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Asymptotic Properties of Monte Carlo Estimators of Derivatives

Author

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  • Jérôme Detemple

    (School of Management, Boston University, 595 Commonwealth Avenue, Boston, Massachusetts 02215, and CIRANO, 2020 University Street, 25th Floor, Montréal, Québec, H3A 2A5 Canada)

  • René Garcia

    (Economics Department, Université de Montréal, Montréal, Québec, H3A 2A5 Canada; CIREQ, Montréal, Canada; and CIRANO, 2020 University Street, 25th Floor, Montréal, Québec, H3A 2A5 Canada)

  • Marcel Rindisbacher

    (J.L. Rotman School of Management, University of Toronto, Toronto, Ontario, Canada, and CIRANO, 2020 University Street, 25th Floor, Montréal, Québec, H3A 2A5 Canada)

Abstract

We study the convergence of Monte Carlo estimators of derivatives when the transition density of the underlying state variables is unknown. Three types of estimators are compared. These are respectively based on Malliavin derivatives, on the covariation with the driving Wiener process, and on finite difference approximations of the derivative. We analyze two different estimators based on Malliavin derivatives. The first one, the Malliavin path estimator, extends the path derivative estimator of Broadie and Glasserman (1996) to general diffusion models. The second, the Malliavin weight estimator, proposed by Fournié et al. (1999), is based on an integration by parts argument and generalizes the likelihood ratio derivative estimator. It is shown that for discontinuous payoff functions, only the estimators based on Malliavin derivatives attain the optimal convergence rate for Monte Carlo schemes. Estimators based on the covariation or on finite difference approximations are found to converge at slower rates. Their asymptotic distributions are shown to depend on additional second-order biases even for smooth payoff functions.

Suggested Citation

  • 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.
  • Handle: RePEc:inm:ormnsc:v:51:y:2005:i:11:p:1657-1675
    DOI: 10.1287/mnsc.1050.0398
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    References listed on IDEAS

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    Cited by:

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    2. Patrick Leoni, 2007. "Monte-Carlo Estimations of the Downside Risk of Derivative Portfolios," Economics Department Working Paper Series n1760607, Department of Economics, National University of Ireland - Maynooth.
    3. Denis Belomestny & Christian Bender & John Schoenmakers, 2009. "True Upper Bounds For Bermudan Products Via Non‐Nested Monte Carlo," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 53-71, January.
    4. Joerg Kampen & Anastasia Kolodko & John Schoenmakers, 2008. "Monte Carlo Greeks for financial products via approximative transition densities," Papers 0807.1213, arXiv.org.
    5. Detemple, Jerome & Rindisbacher, Marcel, 2007. "Monte Carlo methods for derivatives of options with discontinuous payoffs," Computational Statistics & Data Analysis, Elsevier, vol. 51(7), pages 3393-3417, April.
    6. Frazier, David T. & Oka, Tatsushi & Zhu, Dan, 2019. "Indirect inference with a non-smooth criterion function," Journal of Econometrics, Elsevier, vol. 212(2), pages 623-645.
    7. Romuald Elie, 2009. "Double Kernel estimation of sensitivities," Post-Print hal-00416449, HAL.
    8. Romuald Elie, 2009. "Double Kernel estimation of sensitivities," Papers 0909.2624, arXiv.org.
    9. Christian Fries & Joerg Kampen, 2010. "Global existence, regularity and a probabilistic scheme for a class of ultraparabolic Cauchy problems," Papers 1002.5031, arXiv.org, revised Oct 2012.
    10. Boyle, Phelim & Imai, Junichi & Tan, Ken Seng, 2008. "Computation of optimal portfolios using simulation-based dimension reduction," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 327-338, December.

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