A multilevel stochastic approximation algorithm for value-at-risk and expected shortfall estimation

Stéphane Crépey (), Noufel Frikha () and Azar Louzi ()
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Stéphane Crépey: Université Paris Cité
Noufel Frikha: Université Paris 1 Panthéon-Sorbonne
Azar Louzi: Université Paris Cité, CNRS

Finance and Stochastics, 2025, vol. 29, issue 4, No 3, 1015-1074

Abstract: 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 ε $\varepsilon $ , the optimal complexity of a nested stochastic approximation algorithm is shown to be of the order ε − 3 $\varepsilon ^{-3}$ . To estimate the VaR, our MLSA algorithm attains an optimal complexity of the order ε − 2 − δ $\varepsilon ^{-2-\delta }$ , where δ ∈ ( 0 , 1 ) $\delta \in (0,1)$ 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 ε − 2 | ln ε | 2 $\varepsilon ^{-2}|\ln {\varepsilon }|^{2}$ . 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.

Keywords: Value-at-risk; Expected shortfall; Stochastic approximation; Nested Monte Carlo; Multilevel Monte Carlo; Numerical finance; 65C05; 62L20; 62G32; 91G60 (search for similar items in EconPapers)
JEL-codes: C02 C61 G32 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s00780-025-00573-5

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