IDEAS home Printed from https://ideas.repec.org/p/wpa/wuwpfi/0212004.html
   My bibliography  Save this paper

Smart Monte Carlo: Various tricks using Malliavin calculus

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://econwpa.ub.uni-muenchen.de/econ-wp/fin/papers/0212/0212004.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Maria Elvira Mancino & Simona Sanfelici, 2020. "Nonparametric Malliavin–Monte Carlo Computation of Hedging Greeks," Risks, MDPI, vol. 8(4), pages 1-17, November.
    2. Chen, Nan & Glasserman, Paul, 2007. "Malliavin Greeks without Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 117(11), pages 1689-1723, November.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. Anne Laure Bronstein & Gilles Pagès & Jacques Portès, 2013. "Multi-asset American Options and Parallel Quantization," Methodology and Computing in Applied Probability, Springer, vol. 15(3), pages 547-561, September.
    10. Jakša Cvitanić & Jin Ma & Jianfeng Zhang, 2003. "Efficient Computation of Hedging Portfolios for Options with Discontinuous Payoffs," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 135-151, January.
    11. N. Hilber & N. Reich & C. Schwab & C. Winter, 2009. "Numerical methods for Lévy processes," Finance and Stochastics, Springer, vol. 13(4), pages 471-500, September.
    12. Arturo Kohatsu-Higa & Miquel Montero, 2001. "An application of Malliavin Calculus to Finance," Papers cond-mat/0111563, arXiv.org.
    13. Jérôme Detemple & René Garcia & Marcel Rindisbacher, 2005. "Asymptotic Properties of Monte Carlo Estimators of Derivatives," Management Science, INFORMS, vol. 51(11), pages 1657-1675, November.
    14. Coffie, Emmanuel & Duedahl, Sindre & Proske, Frank, 2023. "Sensitivity analysis with respect to a stochastic stock price model with rough volatility via a Bismut–Elworthy–Li formula for singular SDEs," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 156-195.
    15. Fard, Farzad Alavi & Siu, Tak Kuen, 2013. "Pricing participating products with Markov-modulated jump–diffusion process: An efficient numerical PIDE approach," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 712-721.
    16. Bouchard, Bruno & Touzi, Nizar, 2004. "Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 175-206, June.
    17. Ma, Jin & Zhang, Jianfeng, 2005. "Representations and regularities for solutions to BSDEs with reflections," Stochastic Processes and their Applications, Elsevier, vol. 115(4), pages 539-569, April.
    18. Hyungbin Park, 2018. "Sensitivity analysis of long-term cash flows," Finance and Stochastics, Springer, vol. 22(4), pages 773-825, October.
    19. Florian Bourgey & Stefano De Marco & Peter K. Friz & Paolo Pigato, 2023. "Local volatility under rough volatility," Mathematical Finance, Wiley Blackwell, vol. 33(4), pages 1119-1145, October.
    20. Cont, Rama & Lu, Yi, 2016. "Weak approximation of martingale representations," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 857-882.

    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wpa:wuwpfi:0212004. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: EconWPA (email available below). General contact details of provider: https://econwpa.ub.uni-muenchen.de .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.