## Team : mathfi

## Section: New Results

**Keywords: ***Malliavin calculus*, *jump diffusions*.

## Malliavin calculus for jump diffusions

**Participants:**V. Bally, M.P. Bavouzet, M. Messaoud.

One of the financial numerical applications of the Malliavin calculus is the computation of the sensitivities (the Greeks) and the conditional expectations.

In the Wiener case (when the asset follows a log-normal type diffusion for example), Fournié, Lasry, Lebouchoux, Lions and Touzi have developped a methodology based on the Malliavin calculus. The main tool is an integration by parts formula which is strongly related to the Gaussian law (since the diffusion process is a functional of the Brownian motion).

In a first stage, V. Bally has worked in collaboration with Lucia Caramellino (University Rome 2) and Antonino Zanette (University of Udine, Italy) on pricing and hedging American options in a local Black Scholes model driven by a Brownian motion, by using classical Malliavin calculus [50]. The results of this work gave rise to algorithms which have been implemented in PREMIA.

V. Bally, M.P .Bavouzet and M. Messaoud use the Malliavin calculus for Poisson processes in order to compute the Greeks (the Delta for example) of European options with underlying following a jump type diffusion. Imitating the methodology of the Wiener case, the key point is to settle, under some appropriate non-degenerency condition, an integration by parts formula for general random variables. Actually, the random variables on which the calculus is based may be the amplitudes of the jumps, the jump times and the Brownian increments.

On the one hand, M.P. Bavouzet and M. Messaoud deal with pure jump diffusion models and Merton model, where the law of the jump amplitudes has smooth density. One differentiates with respect to the amplitudes of the jumps only (pure jump diffusion) or to both jump amplitudes and Wiener increments (Merton model). Under some non-degenerency condition, one defines all the differential operators involved in the integration by parts formula.

Numerical results show that using Malliavin approach becomes extremely efficient for a discontinuous payoff. Moreover, some localization techniques may be used to reduce the variance of the Malliavin estimator. In the case of the Merton model, it is better to use the two sources of randomness, especially when there are more jumps.

On the other hand, V. Bally, MP. Bavouzet and M. Messaoud deal with pure jump diffusion models but differentiate with respect to the jump times. This case is more difficult because the law of the jump times has not smooth density, so that some border terms appear in the integration by parts formula. Thus, one introduces some weight functions in the definition of the differential operators in order to cancel these border terms. But, in this case, the non-degenerency condition is more difficult to obtain.

Another application of the Malliavin's integration by parts formula is to prove that, under appropriate hypothesis, a large variety of functionals on the Wiener space (like solutions of Stochastic Partial Differential Equations) have absolute continuous laws with smooth density. Under uniform ellipticity assumption, A. Kohatsu-Higa developped a methodology which permits to compute lower bounds of the density. Then, V. Bally relaxed this hypothesis, replacing the uniform ellipticity by only local ellipticity around a deterministic curve. Following the work of V. Bally, M.P. Bavouzet is working on an extension of his results to jump diffusion case (driven by a Brownian motion and a Poisson process).