Stochastic approximations for financial risk computations.

Summary In this thesis, we examine several stochastic approximation methods for both the computation of financial risk measures and the pricing of derivatives.Since explicit formulas are rarely available for such quantities, the need for fast, efficient and reliable analytical approximations is of paramount importance to financial institutions.In the first part, we study several multilevel Monte Carlo approximation methods and apply them to two practical problems: the estimation of quantities involving nested expectations (such as initial margin) and the discretization of integrals appearing in rough models for the forward variance for VIX option pricing.In both cases, we analyze the asymptotic optimality properties of the corresponding multilevel estimators and numerically demonstrate their superiority over a classical Monte Carlo method.In the second part, motivated by the numerous examples from credit risk modeling, we propose a general metamodeling framework for large sums of weighted Bernoulli random variables, which are conditionally independent with respect to a common factor X. Our generic approach is based on the polynomial decomposition of the chaos of the common factor and on a Gaussian approximation. L2 error estimates are given when the factor X is associated with classical orthogonal polynomials.Finally, in the last part of this thesis, we focus on the short-time asymptotics of U.S. implied volatility and U.S. option prices in local volatility models. We also propose a law approximation of the VIX index in rough models for forward variance, expressed in terms of lognormal proxies, and derive expansion results for VIX options with explicit coefficients.