . ,aL}where al, . . . , a are different real numbers. Let 6 be an E-valued random variable such that P(6 = a,) > 0 for all I, and E(5) = 0. We denote o2 = E(52). d We consider an i.i.d. sequence 51. - - . ,tNsuch that Sk = 5, k = 1, . . .,N, and define the process X = (Xn)oSnsN and F = (Fn)O<nsN in the same way as above so that F is the natural filtration of X. So, as above, X is an Fmartingale which generates the filtration F. However, Theorem 1.1 does not hold in this case, that is, there exists an F-martingale which cannot be given as a martingale transform by X. We now obtain a martingale representation theorem in this case. The set of all real functions on E forms a real vector space of dimension L which is a real Hilbert space with respect to the L2-inner product by the law p of 5. Let fi(a), 1 = 0,1,. . ., L - 1, be functions on E which form an orthonormal basis in this Hilbert space. We choose, as we may, fo(a) = 1, and fl(a1) = allo, I = 1,.. .,L. Then, random variables YI = fi(5), 1 = 0,. . .,L - 1, are orthonormal in L2(n,P) and oY1 = 5. Set, for each 1 = 1,. . .,L - 1, an F-adapted stochastic process Y(') = ( Y ~ ~ ) ) by O~,~N
Then, Y(') are all F-martingales and, obviously, = X. It is easy to see that (Y('),Y(')), = dlJn, 0 I n < N for every 1,l' = 1,.. .,L - 1. We can show by the same idea as above that every F-martingale M = (M,,) with MQ= 0 has a representation as a sum of martingale transforms by these martingales in which F-predictable integrands @(I) = (@:))lsksN, 1 = 1,.. . ,L - 1, are uniquely determined:
Thus, we see that the martingale representation by a single basic martingale X is impossible. In this example, we consider S, = eRXn+Pn, (a, ,5; constants), as a stock is an price in a financial market. We choose constants so that (Sn)O
market is complete; an equivalent martingale measure is unique, which is a key in Proof B. 2. The Case of Continuous Time In continuous time, we have several technical difficulties in defining correspondingnotions such as predictability, martingale transforms which are called in this case as stochastic integrals, and so on. We would summarize here some of basic notions, cf. e.g., [IW], [KK], [PI, [RW], [RY]. 2.1 Fundamental notions The time parameter set T is usually the interval [0, w); we often take it to be a n finite interval [0, TI for some T > 0. A filtration F = (EltET is an increasing family of sub a-fields of events. We assume the usual condition unless otherwise stated:
(i) The probability space (a, F,P) is complete and every f i contains all P-null sets. (ii) F is right-continuous, that is, 7;+(:=n,,07;+,) = Ft for every t
E
T
We introduce the following notations. M2 = M2,10c=
( M = (Mt); a square-integrable F - martingale, Mo = 0 ] ( M = (Mt); a locally square-integrable F - martingale, Mo = 0 ]
4 = { M E M;! I t H Mt is continuous, as. ) Mi,,oc= { M E M2,locI t H Mt is continuous, as. 1 A stochastic process X = (Xt(w))is a function of (t, w) E T x ~It.is called F-adapted if Xt is 7;-measurable for every t E T. We identify two processes X = (Xt(w))and Y = (Yt(w))and write X = Y, if P((wl3t E T, Xt(w) # Yt(w)}) = 0. A real process Q, = (Qt(w))is called a simple F-predictable process if it is given, by a sequence 0 = so < sl < . . . < s, of times and bounded real Kt-measurable random variables fi, as
The smallest 0-field P on T x In with respect to which all simple Fpredictable processes are measurable, is called the predictable a-field with respect thefiltration F, or simply, F-predictable o-field. A process 0 = (at(w)) is called F-predictableif it is P-measurable. Every left-continuous F-adapted process is F-predictable. A real process A = (At(o)) for which t H At is right-continuous and increasing (in the wide sense) is called an increasing process, and a real process which can be represented as a difference of two increasing processes
is called a process of bounded variation. The following fact holds: Given M, N E M2,10c, there exists a unique F-predictable process of bounded variation (M, N) = ((M,N)t) such that the process (MtNt - (M, N)t) is an F-local martingale. We denote (M,M) simply by (M). (M) is an F-predictable increasing process. Furthermore, if, at least, one of M and N is continuous, then (M, N) is a continuous process. Let the samplewise total variation of s E [0, t] ++ (M, N), be denoted a.s. Hence, if by II(M, N)llt. Then it holds that II(M, N)llt I I-,/ M, N E M2,then E [II(M, N)llt] 5 E[M:]'/~E[N:]'/~ < m for every t E T. Given M E M2,10c,we denote by L2,10c(M) (L2(M))the totality of real F-predictable processes Q, = ( a t ) such that --f IQ12d(M), < m, a.s. for every
[A~
t E T, (rap. E I @ ~ P ~ ( M<) , m ] for every t E T).' Then, for given M E M2,10cand 0 E L2,10c(M),there exists a unique N E MZlocsuch that the following holds for any L E M2,10c: (N, L), = Jo Qsd(M,L), a.s. for every t E T. We call this N = (Nt) the stochastic integral of 0 by M and denote it by N, = Q ~ ~ Mor , , simply, by N = J Q ~ M . J Q ~ ME M ; if~M~c M;,,~~, ~ andJQ,dE ~ M2ifQ E L ~ ( M ) . If Q, is given by a simple F-predictable process as given by (2.1), then 0 E L2,10c(M) for every M E M2,10c and J Q ~ Mis given by the following Riemann sum:
When a right-continuous process X = (Xt) is such that MX E M2,10c where MXis defined by Mf = Xt - Xo, then the stochastic integral J 0 d ~ is also denoted by J 0dX. We call it also the stochastic integral of Q, by X. When an F-predictable process 0 = (Q) is given by Q,t =![,2tl where o E T U (m}is an F-stopping time, the process (XO+ JtQ,,dX,)t.r coincides with the stopped process Xu = (XtA0)of X by o. To realize the idea of Proof B above in the continuous time case, we introduce the following notion. We consider the case of M2, the case of M2,10c can be discussed similarly by an obvious modification. For MI, M2 E M2, we say that M1 and M2 are orthogonal, and denote this by MllM2, 'TWO @,YE L2,~,JM) are identified and denoted by 4, = Y if for every t.
,tI@,
- Y,I~~(M), = 0,as.
~
if (Ml,M2) = 0. If M l l M 2 and Oi E -&(Mi), i = l,2, then it holds that @ l d ~Il a l d ~ 2 .We have the following orthogonal decomposition result (cf. [KW], [KK], [PI).
S
S
Prop 2.1. Let MI,. . .,M, E M2 and MilMi if i f j. Then, for every M E M2, there exists unique2 Qi E L(Mi), i = 1,. . .,n, and N E MZsuch that
ai is an F-predictable process satisfying (M,Mi)t
= J~(B;),~(M;),,i =
1, ..., n.
2.2 The case of Brownian filtrations Let W = (Wt)be the d-dimensional standard Brownian motion (Wiener process) such that Wo = 0. Let FW= (7;)be the natural filtration, that is, is the o-field generated by W,, 0 I s I t (and P-null sets). Then it is known that it satisfies the usual condition. FWis called the (d-dimensional) Brownian filtration. Denoting Wt = (W:, . . . ,w:) by its components, W' = (w:) E M ~ ( F ~ ) , i = 1,. . .,d. This system of martingales satisfies (W;, Wj)t = dirjt, so that, in particular, (Wi)t t, i, j = 1,. . .,d. (2.3)
-
Hence, L2,10c(V?)(:=L2,10c)= ( Q) = ( ~ t )~; ~ - ~ r e d i c t a and ble
S
@:ds < m, a.s. Vt }
and L ~ ( w ~ ) ( :L2) = = ( O = (at); ~ ~ - ~ r e d i c t a band le E
[I' I
e d s < m, a s . Vt 1.
For O = (at= (a:. . . .,@f)) E ( L ~ , ~ ~the ~ )stochastic ", integral J@dw E M!$oc(FW, is defined by
The martingale representation theorem for the Brownian filtration FW asserts that every martingale in M2,1,,(FW)can be represented by such a 2 ~ fFootnote . 1.
stochastic integral. This results implies, in particular, that every martingale with respect to the Brownian filtration is necessarily continuous. Namely, we have
Theorem 2 . t (i) For every M E M2,10c(~W), (M~(FW)), there exist unique Qi E &,roc (resp. L2), i = I,. . .,d, such that
In particular, it holds that (2.5)
M2,,oc(~W) = Mc,,,~(F~) and
MdFW)= Mi(FW).
(ii) Every Fw-locallysquare integrable martingale X = (Xt) can be represented in theform
by some Q E (f2,roc)hnd Xo is a constant. In particular, X is continuous. (iii) Every ~ ~ - 1 o cmartingale al X = (Xt)is continuous so that it is locally square integrable. Hence the result (ii) can be applied to obtain the representation (2.6). We give two proofs (which correspond to Proofs A and B of Section 1) in the case of M E M2(FW).The case of M E M2,,,,(FW)cafi be treated by a standard localization argument. (ii) is obviously deduced from (i), and (iii) can be shown by approximating X by bounded martingales which are necessarily continuous by (ii). Such an approximation is easily obtained by applying a standard maximal inequality for martingales. Proof A. This is based on the following It6 representation theorem of which we mentioned in Introduction.
Theorem 2.2. Let T > 0 be arbitrary butfixed. Let Y E L2(7?). ) ~ that exists a unique = (ai)E ( ~ 2 such
Then there
Proof. The proof is based on the well-known Wiener chaos expansion (the expansion by multiple Wiener-It6 integrals) as will be given by Theorem 2.4 below. (2.8)
.kt,<,.,
Y = E[YI + ZE1J . .
-
dW(fn).
Here f;, E L2 (A!) ( R ~ ) @A?) ~ ) , := ((11, . . ,ft1)10< tl < . . . < t,, < TI, and E (R"@" and we use the following notational convention; for A = (A'~,"','u)
a1 = (a;)
E
Rd, ..., a, = (a:,) E RdlAal . . .a,, is a real number defined by
. .
11,"',l,,-l
Setting ( ~ ~ , ( t l. . ,. tn-1))
= (fn(tl,
.
.
.
. . ,tn-I, f))",'~t'u-lt' ,define
Then @' = (of)has a version as an F-predictable process so that Q = (0')E (L2)d and (2.7) follows from (2.8). The proof of Theorem 2.1 (i) follows at one from Theorem 2.2. Proof B. By Prop. 2.1, it is enough to show that every N E M ~ ( Fwhich ~) is orthogonal to every Wi, i = 1,. . .,d must be 0. When NT E ~ ~ ( 7 y ) , we can choose c > 0 such that 1 + cNr is bounded from above and below by positive constants, so that a probability Q on (Q,Fy) can be defined which is equivalent to P with density 1+cNT We deduce that any stochastic integral S@dWfor Q E L2is in M ~ ( Fwith ~ )respect to Q. In particular, Wf and W: W! - dyf are all Q-martingales. This implies, by Levy's martingale characterization of Wiener process, that W = (w') is also a d-dimensional Wiener process with respect to Q, that is, P = Q on (QF?). Hence, 1 + CNT= 1, that is, NT = 0, a.s. Then Nt = E(N~17t)= 0, a.s. for 0 I tI T. In the general case of unbounded N, the proof can be reduced to the bounded case by approximation (cf. e.g. [IW], p. 82, [RY], p. 210). 2.3 The Clark-Bismut-Ocone Formula and Wiener chaos expansion. The Clark-Bismut-Ocone Formula is concerned with an expression of the integrand @ in the It6 representation (2.7). In discussing this, we apply some functional analysis (Malliavin calculus) on Wiener process so that it is convenient to set up the Wiener process canonically: We take, as our basic probability space (0,F ) , the path space Wo(Rd) := ( w; [0, T] 3 t I-+ w(t) E Rd, continuous, ~ ( 0=) 0 1, which is a Banach space with the usual maximum norm, endowed with the d-dimensional Winer measure on the o-field 7 of Pw-measurable sets. The natural filtration F~ = (qW} is defined as usual, so that 7 = ~ ( tW), := ~ ( f )w, E WO(Rd),is the canonical realization of d-dimensional Wiener process. Let H c wo(Rd)be the Cameron-Martin subspace;
77.
A pw-measurable function on W~(R*)is called a Wiener functional. The Malliavin calculus is a differential and integral calculus for Wiener functionals (cf. e.g. [IW], [MI). Typical differential operators are, the gradient operator or Gross-Malliavin-Shigekawa operator D which sends a real Wiener functional to an H-valued Wiener functional, its dual operator or Skorohod operator D* and the Ornstein-Uhlenbeck operator L = -D*D. D is defined formally, for a Wiener functional F = (F(w)),by
+ eh) - F(w))/E, h (DF, h ) =~ Olim(F(w E'
E
H.
For a real separable Hilbert space E, we denote by LP(E), 1 < p < w, the usual LP-space of E-valued Wiener functionals. In the Malliavin calculus, a family of Sobolev-type spaces D!(E), 1 I p < co,k = 0,1,. . .,is introduced; F LP(E @ Hak). When E = R, LP(E) roughly, F E q ( ~if )F E LP(E) and D ~ E and D:(E) are denoted simply by LP and respectively. Let F E D:. Then DF E LZ(H)and we denote DtF = (DF)'(t), that is,
D~E,
(DF, h ) =~ J ~ ( D ~ht(t))&t, F, for every h E H. The Clark-Bismut-Ocone formula may be stated as follows:
Theorem 2.3. Let F E D:(c L2). Then we can define a version Qt of E(DtF1CW) suitably such that Q = (Qt)E (L2)d and it is the infegrand of the It6 representation (2.7) for F. Thus, we may write formally
A proof based on the Wiener chaos expansion will be given below.
Example 2.1. Let d = 1. For the canonical representation of onedimensional Wiener process w(t) = x(t,w), we define m(t) = m(t, w ) := maxo<,
2
9.
-
sr(') dx
-m
JT
(exp
2z(T - t)
{- L } exp { 2(T t ) -
-
- 12r(R- ' I 2 ) )
2(T - t)
In particular, iff (x,y) = g(x), then
and, if f (x,y) = g(y), then
A proof can be deduced from the following facts: If F(w) = m(T),then F E D: and Dt = llO,Tl(t), where T is a unique random time in [0, T] such that W ( T ) = m(T). The condition T > t is equivalent to the condition that m(T) > m(t). If, for each 0 < t < T , a continuous path w: is defined by w:(s) = w(t + s) - w(t),0 I s < T - t, then w(T) = w(t)+ x(T - t, w:), and
m(T) = m(t)l[m(t)=m(~)] +( ~ ( + t )m(T - t, w:)) l[m(t)<m(T)] = m(t)l[m(~-t,w:)
cW
We use the independenceof and w:, and also the following well-known formula for the joint distribution of w(t)and m(t),cf. [IM], ~ ( w ( tt)dx, m(t) t dy) = 1 ~,,,,,ol
exp
{- I2';
( 2 -~x)dxdy.
The next example is concerned with a stochastic differential equation (SDE).Consider the following SDE for a diffusion process X = (XP)on Rm with the initial point x = (xu): (2.10) dX; = oy (Xi)dwi(t)+ bff(Xt)df, X; = x",
a = 1, . . .,m.
Here, coefficients 0; and b" are assumed to be smooth with bounded first order derivatives. Then a solution Xx = (X:(w))exists, unique in pathwise, so that X: definesan 7;W-measurableWiener functional with values in Rm. Let Pt be the transition operator: Pt f ( x ) = E [ f (X:)],acting on E(R"). Then, iff E E2(Rm),Ptf E C2(Rm),and (t,x) H Ptf ( x )is C' in t and C2 in x. Here, C(Rm)= ( f ; real, continuous on Rm having limits as 1x1 + m), and c ~ ( R "= ) (f E E(Rm);derivatives up to the order k are in C ( R ~ ) } . Then, u(t,x) := Ptf (x),f E E 2 ( ~ m ) satisfies , the heat equation:
-
where L is a second order differential operator given by (2.12)
& + bff(x)6
L = f affp(x)
with affP(x)= ~
f cr~(x)$(x). = ~
Example 2.2. Consider the case when F E L2 is given by F(w) = f (X:) where f E E2(Rm).Then the It6 representation (2.7) for F is given by (2.13)
F(w) = PTf ( x )+ J~y ( ~ : ) o : ( ~ f ) d w ' ( t ) .
It is well-known that F E D: and, by (2.13),we see that the Clark-BismutOcone density E ((DJ)'I%)is given by
This follows at once from the It8 formula: If u(t,x) := Ptf (x),then
3We omit the summation sign by following the usual convention.
Noting that u satisfies (2.11), (2.13)follows readily by letting f + T.
v,
Wiener chaos expansion. Consider the space L2 = L2(w0(Rd), PW) of square-integrableWiener functionals. Then we have its Wiener chaos decomposition into sum of mutually orthogonal subspacesC!!), n = 0,1,. . .
c?)is one-dimensional, consisting of all constant functionals. C(Tn) for n 2 1 is usually described by ItB's multiple Wiener integrals (cf. [I-11) and, as is remarked in this paper, a multiple Wiener integral may be defined by an iteration of It6's stochastic integrals as follows: Let, as above, A?) := {(tl,.. . ,tn)10 < tl < . . . < t,, < T ] c [0,TIn and L ~ ( A ? )be ) the usual L2-space of real functions f on A?) such that
Let
f
=
(f".-.,i..)
E
2
( A+)
,
so that
,.-,i,,
=
(f"l'""~l(tl,. . . ,t n ) )E L ~ ( A ? )for ) every (il,. . . ,in) E (1,. . . ,dln. We define lcn)(f)E L2 by the following iterated It6 stochastic integral^:^
As in (2.8), the right-hand side is denoted simply by
It holds that
Theorem 2.4. For each n, the subspace c!) is isomefricallyisomorphic to by ~fhe) correspondence In( f ) + f,so that every F E L2 has L2 (A!) + ( R ~ ) @
we omit the summation sign for i l , .. . ,in.
an orthogonal expansion by uniquely determined f, E L2 (A!) 1,2,. . ., :
+ (Rd)@"), n=
We have by (2.15) that
Example 2.3. (Veretennikov-Krylov expansion [VK]) We consider the SDE (2.10) and consider the same F(w) = f(X+) as in Example 2.2. Let the transition operator Pt, t E [O, q,be defined as in Example 2.2 by Pt f ( x )= E[f (X:)], and define the operator Qf, t E [0,TI, i = 1, . . . ,d, by
Take f E E2(Rm).Then a smoothness of Ptf is guaranteed as in Example 2.2: If of,is nondegenate so that the operator L is elliptic, we may take any f E Z ( R ~ Then ) . the Wiener chaos expansion of F(w) = f (XX,),for a fixed x E Rm,is given by the following Veretennikov-Kylovformula: E(F) = PTf ( x ) and, for n = 1,2,. . .,
.
f ; ~ (tl ~ '. .~. ,~tn)= fi,
(a::-,,(. ..(a$-,,f ) . . .)) (x).
A proof can be given by applying the following formula successively: If g E E 2 ( ~ m ) then , d
J sQ:-,g(~f)dw'(t) for every
g ( X 3 - Psg(x) = i=l
o <s
0
This formula is obtained from the Itd formula applied to u(t,XX(t)),t E [0,s], where u(t,x) = P,-,g(x), for each fixed 0 < s < T .
Proof of the Clark-Bismut-Ocone formula (Theorem 2.3). It is enough to show this when F is in the n-th order Wiener chaos, i.e. F t C(,"', for n 2 1. So we assume that F = J . . . J (f(tl,...,t,)dw(tl)...dw(t,), f E
AT'"
L2 (A::"' + (Rd)@,).Then, for each 0 < s < T and i = 1,. . .,d. (DF,)' in the (n - 1)-th order Wiener chaos given by
f (4, (
)=.
. .i7-l)
when n = 1 7ti(tl,.. . ,tn-l)dw(tl). . . d ~ ( f , - ~when ), n>1
where, if n > 1,
pie L2 (A!-')
+ (R~)""-~))is defined by
with the conventions to = 0 and t, = T. Then,
because, if (tl,. . . ,tn-l) E A:-'),
that is, if 0 < tl < . . . < t,l-l < s, then
We can now conclude that
2.4 The case of one-dimensional diffusion processes. Let = -w 5 1 < r 5 co be given and fixed. A most general regular diffusion process X = (X(t))on the interval I = [I, r] is determined by a canonical scale s(x), a speed measure dm(x) and, when the boundary 1 or r is regular, a Feller bounday condition on the boundary: s(x) is a strictly increasing continuous function on I0 = (I, r), dm(x) is an everywhere positive Radon measure on P = (I, r), and I (r), is regular if -co < s(l+) + m((1,c]), (resp. s(r-) + m[c, r) < co). Here (and below), we fix a point c such that 1 < c < r. The local generator of X is given by a generalized second-order differential operator (cf. [IM], [I]). By changing the coordinate by the scale function, we may and do assume from now that s(x) = x, unless otherwise stated. If X(t) starting at a point in locannot hit a boundary in a finite time, we may delete the boundary from the state space I. For simplicity, we assume that every bounday point is a trap5. That is, we assume X(t) = X(ma)for all
& 2,
5cf. Remark 2.1 below for the case of more general boundary conditions.
t > ma when ma < m~ = m l Am,.
oo, where we
set ma = inf( t I X(t) = a ] for a E 1 and
It is well-known that, if X(0) = a, a E lo, X(t) is obtained from one-dimensional Brownian motion by a time change: Let W(t) be a one-dimensional Wiener process with W(0) = 0, Ba(t) = a + W(t) and t I(t,x) := lim,lo(4e)-1 lg-,,x+,)(Ba(s))dsbe the local time of Ba = (Ba(t)). Define an increasing process A(t) by
where m; is the hitting time to the boundaries I and r for Ba = (Ba(t)) defined similarly as above. Let t H .rt be the right-continuous inverse of t H A(t).Then, X(t) = Ba(.rt). We fix a E lo. Let FX = be the natural filtration of X = (X(t))with X(0) = a. From the above time change expression, we see that X = (X(t))is a local martingale.
(e}
Theorem 2.5. Every M = (M(t))E M2,r,,(~X) has thefollowing representation
Proof. Let C(1)be the space of all real continuous functions on I = [I, r]; I is compact even in the case of unbounded boundaries. The resolvent operator GA,A > 0, on C(1) is defined, as usual, by GAf (b) = E e-*'f (xb(t))dt], where xb = (xb(t))denote the diffusion starting at b E I. The generator L of the diffusion is defined (independently of A) by L = A - G,' with domain D(L) = Gn(C(1)). Then, for u E D(L), Mu(t) = u(xb(t))- u(b) ~ u ( ~ ~ ( s )E) dM~(F"), s so that the semimartingale decomposition of u(xb(t)) is given by
[Am
$
On the other hand, u E D(L) is absolutely continuous on loand the Radon-Nikodym derivative u' is a function of bounded variation. Therefore, we can apply an extension of ItB's formula (ItB-Tanaka formula, cf. e.g. [RW], [RY]) to obtain
Then, by the time change t
H T',
and we have (cf. [IW], p. 102) that f U ' ( B ~ ( S ) ) ~ B = ~J' ( Su'(x~(s))~x~(s). ) Comparing this with the semimartingale decomposition (2.18),we readily see that MU(t)= J' u'(x~(s))~x~(s). which coincides with Jtuf(X(s))dX(s) when we take b = a. We have shown in [KW] that {MU; u E D(L)}generates the space M2(FX6); any M E M2(FXb)), which is orthogonal to every Mu, u E D(L), must be zero. Theorem follows at once from this. We would ask the following question: When is the local martingale X = X(t) with X(0) = a E loa true martingale? In order to answer this, we first introduce the well-known classification of boundary points. Define
In the following, we state the classification in the case of the boundary I only; the case of the boundary r can be stated similarly by replacing o(1) and ~ ( 1with ) u(r) and ,u(r), respectively.
Definition 2.1. The boundary 1 is called (i)
) co and p(1) < w, regular if ~ ( 1 <
(ii)
) m and p(1) = m, exit if ~ ( 1 <
(iii)
entrance if u(1) = w and p(1) < co,
(iv)
natural if a(1) = m and p(1) = co.
The following criteria are sometimes more practical:
< 1 and m((1,c]) < m.
a
1 is regular if and only if
a
1 is exit if and only if m((1,c]) = w and
a
1 is entrance if and only if 1 = -m
-m
fm((x,cl)dx < w. and fIxldm(x) < m.
The following result is due to S. Kotani ([K]): Theorem 2.6. X = (X(t))(with Xo = a E lo)is a martingale ifand only if the both boundaries I and r are not entrance. In other words, an existence of entrance boundary, and only this, can destroy the true martingale character of the local martingale X. Before giving a proof, we discuss some examples. Example 2.4. (Two-dimensional Bessel diffusion process.) A twodimensional Bessel diffusion process, that is, the radial motion of a twodimensional Brownian motion, is a diffusion X = (X(t))on (0, M)with the dZ generator L = 1 (z +f A canonical scale is given by s(x) = logx, so that, choosing this as the coordinate, we are considering a diffusion Y = (Y(t)) on (-w, M) with the canonical scale s(y) = y and the speed measure dm@)= Zewdy, i.e. the generator given by L = ke-2y&. By the above criterion, the boundary -m is entrance and the boundary m is natural. Then, by the theorem, the local martingale Y(t)(= log(X(t)))is not a true martingale. This is a little surprising, as the all moments of Y(t) are finite.
A).
Example 2.5. (Positive diffusions.) Motivated by a problem in mathematical finance, Delbaen and Shirakawa (cf. [DS]) studied the following diffusion process X = (X(t))on [0, w) which is given by a solution of the following SDE: dXt = 0(Xt)dWt, Xo = 1.
The coefficient u(x) is assumed to be 0 when x 5 0, and is strictly positive and continuous for x > 0. Then a solution exists, unique in the law sense, until it reaches the origin and, we assume that it is stopped as soon as the origin is reached in a finite time. This is equivalent to consideringa diffusionprocess X = (X(t)), X(0) = 1, on [0, m), that is, 1 = 0 and r = M, with the scale function s(x) E x and the speed measure dm(x) = I d x . The origin is assumed to be a trap when it can be reached in a finite time. According to the above definition and 1 1 criteria, the origin 0 is regular if -dx < M, exit if S, &dx = w but
4
1
1
1
&dx = m. It cannot be entrance. As for -dx < m, and natural if the boundary point M, it is either entrance or natural, and it is so according asJ;OD&dx<mor~&dx=m. It is known (cf. [IM], [IW, p. 4501) that X reaches the origin in a finite 1 time as. if the origin is either regular or exit, equivalently, if S, &dx < M, and X cannot reach the origin in a finite time, if the origin is natural, 1 equivalently, if S, -&dx = w.
Also, by the theorem above, we can conclude that X(t) is a true martingale if and only if m is natural, equivalently, if $+dx = m. In [DS], these results are rediscovered by a method based on the RayKnight Theorem for Brownian local times.
Am
Proof of Theorem 2.6. First, we note that X = ( X ( t ) )is a martingale if and only if xb(t)is integrable and E [ x ~=( b~for ) ]any t > 0 and b E lo. This is a simple consequence of the Markov property of the diffusion. By taking the Laplace transform, these conditions can be equivalently stated in terms of the resolvent kernel: Let, as above, GAf ( x ) = e - A tU ~ ( X X ( t ) )dt, ] f E C(l). Then, GAf ( x ) = GA(x,dy)f ( y )by the resolvent kernel GA(x,dy). X = ( X ( t ) ) is a martingale if and only J G A ( Xdy)lyl , < DJ and J G*(x, dy)y = for every A > 0 and x E lo. Let ul(x) be a positive and increasing solution of the homogeneous equation Au - Lu = 0 on lo, where L = is the local generator of X. When I is regular, we impose the condition that u(l+) = 0. Then ul exists and it is unique up to a multiplicative constant. Similarly, let u2(x) be a positive and decreasing solution of the homogeneous equation Au - Lu = 0 on lowith the condition u(r-) = 0 when r is regular. Then u2 exists and it is unique up to a multiplicative constant. The following property of functions ul and u2 is a key in the proof, cf. [IM], [I]; a nice and detailed proof can be found in the latter reference. We state it for the increasing solution ul and its derivative u;: For the decreasing solution u;! and -u;, it can be stated similarly by exchanging the role of boundaries 1 and r.
1
:,
&6
ul(l+) > 0 if and only if 1 is entrance; so, if I is regular, exit or natural, then ul(I+) = 0. ul(r-) < oo if and only if r is regular or exit; so, if r is entrance or natural, ul(r-) = m. u;(l+) > 0 if and only if 1 is regular or exit; so, if I is entrance or natural, u;(l+) = 0. a
u;(r-) < oo if and only if r is regular or entrance; so, if r is exit or natural, u;(r-) = 03.
Define ''(*'
0,
if r is regular or exit, if r is entrance or natural,
and similarly t$)),
0,
if 1 is regular or exit, if I is entrance or natural.
We define the A-order Green function gA(x,y), x, y
E
1°, by
where h(A) = u ; ( x ) u ~ ( Y-)u ~ ( x ) u ; ( xis) the Wronskian of ul and uz (which is independent of x). Then the resolvent kernel GA(x,dy), x E lo, d y c I, is given by
@1(x) GA(x,dy) = l ~ ( y )(x, g y)dm(y) ~ +n . dr(dy)+
2A0 . bl(dy).
Note that, for a fixed x E 1°, the measure GA(x,dy) has its full measure 111 on I n (-CO, w). Now we can show that JGA(x,dy)lyl < m, and the following relation:holds:
For this, we need the following lemma: Lemma 2.1. (1)
-
-
x
&,,u ~ ( y ) d m ( y-)
x
ul(y)dm(y)-
x
u2(y)dm(y)+
(
{
&,r) uz(y)dm(y)+
u,(x)
u{(l+)(x-O
+ A , i f 1 is regular or exit, + F, i f 1 is entrance or natural.
+ -,
, uz(y)
UZ(T--)
-T
,
yr
is regular or exit,
i f r is entrance or natural.
Note also that h(A) if 1 is regular or exit, u;(l+) = uz(l+)
and h(4 if r is regular or exit. ui(r-) = -u1(r-1 Proofof Lemma 2.1. Suppose that 1 is regular or exit. Then ul(l+) = 0 and we have
Since u l is a solution of Au - Lu = 0, we have du;(y) = Aul(y)dm(y).Hence,
From this, we obtain
The proof in the case when I is entrance or natural is similar: We note, in this case, that ~ ( ~ ( l2+0,) (u(l(l+) > 0 if and only if 1 is entrance), and u;(l+) = 0 always. The proof in the case of the right boundary point r is the same if the role of 1 and r are exchanged. We have
io
and, by noting g(x, y)dm(y) 5 l / A and using Lemma 2.1, we can estimate this as O(1 + 1x1). We have also,
Substituting the expressions in Lemma 2.1 to the integrals in the right-hand side, we can obtain (2.19). From (2.19), we see that Gn(x,dy)y = xlh, x E lo, if and only if u ~ ( I + ) u ~( xu2(r-)ul(x) ) = 0 on lo.We can easily deduce that this holds if
and only if ul(l+) = u2(r-) = 0. This holds if and only if neither 1 nor entrance. This completes the proof of Theorem 2.6.
r is
Remark 2.1. When I or r is regular, w e assumed it to be a trap for our diffusion. Of course, there is a variety of possible boundary behaviors and X(t) is generally not a local martingale, any more. References DS. F. Delbaen and H. Shirakawa, No arbitrage condition for positive diffusion price processes, Asia-Pacijic Financial Markets 9 (2002), 159-168. IW. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Second Edition, North-Holland/Kodansha,Amsterdaflokyo, 1988. 1-1. K. It6, Multiple Wiener Integral, J. Math. Soc. Japan, 3 (1951), 157-169. 1-2. K. It6, Kakuritu Katei 11 (Stochastic Processes, 11), Iwanami Shoten, Tokyo, 1957 (in Japanese); English Translation by Yuji Ito, Yale University, 1961. IM. K. It6 and H. P. McKean, Jr., Diffusion Processes and Their Sample Paths, Springer, Berlin, 1965, Second Printing 1974, in Classics in Mathematics, 1996. KK. G. Kallianpur and R. L. Karandikar, introduction to Option Pricing Theory, Birkhauser, Boston/Basel/Berlin,2000. K. S. Kotani, On a condition that one-dimensional diffusion processes are martingales, 2003. KW. H. Kunita and S. Watanabe, On square integrable martingales, Nagoya Math. 1. 30 (1967), 209-245. M. P. Malliavin, Stochastic Analysis, Springer, Berlin, 1997. P. P. Protter, Stochastic integration and Differential Equations, A New Approach, Springer Verlag, Berlin/Heidelberg/NewYork, 1990. RW. L. C. G. Rogers and D. Williams, DzJusion, Markov Processes, and Martingales, Vol. 2, It6 Calculus, John Wiley & Sons, Chichesterrnew York/Brisbane~oronto/ Singapore, 1987 S. D. W. Stroock, Markov Processesfrom K. Itb's Perspective, Annals of Mathematical Studies 155 (2003), Princeton University Press, Princeton/Oxford. VK. A. Ju. Veretennikov and N. V. Krylov, On explicit formulas for solutions of stochastic differential equations, Math. USSR Sbornik 29 (1976), 239-256.
Based around recent lectures given at the prestigious Ritsumei kan conference, the t u t o r i a l and expository articles contained in this volume are an essential guide for practitioners and graduates alike who use stochastic calculus in finance.
---I * * wwn
-> -+
-h
-
b
>-
-1
n
0
tll-
-
. Among the eminent papers are:
Harmonic Analysis Methods for Nonparametric Estimation of Volatility: Theory and Applications by E. Barucci, P. Malliavin, and M. E. Mancino Hedging of Credit Derivatives in
Models with Totally Unexpected Default by T. R . Bielecki, M. -~eanblanc, and M. Rutkowski Martingale Representation Theorem and Chaos Expansion by S. Watanabe
5958 hc
