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

St\'ephane Cr\'epey, Noufel Frikha and Azar Louzi
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St\'ephane Cr\'epey: LPSM
Noufel Frikha: CES
Azar Louzi: LPSM

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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 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 stochastic approximation problems with biased innovations. In this framework, for a prescribed accuracy $\epsilon$, the optimal complexity of a nested 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 $\delta$ \

Date: 2023-03, Revised 2024-07
New Economics Papers: this item is included in nep-rmg
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