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A framework for adaptive Monte Carlo procedures

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  • Lapeyre Bernard

    (Université Paris-Est, CERMICS, Projet MathFi ENPC-INRIA-UMLV, 6 et 8 avenue Blaise Pascal, 77455 Marne La Vallée, Cedex 2, France.)

  • Lelong Jérôme

    (Laboratoire Jean Kuntzmann, Université de Grenoble et CNRS, 51, rue des mathématiques BP 53, 38041 Grenoble Cédex 9, France.)

Abstract

Adaptive Monte Carlo methods are recent variance reduction techniques. In this work, we propose a mathematical setting which greatly relaxes the assumptions needed by for the adaptive importance sampling techniques presented in [Arouna, Monte Carlo Methods Appl. 10: 1–24, 2004, Arouna, The Journal of Computational Finance 7: Winter 2003/2004, Su and Fu, Journal of Computational Finance 5: 27–50, 2002, Vázquez-Abad and Dufresne, Accelerated simulation for pricing asian options: 1493–1500, IEEE Computer Society Press, 1998]. We establish the convergence and asymptotic normality of the adaptive Monte Carlo estimator under local assumptions which are easily verifiable in practice. We present one way of approximating the optimal importance sampling parameter using a randomly truncated stochastic algorithm. Finally, we apply this technique to some examples of valuation of financial derivatives.

Suggested Citation

  • Lapeyre Bernard & Lelong Jérôme, 2011. "A framework for adaptive Monte Carlo procedures," Monte Carlo Methods and Applications, De Gruyter, vol. 17(1), pages 77-98, January.
    • Handle: RePEc:bpj:mcmeap:v:17:y:2011:i:1:p:77-98:n:2
    DOI: 10.1515/mcma.2011.002
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    References listed on IDEAS

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    1. Lelong, Jérôme, 2008. "Almost sure convergence of randomly truncated stochastic algorithms under verifiable conditions," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2632-2636, November.
    2. Sujin Kim & Shane G. Henderson, 2007. "Adaptive Control Variates for Finite-Horizon Simulation," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 508-527, August.
    3. Paul Glasserman & Philip Heidelberger & Perwez Shahabuddin, 1999. "Asymptotically Optimal Importance Sampling and Stratification for Pricing Path‐Dependent Options," Mathematical Finance, Wiley Blackwell, vol. 9(2), pages 117-152, April.
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