American Options Based on Malliavin Calculus and Nonparametric Variance Reduction Methods

Lokman Abbas-Turki, Bernard Lapeyre
Universite Paris-Est, Laboratoire d’Analyse et de Mathematiques Appliquees, Champs-surMarne, 77454 Marne-la-Vallee Cedex2, France
arXiv:1104.5131v2 [q-fin.PR] (28 Apr 2011)


   author={Abbas-Turki}, L. and {Lapeyre}, B.},

   title={“{American Options Based on Malliavin Calculus and Nonparametric Variance Reduction Methods}”},

   journal={ArXiv e-prints},




   keywords={Quantitative Finance – Pricing of Securities, Mathematics – Probability},




   adsnote={Provided by the SAO/NASA Astrophysics Data System}


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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.
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