A Multilevel Stochastic Approximation Algorithm for Value-at-Risk and Expected Shortfall Estimation

TL;DR

This paper introduces a multilevel stochastic approximation algorithm designed to efficiently estimate Value-at-Risk and Expected Shortfall in financial risk management, addressing the challenges of nested simulation problems.

Contribution

The authors develop an MLSA algorithm that achieves near-optimal complexity for estimating VaR and ES, improving computational efficiency in nested stochastic approximation tasks.

Findings

The MLSA algorithm attains an optimal complexity of order ε^{-2-δ} for VaR estimation.

For ES estimation, the algorithm achieves an optimal complexity of order ε^{-2} |ln ε|^2.

Numerical results show the MLSA algorithm significantly reduces performance loss due to nesting.

Abstract

We propose a multilevel stochastic approximation (MLSA) scheme for the computation of the value-at-risk (VaR) and expected shortfall (ES) of a financial loss, which can only be computed via simulations conditionally on the realisation 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 stochastic approximation problems with biased innovations. In this framework, for a prescribed accuracy $\varepsilon$, the optimal complexity of a nested stochastic approximation algorithm is shown to be of the order $\varepsilon^{-3}$. To estimate the VaR, our MLSA algorithm attains an optimal complexity of the order $\varepsilon^{-2-\delta}$, where $\delta\in(0,1)$ is some parameter depending on the integrability degree of the loss, while to estimate the ES, the algorithm achieves an optimal complexity of the order $\varepsilon^{-2}|\ln{\varepsilon}|^2$. 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 performance lost due to the nested nature of the problem.