Stochastic Calculus of Variations in Mathematical Finance
Description
Malliavin calculus provides an infinite-dimensional differential calculus in the context of continuous paths stochastic processes. The calculus includes formulae of integration by parts and Sobolev spaces of differentiable functions defined on a probability space. This new book, demonstrating the relevance of Malliavin calculus for Mathematical Finance, starts with an exposition from scratch of this theory. Greeks (price sensitivities) are reinterpreted in terms of Malliavin calculus. Integration by parts formulae provide stable Monte Carlo schemes for numerical valuation of digital options. Finite-dimensional projections of infinite-dimensional Sobolev spaces lead to Monte Carlo computations of conditional expectations useful for computing American options. The discretization error of the Euler scheme for a stochastic differential equation is expressed as a generalized Watanabe distribution on the Wiener space. Insider information is expressed as an infinite-dimensional drift. The last chapter gives an introduction to the same objects in the context of jump processes where incomplete markets appear.
Reviews / Votes
From the reviews:
"This short book introduces Malliavin calculus and illustrates important applications in finance. . For readers with the necessary mathematical skills, this is a valuable introduction to the mathematics and financial applications of Malliavin calculus. . it provides a direct gateway to the relevant literature." (www.riskbook.com, November, 2006)
"The book under review is on the applications of the Malliavin calculus to financial mathematics. . The authors have written a short book introducing the reader efficiently to the key points of the Malliavin calculus in mathematical finance. . Also the list of references is comprehensive and updated, and gives a clear picture of the activity and relevance of this approach to many financial problems. . This book is recommended to all researchers in mathematical finance." (MathSciNet, February, 2007)
"The book under review is on the applications of the Malliavin calculus to financial mathematics. . The compact form is to the advantage of the reader, who is led to the applications rather quickly. . This book is recommended to all researchers in mathematical finance. It shows how advanced mathematics can play an important role in solving practical financial problems as well as developing new understanding and concepts." (Fred Espen Benth, Mathematical Reviews, Issue 2007 b)
"The book under review demonstrates the power and versatility of the Malliavin calculus in a variety of problems arising in Mathematical Finance. Despite being mathematically demanding, it is directed not only towards researchers in mathematics, but also to practitioners . . The book will certainly address in the first place researchers in mathematical finance. It can however be recommended to a much wider public in mathematics beyond probability . ." (Peter Imkeller, Zentralblatt MATH, Vol. 1124 (1), 2008)
"This monograph is devoted to an updated presentation, in arigorous mathematical framework, of the applications of the stochastic calculus of variations in mathematical finance. ... In conclusion, this book aims to explain the role played by the stochastic calculus of variations in mathematical finance, and it will be useful for researchers working in these fields." (David Nualart, Bulletin of the American Mathematical Society, Vol. 44 (3), July, 2007)
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Content
In this chapter we drop the ellipticity assumption which served as a basic hypothesis in Chap. 3 and in Chap. 2, except in Sect. 2.2.
We give up ellipticity in order to be able to deal with models with random interest rates driven by Brownian motion (see [61] and [104]). The empirical market of interest rates satis.es the following two facts which rule out the ellipticity paradigm:
1) high dimensionality of the state space constituted by the values of bonds at a large numbers of distinct maturities;
2) low dimensionality variance which, by empirical variance analysis, within experimental error of 98/100, leads to not more than 4 independent scalar-valued Brownian motions, describing the noise driving this highdimensional system (see [41]).
Elliptic models are therefore ruled out and hypoelliptic models are then the most regular models still available. We shall show that these models display structural instability in smearing instantaneous derivatives which implies an unstable hedging of digital options.
Practitioners hedging a contingent claim on a single asset try to use all trading opportunities inside the market. In interest rate models practitioners will be reluctant to hedge a contingent claim written under bounds having a maturity less than .ve years by trading contingent claims written under bounds of maturity 20 years and more. This quite di.erent behaviour has been pointed out by R. Cont [52] and R. Carmona [48].
R. Carmona and M. Tehranchi [49] have shown that this empirical fact can be explained through models driven by an in.nite number of Brownian motions. We shall propose in Sect. 5.6 another explanation based on the progressive smoothing e.ect of the heat semigroup associated to a hypoelliptic operator, an e.ect which we call compartmentation.
This in.nite dimensionality phenomena is at the root of modelling the interest curve process: indeed it has been shown in [72] that the interest rate model process has very few .nite-dimensional realizations.
Section 5.7 develops for the interest rate curve a method similar to the methodology of the price-volatility feedback rate (see Chap. 3). We start by stating the possibility of measuring in real time, in a highly traded market, the full historical volatility matrix: indeed cross-volatility between the prices of bonds at two di.erent maturities has an economic meaning (see [93, 94]). As the market is highly non-elliptic, the multivariate price-volatility feedback rate constructed in [19] cannot be used. We substitute a pathwise econometric computation of the bracket of the driving vector of the di.usion. The question of e.ciency of these mathematical objects to decipher the state of the market requires numerical simulation on intra-day ephemerides leading to stable results at a properly chosen time scale.
