## Section: New Results

Keywords : Malliavin calculus, jump diffusions, greecks.

### Computation of sensitivities (Greeks) and conditional expectations using Malliavin calculus

Participants : V. Bally, M.P. Bavouzet, L. Caramellino, J. Da Fonseca, M. Messaoud, A. Zanette.

**Malliavin calculus for jump diffusions.**V. Bally, L. Caramellino (University Rome 2) and A. Zanette worked on pricing and hedging American options in a local Black Scholes model driven by a Brownian motion by using the classical Malliavin calculus
[23] . The results of this work gave rise to algorithms which have been implemented in PREMIA. Moreover V. Bally, M.
Massoud and M.P. Bavouzet are working on the application of the Malliavin calculus for jump type processes for computing greeks in jump type models as Merton's model. Two papers have been written, one on differentiation with respect to the jump amplitude and one on differentiation with
respect to the jump times. The second subject is much more subtile and needs some theoretical developments which are not standard up to now
[48] ,
[47] .

**Malliavin calculus for the Libor Market Model.**M. Messaoud and J. Da Fonseca study an application of Malliavin Calculus for the computation of the Greeks for European interest rate derivative products
[58] . Within the Libor market model framework, which is widely used in practice, they can still apply integration by
parts even if the Malliavin covariance matrix of the diffusion does not satisfy the non degeneracy condition. They find various non degeneracy conditions on some functional that aggregates the multidimensional diffusion which allow them to integrate by parts. They provide the Malliavin
estimators for the delta, the gamma and the global vega for an European swaption. The results can be easily extended to other products. They use localization techniques for variance reduction purpose. Numerical results show that Malliavin estimators outperform substantially finite difference
for a discontinuous payoff.