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Optimal Malliavin Weighting Function for the Computation of the Greeks

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  • Eric Benhamou

Abstract

This paper reexamines the Malliavin weighting functions introduced by Fournié et al. (1999) as a new method for efficient and fast computations of the Greeks. Reexpressing the weighting function generator in terms of its Skorohod integrand, we show that these weighting functions have to satisfy necessary and sufficient conditions expressed as conditional expectations. We then derive the weighting function with the smallest total variance. This is of particular interest as it bridges the method of Malliavin weights and the one of likelihood ratio, as introduced by Broadie and Glasserman (1996). The likelihood ratio is precisely the weighting function with the smallest total variance. We finally examine when to use the Malliavin method and when to prefer finite difference.

Suggested Citation

  • Eric Benhamou, 2003. "Optimal Malliavin Weighting Function for the Computation of the Greeks," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 37-53, January.
    • Handle: RePEc:bla:mathfi:v:13:y:2003:i:1:p:37-53
    DOI: 10.1111/1467-9965.t01-1-00004
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    References listed on IDEAS

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    1. Mark Broadie & Paul Glasserman, 1996. "Estimating Security Price Derivatives Using Simulation," Management Science, INFORMS, vol. 42(2), pages 269-285, February.
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    Cited by:

    1. Kloeden Peter E. & Sanz-Chacón Carlos, 2011. "Efficient price sensitivity estimation of financial derivatives by weak derivatives," Monte Carlo Methods and Applications, De Gruyter, vol. 17(1), pages 47-75, January.
    2. Hyungbin Park, 2018. "Sensitivity analysis of long-term cash flows," Finance and Stochastics, Springer, vol. 22(4), pages 773-825, October.
    3. Elisa Alòs & Christian-Olivier Ewald, 2005. "A note on the Malliavin differentiability of the Heston volatility," Economics Working Papers 880, Department of Economics and Business, Universitat Pompeu Fabra.
    4. Chen, Nan & Glasserman, Paul, 2007. "Malliavin Greeks without Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 117(11), pages 1689-1723, November.
    5. Robert de Rozario, 2003. "On Higher Derivatives of Expectations," Risk and Insurance 0308001, University Library of Munich, Germany.
    6. Leão, Dorival & Ohashi, Alberto, 2012. "Weak Approximations for Wiener Functionals," Insper Working Papers wpe_276, Insper Working Paper, Insper Instituto de Ensino e Pesquisa.
    7. Gilles Pag`es & Olivier Pironneau & Guillaume Sall, 2016. "Vibrato and automatic differentiation for high order derivatives and sensitivities of financial options," Papers 1606.06143, arXiv.org.
    8. D. Baños & T. Meyer-Brandis & F. Proske & S. Duedahl, 2017. "Computing deltas without derivatives," Finance and Stochastics, Springer, vol. 21(2), pages 509-549, April.
    9. Roberto Daluiso & Giorgio Facchinetti, 2018. "Algorithmic Differentiation For Discontinuous Payoffs," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(04), pages 1-41, June.
    10. Hyungbin Park, 2021. "Influence of risk tolerance on long-term investments: A Malliavin calculus approach," Papers 2104.00911, arXiv.org.
    11. Jiun Hong Chan and Mark Joshi, 2012. "Optimal Limit Methods for Computing Sensitivities of," Department of Economics - Working Papers Series 1142, The University of Melbourne.
    12. Muroi, Yoshifumi & Suda, Shintaro, 2013. "Discrete Malliavin calculus and computations of greeks in the binomial tree," European Journal of Operational Research, Elsevier, vol. 231(2), pages 349-361.
    13. Eric Benhamou, 2002. "Smart Monte Carlo: various tricks using Malliavin calculus," Quantitative Finance, Taylor & Francis Journals, vol. 2(5), pages 329-336.
    14. Muroi, Yoshifumi & Suda, Shintaro, 2017. "Computation of Greeks in jump-diffusion models using discrete Malliavin calculus," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 140(C), pages 69-93.
    15. Maria Elvira Mancino & Simona Sanfelici, 2020. "Nonparametric Malliavin–Monte Carlo Computation of Hedging Greeks," Risks, MDPI, vol. 8(4), pages 1-17, November.
    16. Leão, Dorival & Ohashi, Alberto, 2010. "Weak Approximations for Wiener Functionals," Insper Working Papers wpe_215, Insper Working Paper, Insper Instituto de Ensino e Pesquisa.
    17. Yuh-Dauh Lyuu & Huei-Wen Teng, 2011. "Unbiased and efficient Greeks of financial options," Finance and Stochastics, Springer, vol. 15(1), pages 141-181, January.
    18. Christian P. Fries & Joerg Kampen, 2005. "Proxy simulation schemes using likelihood ratio weighted Monte Carlo for generic robust Monte-Carlo sensitivities and high accuracy drift approximation (with applications to the LIBOR Market Model)," Finance 0504010, University Library of Munich, Germany.
    19. Fabrice Borel-Mathurin & Nicole El Karoui & Stéphane Loisel & Julien Vedani, 2020. "Locality in time of the European insurance regulation "risk-neutral" valuation framework, a pre-and post-Covid analysis and further developments," Working Papers hal-02905181, HAL.
    20. Joshi, Mark S. & Zhu, Dan, 2016. "An exact method for the sensitivity analysis of systems simulated by rejection techniques," European Journal of Operational Research, Elsevier, vol. 254(3), pages 875-888.
    21. Shaolong Tong & Guangwu Liu, 2016. "Importance Sampling for Option Greeks with Discontinuous Payoffs," INFORMS Journal on Computing, INFORMS, vol. 28(2), pages 223-235, May.

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