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Asymptotic Error Analysis of Multilevel Stochastic Approximations for the Value-at-Risk and Expected Shortfall

Author

Listed:
  • Stéphane Crépey

    (LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité)

  • Noufel Frikha

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

  • Azar Louzi

    (LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité)

  • Gilles Pagès

    (LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité)

Abstract

Crépey, Frikha, and Louzi (2023) introduced a nested stochastic approximation algorithm and its multilevel acceleration to compute the value-at-risk and expected shortfall of a random financial loss. We hereby establish central limit theorems for the renormalized estimation errors associated with both algorithms as well as their averaged versions. Our findings are substantiated through a numerical example.

Suggested Citation

  • 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.
    • Handle: RePEc:hal:cesptp:hal-04304985
    Note: View the original document on HAL open archive server: https://hal.science/hal-04304985v2
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    References listed on IDEAS

    as
    1. Stéphane Crépey & Noufel Frikha & Azar Louzi, 2024. "A Multilevel Stochastic Approximation Algorithm for Value-at-Risk and Expected Shortfall Estimation," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-04037328, HAL.
    2. Michael B. Giles & Abdul-Lateef Haji-Ali, 2018. "Multilevel nested simulation for efficient risk estimation," Papers 1802.05016, arXiv.org, revised Feb 2019.
    3. Bernard Bercu & Sébastien Gadat & Manon Costa, 2021. "Stochastic approximation algorithms for superquantiles estimation," Post-Print hal-03352812, HAL.
    4. 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, revised Jul 2024.
    5. Daphné Giorgi & Vincent Lemaire & Gilles Pagès, 2020. "Weak Error for Nested Multilevel Monte Carlo," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1325-1348, September.
    6. Michael B. Giles & Abdul-Lateef Haji-Ali & Jonathan Spence, 2023. "Efficient Risk Estimation for the Credit Valuation Adjustment," Papers 2301.05886, arXiv.org, revised May 2024.
    7. 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.
    8. Stéphane Crépey & Noufel Frikha & Azar Louzi, 2024. "A Multilevel Stochastic Approximation Algorithm for Value-at-Risk and Expected Shortfall Estimation," Working Papers hal-04037328, HAL.
    9. 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 & Jonathan Spence, 2024. "Adaptive Multilevel Stochastic Approximation of the Value-at-Risk [Approximation stochastique adaptative à plusieurs niveaux de la valeur à risque]," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-04670735, HAL.
    2. St'ephane Cr'epey & Noufel Frikha & Azar Louzi & Jonathan Spence, 2024. "Adaptive Multilevel Stochastic Approximation of the Value-at-Risk," Papers 2408.06531, arXiv.org.

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    More about this item

    Keywords

    value-at-risk; expected shortfall; stochastic approximation; multilevel Monte Carlo; Polyak-Ruppert averaging; central limit theorem;
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