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Smart Monte Carlo: Various tricks using Malliavin calculus

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  • Eric Benhamou

    (Goldman Sachs International)

Abstract

Current Monte Carlo pricing engines may face computational challenge for the Greeks, because of not only their time consumption but also their poor convergence when using a finite difference estimate with a brute force perturbation. The same story may apply to conditional expectation. In this short paper, following Fournié et al. (1999), we explain how to tackle this issue using Malliavin calculus to smoothen the payoff to estimate. We discuss the relationship with the likelihood ration method of Broadie and Glasserman (1996). We show on numerical results the efficiency of this method and discuss when it is appropriate or not to use it. We see how to apply this method to the Heston model.

Suggested Citation

  • Eric Benhamou, 2002. "Smart Monte Carlo: Various tricks using Malliavin calculus," Finance 0212004, University Library of Munich, Germany.
    • Handle: RePEc:wpa:wuwpfi:0212004
    Note: Type of Document - PDF; prepared on windows; pages: 126
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    References listed on IDEAS

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    1. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux, 2001. "Applications of Malliavin calculus to Monte-Carlo methods in finance. II," Finance and Stochastics, Springer, vol. 5(2), pages 201-236.
    2. Eric Benhamou, 2003. "Optimal Malliavin Weighting Function for the Computation of the Greeks," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 37-53, January.
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    Cited by:

    1. Bilgi Yilmaz, 2018. "Computation of option greeks under hybrid stochastic volatility models via Malliavin calculus," Papers 1806.06061, arXiv.org.
    2. T. R. Cass & P. K. Friz, 2006. "The Bismut-Elworthy-Li formula for jump-diffusions and applications to Monte Carlo pricing in finance," Papers math/0604311, arXiv.org, revised May 2007.
    3. Aprahamian, Hrayer & Maddah, Bacel, 2015. "Pricing Asian options via compound gamma and orthogonal polynomials," Applied Mathematics and Computation, Elsevier, vol. 264(C), pages 21-43.
    4. Delphine David & Nicolas Privault, 2009. "Numerical computation of Theta in a jump-diffusion model by integration by parts," Quantitative Finance, Taylor & Francis Journals, vol. 9(6), pages 727-735.
    5. Yeliz Yolcu-Okur & Tilman Sayer & Bilgi Yilmaz & B. Alper Inkaya, 2018. "Computation of the Delta of European options under stochastic volatility models," Computational Management Science, Springer, vol. 15(2), pages 213-237, June.
    6. Tebaldi, Claudio, 2005. "Hedging using simulation: a least squares approach," Journal of Economic Dynamics and Control, Elsevier, vol. 29(8), pages 1287-1312, August.

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    More about this item

    Keywords

    Monte-Carlo; Quasi-Monte Carlo; Greeks; Malliavin Calculus; Wiener Chaos.;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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