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Adaptive Multilevel Stochastic Approximation of the Value-at-Risk
[Approximation stochastique adaptative à plusieurs niveaux de la valeur à risque]

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)

  • Jonathan Spence

    (Maxwell Institute for Mathematical Sciences, School of Mathematics - University of Edinburgh - Edin. - University of Edinburgh)

Abstract

Crépey, Frikha, and Louzi (2023) introduced a multilevel stochastic approximation scheme to compute the value-at-risk of a financial loss that is only simulatable by Monte Carlo. The optimal complexity of the scheme is in $O(\varepsilon^{-5/2})$, $\varepsilon>0$ being a prescribed accuracy, which is suboptimal when compared to the canonical multilevel Monte Carlo performance. This suboptimality stems from the discontinuity of the Heaviside function involved in the biased stochastic gradient that is recursively evaluated to derive the value-at-risk. To mitigate this issue, this paper proposes and analyzes a multilevel stochastic approximation algorithm that adaptively selects the number of inner samples at each level, and proves that its optimal complexity is in $O(\varepsilon^{-2}|\ln{\varepsilon}|^{5/2})$. Our theoretical analysis is exemplified through numerical experiments.

Suggested Citation

  • 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.
    • Handle: RePEc:hal:cesptp:hal-04670735
    Note: View the original document on HAL open archive server: https://hal.science/hal-04670735v1
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    References listed on IDEAS

    as
    1. 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.
    2. Aharon Ben‐Tal & Marc Teboulle, 2007. "An Old‐New Concept Of Convex Risk Measures: The Optimized Certainty Equivalent," Mathematical Finance, Wiley Blackwell, vol. 17(3), pages 449-476, July.
    3. 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.
    4. 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.
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    Cited by:

    1. Martin Arnaiz Iglesias & Adil Rengim Cetingoz & Noufel Frikha, 2024. "Mirror Descent Algorithms for Risk Budgeting Portfolios," Papers 2411.12323, arXiv.org.

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