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Malliavin calculus and applications in finance

Participants : Vlad Bally, Arturo Kohatsu-Higa, Agnès Sulem, Antonino Zanette.

The original Stochastic Calculus of Variations, now called the Malliavin calculus, was developed by Paul Malliavin in 1976 [81] . 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 [83] , D. Bell [66] D. Ocone [85] , B. Øksendal [91] ) give expositions of the subject. See also V. Bally [64] 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 [84] .

Since then, the Malliavin calculus has raised increasing interest and subsequently many other applications to finance have been found [82] , 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 $E\left(f^{\text{'}}\left(X\right)\right)=E\left(f\left(X\right)Q\right)$. 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 $X=\sigma \Delta$ where $\Delta$ 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 consequences of the integration by parts formula. If one considers the Dirac function ${\delta }_x\left(y\right)$, then ${\delta }_x\left(y\right)=H^{\text{'}}(y-x)$ where $H$ is the Heaviside function and the above integration by parts formula reads $E\left({\delta }_x\left(X\right)\right)=E\left(H(X-x)Q\right),$ where $E\left({\delta }_x\left(X\right)\right)$ 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 $E\left({\delta }_x\left(y\right)\right)\sim \frac{1}{M}{\sum }_{i=1}^M{\delta }_x\left(X^i\right)$ where $X^1,...,X^M$ 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 $X^i$ 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 $E\left({\delta }_x\left(X\right)\right)=E\left(H(X-x)Q\right)\sim \frac{1}{M}{\sum }_{i=1}^{M}E\left(H(X^i-x)Q^i\right).$ 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 [73] and [72] and the papers by Bally et al., Benhamou, Bermin et al., Bernis et al., Cvitanic et al., Talay and Zheng and Temam in [79] .

L. Caramellino, A. Zanette and V. Bally have been concerned with the computation of conditional expectations using Integration by Parts formulas and applications to the numerical computation of the price and the Greeks (sensitivities) of American or Bermudean options. The aim of this research was to extend a paper of Reigner and Lions who treated the problem in dimension one to higher dimension - which represent the real challenge in this field. Significant results have been obtained up to dimension 5 [65] and the corresponding algorithms have been implemented in the Premia software.

Moreover, there is an increasing interest in considering jump components in the financial models, especially motivated by calibration reasons. Algorithms based on the integration by parts formulas have been developed in order to compute Greeks for options with discontinuous payoff (e.g. digital options). Several papers and two theses (M. Messaoud and M. Bavouzet defended in 2006) have been published on this topic and the corresponding algorithms have been implemented in Premia. Malliavin Calculus for jump type diffusions - and more general for random variables with localy smooth law - represents a large field of research, also for applications to credit risk problems.

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 [79] ). 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 [79] ).

A. Kohatsu Higa and A. Sulem have studied a controlled stochastic system whose state is described by a stochastic differential equation with anticipating coefficients. These SDEs can be interpreted in the sense of forward integrals, which are the natural generalization of the semimartingale integrals, as introduced by Russo and Valois [87] . This methodology has been applied for utility maximization with insiders.