BFS 2002

Contributed Talk

VaR Approximation

Ziyu Zheng, Denis Talay

We study the convergence rate of the numerical approximation of the quantiles of the marginal laws of $(X_t)$, where $(X_t)$ is a diffusion process, when one uses a Monte Carlo method combined with the Euler discretization scheme. Our convergence rate estimates are obtained under two sets of hypotheses: either $(X_t)$ is uniformly hypoelliptic, or the inverse of the Malliavin covariance of the marginal law under consideration satisfies a Condition (M). We then show that Condition (M) seems widely satisfied in the applied contexts. We particularly study two financial applications: 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. In addition, we precise the constants in the convergence rate estimate by proving an accurate estimate from below for the density of the Profit and Loss.