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Multidimensional Quasi-Monte Carlo Malliavin Greeks

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  • Nicola Cufaro Petroni
  • Piergiacomo Sabino
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    Abstract

    We investigate the use of Malliavin calculus in order to calculate the Greeks of multidimensional complex path-dependent options by simulation. For this purpose, we extend the formulas employed by Montero and Kohatsu-Higa to the multidimensional case. The multidimensional setting shows the convenience of the Malliavin Calculus approach over different techniques that have been previously proposed. Indeed, these techniques may be computationally expensive and do not provide flexibility for variance reduction. In contrast, the Malliavin approach exhibits a higher flexibility by providing a class of functions that return the same expected value (the Greek) with different accuracies. This versatility for variance reduction is not possible without the use of the generalized integral by part formula of Malliavin Calculus. In the multidimensional context, we find convenient formulas that permit to improve the localization technique, introduced in Fourni\'e et al and reduce both the computational cost and the variance. Moreover, we show that the parameters employed for variance reduction can be obtained \textit{on the flight} in the simulation. We illustrate the efficiency of the proposed procedures, coupled with the enhanced version of Quasi-Monte Carlo simulations as discussed in Sabino, for the numerical estimation of the Deltas of call, digital Asian-style and Exotic basket options with a fixed and a floating strike price in a multidimensional Black-Scholes market.

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    File URL: http://arxiv.org/pdf/1103.5722
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    Paper provided by arXiv.org in its series Papers with number 1103.5722.

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    Date of creation: Mar 2011
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    Handle: RePEc:arx:papers:1103.5722

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    1. Arturo Kohatsu & Montero Miquel, 2003. "Malliavin calculus in finance," Economics Working Papers 672, Department of Economics and Business, Universitat Pompeu Fabra.
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