Quantiles of the Euler Scheme for Diffusion Processes and Financial Applications

In this paper we briefly present the results obtained in our paper ( Talay and Zheng 2002a ) on the convergence rate of the approximation of quantiles of the law of one component of  ( X_t )  , where  ( X_t )  is a diffusion process, when one uses a Monte Carlo method combined with the Euler discretization scheme. We consider the case where  ( X_t )  is uniformly hypoelliptic (in the sense of Condition (UH) below), or the inverse of the Malliavin covariance of the component under consideration satisfies the condition (M) below. We then show that Condition (M) seems widely satisfied in applied contexts. We particularly study financial applications: the computation of quantiles of models with stochastic volatility, the computation of the VaR of a portfolio, and the computation of a model risk measurement for the profit and loss of a misspecified hedging strategy.