Recursive Computation of Value-at-Risk and Conditional Value-at-Risk using MC and QMC

Olivier Bardou (), Noufel Frikha () and Gilles Pagès ()
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Olivier Bardou: Laboratoire de Probabilités et Modèles aléatoires

A chapter in Monte Carlo and Quasi-Monte Carlo Methods 2008, 2009, pp 193-208 from Springer

Abstract: Abstract Value-at-Risk (VaR) and Conditional-Value-at-Risk (CVaR) are two widely-used measures in risk management. This paper deals with the problem of estimating both VaR and CVaR using stochastic approximation (with decreasing steps): we propose a first Robbins-Monro (RM) procedure based on Rockafellar-Uryasev’s identity for the CVaR. The estimator provided by the algorithm satisfies a Gaussian Central Limit Theorem. As a second step, in order to speed up the initial procedure, we propose a recursive and adaptive importance sampling (IS) procedure which induces a significant variance reduction of both VaR and CVaR procedures. This idea, which has been investigated by many authors, follows a new approach introduced in Lemaire and Pagès 20. Finally, to speed up the initialization phase of the IS algorithm, we replace the original confidence level of the VaR by a deterministic moving risk level. We prove that the weak convergence rate of the resulting procedure is ruled by a Central Limit Theorem with minimal variance and we illustrate its efficiency by considering typical energy portfolios.

Keywords: Importance Sampling; Stochastic Approximation; Variance Reduction; Asymptotic Variance; Stochastic Approximation Algorithm (search for similar items in EconPapers)
Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-04107-5_11

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DOI: 10.1007/978-3-642-04107-5_11

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