, Volume 5, Issue 2, pp 201-236

Applications of Malliavin calculus to Monte-Carlo methods in finance. II

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Abstract.

This paper is the sequel of Part I [1], where we showed how to use the so-called Malliavin calculus in order to devise efficient Monte-Carlo (numerical) methods for Finance. First, we return to the formulas developed in [1] concerning the “greeks” used in European options, and we answer to the question of optimal weight functional in the sense of minimal variance. Then, we investigate the use of Malliavin calculus to compute conditional expectations. The integration by part formula provides a powerful tool when used in the framework of Monte Carlo simulation. It allows to compute everywhere, on a single set of trajectories starting at one point, solution of general options related PDEs.

Our final application of Malliavin calculus concerns the use of Girsanov transforms involving anticipating drifts. We give an example in numerical Finance of such a transform which gives reduction of variance via importance sampling.

Finally, we include two appendices that are concerned with the PDE interpretation of the formulas presented in [1] for the delta of a European option and with the connections between the functional dependence of some random variables and their Malliavin derivatives.

Manuscript received: February 1999; final version received: January 2000