A multilevel stochastic approximation scheme for Value-at-Risk and Expected Shortfall estimation

Abstract

We propose a multilevel stochastic approximation (MLSA) scheme for the computation of the Value-at-Risk (VaR) and the Expected Shortfall (ES) of a financial loss, which can only be computed via simulations conditional on the realization of future risk factors. Thus, the problem of estimating its VaR and ES is nested in nature and can be viewed as an instance of a stochastic approximation problem with biased innovation. In this framework, for a prescribed accuracy epsilon, the optimal complexity of a standard stochastic approximation algorithm is shown to be of order epsilon^(-3). To estimate the VaR, our MLSA algorithm attains an optimal complexity of order epsilon^(-2-delta), where delta < 1 is some parameter depending on the integrability degree of the loss, while to estimate the ES, it achieves an optimal complexity of order epsilon^(-2)log(epsilon)^2. Numerical studies of the joint evolution of the error rate and the execution time demonstrate how our MLSA algorithm regains a significant amount of the lost performance due to the nested nature of the problem.