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American Options Based on Malliavin Calculus and Nonparametric Variance Reduction Methods

Author

Listed:
  • Lokman Abbas-Turki

    (LAMA)

  • Bernard Lapeyre

    (CERMICS)

Abstract

This paper is devoted to pricing American options using Monte Carlo and the Malliavin calculus. Unlike the majority of articles related to this topic, in this work we will not use localization fonctions to reduce the variance. Our method is based on expressing the conditional expectation E[f(St)/Ss] using the Malliavin calculus without localization. Then the variance of the estimator of E[f(St)/Ss] is reduced using closed formulas, techniques based on a conditioning and a judicious choice of the number of simulated paths. Finally, we perform the stopping times version of the dynamic programming algorithm to decrease the bias. On the one hand, we will develop the Malliavin calculus tools for exponential multi-dimensional diffusions that have deterministic and no constant coefficients. On the other hand, we will detail various nonparametric technics to reduce the variance. Moreover, we will test the numerical efficiency of our method on a heterogeneous CPU/GPU multi-core machine.

Suggested Citation

  • Lokman Abbas-Turki & Bernard Lapeyre, 2011. "American Options Based on Malliavin Calculus and Nonparametric Variance Reduction Methods," Papers 1104.5131, arXiv.org, revised Apr 2011.
    • Handle: RePEc:arx:papers:1104.5131
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    References listed on IDEAS

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    1. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    2. Broadie, Mark & Detemple, Jerome, 1996. "American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods," The Review of Financial Studies, Society for Financial Studies, vol. 9(4), pages 1211-1250.
    3. Stephane Villeneuve & Antonino Zanette, 2002. "Parabolic ADI Methods for Pricing American Options on Two Stocks," Mathematics of Operations Research, INFORMS, vol. 27(1), pages 121-149, February.
    4. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
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