Recursive Computation of Value-at-Risk and Conditional Value-at-Risk using MC and QMC

Value-at-Risk (VaR) and Conditional-Value-at-Risk (CVaR) are two widely-used measures in risk management. This paper deals with the problem of estimating both VaR and CVaR using stochastic approximation (with decreasing steps): we propose a first Robbins-Monro (RM) procedure based on Rockafellar-Uryasev's identity for the CVaR. The estimator provided by the algorithm satisfies a Gaussian Central Limit Theorem. As a second step, in order to speed up the initial procedure, we propose a recursive and adaptive importance sampling (IS) procedure which induces a significant variance reduction of both VaR and CVaR procedures. This idea, which has been investigated by many authors, follows a new approach introduced in Lemaire and Pages [20]. Finally, to speed up the initialization phase of the IS algorithm, we replace the original confidence level of the VaR by a deterministic moving risk level. We prove that the weak convergence rate of the resulting procedure is ruled by a Central Limit Theorem with minimal variance and we illustrate its efficiency by considering typical energy portfolios.