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

Stéphane Crépey (), Noufel Frikha () and Azar Louzi ()
Additional contact information
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

Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) from HAL

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 $\varepsilon$, the optimal complexity of a nested stochastic approximation algorithm is shown to be of order $\varepsilon^{-3}$. To estimate the VaR, our MLSA algorithm attains an optimal complexity of order $\varepsilon^{-2-\delta}$ , where $\delta<1$ is some parameter depending on the integrability degree of the loss, while to estimate the ES, it achieves an optimal complexity of order $\varepsilon^{-2}|\ln{\varepsilon}|^2$. 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 algorithm; Nested Monte Carlo; Multilevel Monte Carlo; numerical finance (search for similar items in EconPapers)
Date: 2024-11-29
New Economics Papers: this item is included in nep-des, nep-ecm and nep-rmg
Note: View the original document on HAL open archive server: https://hal.science/hal-04037328v3
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
https://hal.science/hal-04037328v3/document (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:hal:cesptp:hal-04037328

Access Statistics for this paper

More papers in Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) from HAL
Bibliographic data for series maintained by CCSD ().