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

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  • St'ephane Cr'epey

    (LPSM)

  • Noufel Frikha

    (CES)

  • Azar Louzi

    (LPSM)

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 $\delta$

Suggested Citation

  • St'ephane Cr'epey & Noufel Frikha & Azar Louzi, 2023. "A Multilevel Stochastic Approximation Algorithm for Value-at-Risk and Expected Shortfall Estimation," Papers 2304.01207, arXiv.org.
    • Handle: RePEc:arx:papers:2304.01207
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    References listed on IDEAS

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    1. O. Bardou & N. Frikha & G. Pagès, 2016. "CVaR HEDGING USING QUANTIZATION-BASED STOCHASTIC APPROXIMATION ALGORITHM," Mathematical Finance, Wiley Blackwell, vol. 26(1), pages 184-229, January.
    2. Albanese Claudio & Armenti Yannick & Crépey Stéphane, 2020. "XVA metrics for CCP optimization," Statistics & Risk Modeling, De Gruyter, vol. 37(1-2), pages 25-53, January.
    3. Michael B. Giles & Abdul-Lateef Haji-Ali, 2018. "Multilevel nested simulation for efficient risk estimation," Papers 1802.05016, arXiv.org, revised Feb 2019.
    4. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
    5. Mark Broadie & Yiping Du & Ciamac C. Moallemi, 2015. "Risk Estimation via Regression," Operations Research, INFORMS, vol. 63(5), pages 1077-1097, October.
    6. Michael B. Gordy & Sandeep Juneja, 2010. "Nested Simulation in Portfolio Risk Measurement," Management Science, INFORMS, vol. 56(10), pages 1833-1848, October.
    7. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
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    Cited by:

    1. Stéphane Crépey & Noufel Frikha & Azar Louzi & Gilles Pagès, 2023. "Asymptotic Error Analysis of Multilevel Stochastic Approximations for the Value-at-Risk and Expected Shortfall," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-04304985, HAL.
    2. Abdul-Lateef Haji-Ali & Jonathan Spence, 2023. "Nested Multilevel Monte Carlo with Biased and Antithetic Sampling," Papers 2308.07835, arXiv.org.

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