Section: New Results

Numerical probability

Simulation of stochastic processes

Participants : Aurélien Alfonsi, Benjamin Jourdain, Arturo Kohatsu-Higa [(University of Osaka)] , Abdelkoddousse Ahdida, Stefanp De Marco.

Pathwise convergence of the Euler scheme

A. Alfonsi, B. Jourdain and A. Kohatsu-Higa are studying the convergence for the Wasserstein distance of the Euler scheme towards the limit diffusion. This would be a way to analyze the weak pathwise convergence of this discretization scheme. They have obtained some promising results in the one-dimensional case.

Multi-dimensional models and correlation issues

A. Alfonsi and his PhD Student A. Ahdida have submitted a paper on the simulation of Wishart processes. They have introduced a new family of stochastic differential equations that are defined in the space of correlation matrices and provided high order discretization schemes for such processes [51] . Theuy are currently trying to use this type of matrix valued processes to model the dependence between assets. In particular, they would like to calibrate Index options data. This work is still in progress. The thesis of A. Ahdida has been defended on December 1st [12] .

The PhD student Lokmane Abbas Turki has worked on numerical methods for American optuion pricing based on Malliavin calculus and parallel implementation. He has submitted a paper (co-authored with B. Lapeyre) [50] . He is now working on the dependence of option prices with respect to correlations between stocks in multi-dimensional models.

A. Alfonsi and S. De Marco (postdoc CERMICS) have studied how some option prices (such as spread options) are modified when the correlation between stocks is increased.

Stochastic volatility models

Exotic options and stochastic volatility models is the subject of Sidi Mohamed OULD ALY's thesis, defended on June 16th, 2011. Sidi-Mohamed has results on the effective computation of option prices in a stochastic volatility model, in the context of variance swap modelling. He has worked out a new model, in the spirit of Bergomi's approach. This model has remarkable features in terms of tractability and calibration. S.M. Ould Aly has developed numerical methods and an original variance reduction method for models with a log normal volatility. He also has results on the monotony of option prices with respect to the correlation between the stock price and the volatility in the Heston model. Sidi Mohamed has submitted three papers.