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Section: Scientific Foundations

Keywords : Malliavin calculus, stochastic variations calculus, sensibility calculus, greek computations.

Application of Malliavin calculus in finance

Participants : V. Bally, M.P. Bavouzet, J. Da Fonseca, B. Jourdain, A. Kohatsu-Higa, D. Lamberton, B. Lapeyre, M. Messaoud, A. Sulem, A. Zanette.

The original Stochastic Calculus of Variations, now called the Malliavin calculus, was developed by Paul Malliavin in 1976 [77] . It was originally designed to study the smoothness of the densities of solutions of stochastic differential equations. One of its striking features is that it provides a probabilistic proof of the celebrated Hörmander theorem, which gives a condition for a partial differential operator to be hypoelliptic. This illustrates the power of this calculus. In the following years a lot of probabilists worked on this topic and the theory was developed further either as analysis on the Wiener space or in a white noise setting. Many applications in the field of stochastic calculus followed. Several monographs and lecture notes (for example D. Nualart [79] , D. Bell [50] D. Ocone [81] , B. Øksendal [88] ) give expositions of the subject. See also V. Bally [49] for an introduction to Malliavin calculus.

From the beginning of the nineties, applications of the Malliavin calculus in finance have appeared : In 1991 Karatzas and Ocone showed how the Malliavin calculus, as further developed by Ocone and others, could be used in the computation of hedging portfolios in complete markets [80] .

Since then, the Malliavin calculus has raised increasing interest and subsequently many other applications to finance have been found [78] , such as minimal variance hedging and Monte Carlo methods for option pricing. More recently, the Malliavin calculus has also become a useful tool for studying insider trading models and some extended market models driven by Lévy processes or fractional Brownian motion.

Let us try to give an idea why Malliavin calculus may be a useful instrument for probabilistic numerical methods. We recall that the theory is based on an integration by parts formula of the form Im1 ${{E(}f^\#8242 {(X))}=E(f(X)Q)}$ . Here X is a random variable which is supposed to be ``smooth'' in a certain sense and non-degenerated. A basic example is to take X = $ \sigma$$ \upper_delta$ where $ \upper_delta$ is a standard normally distributed random variable and $ \sigma$ is a strictly positive number. Note that an integration by parts formula may be obtained just by using the usual integration by parts in the presence of the Gaussian density. But we may go further and take X to be an aggregate of Gaussian random variables (think for example of the Euler scheme for a diffusion process) or the limit of such simple functionals.

An important feature is that one has a relatively explicit expression for the weight Q which appears in the integration by parts formula, and this expression is given in terms of some Malliavin-derivative operators.

Let us now look at one of the main consequenses of the integration by parts formula. If one considers the Dirac function $ \delta$x(y) , then Im2 ${\#948 _x{(y)}=H^\#8242 {(y-x)}}$ where H is the Heaviside function and the above integration by parts formula reads E($ \delta$x(X)) = E(H(X-x)Q), where E($ \delta$x(X)) can be interpreted as the density of the random variable X. We thus obtain an integral representation of the density of the law of X. This is the starting point of the approach to the density of the law of a diffusion process: the above integral representation allows us to prove that under appropriate hypothesis the density of X is smooth and also to derive upper and lower bounds for it. Concerning simulation by Monte Carlo methods, suppose that you want to compute Im3 ${{E(}\#948 _x{(y))\#8764 }\mfrac 1M\#8721 _{i=1}^M\#948 _x{(}X^i{)}}$ where X1, ..., XM is a sample of X. As X has a law which is absolutely continuous with respect to the Lebesgue measure, this will fail because no Xi hits exactly x. But if you are able to simulate the weight Q as well (and this is the case in many applications because of the explicit form mentioned above) then you may try to compute Im4 ${{E(}\#948 _x{(X))}=E(H(X-x)Q)\#8764 \mfrac 1M\#8721 _{i=1}^M{E(H(}X^i{-x)}Q^i{).}}$ This basic remark formula leads to efficient methods to compute by a Monte Carlo method some irregular quantities as derivatives of option prices with respect to some parameters (the Greeks ) or conditional expectations, which appear in the pricing of American options by the dynamic programming). See the papers by Fournié et al [64] and [63] and the papers by Bally et al, Benhamou, Bermin et al., Bernis et al., Cvitanic et al., Talay and Zheng and Temam in [73] .

More recently the Malliavin calculus has been used in models of insider trading. The "enlargement of filtration" technique plays an important role in the modeling of such problems and the Malliavin calculus can be used to obtain general results about when and how such filtration enlargement is possible. See the paper by P.Imkeller in [73] ). Moreover, in the case when the additional information of the insider is generated by adding the information about the value of one extra random variable, the Malliavin calculus can be used to find explicitly the optimal portfolio of an insider for a utility optimization problem with logarithmic utility. See the paper by J.A. León, R. Navarro and D. Nualart in [73] ).


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