Elements of Malliavin calculus
This chapter offers a brief introduction to Malliavin calculus and its applications to mathematical finance, in particular the computation of the Greeks by the Monte Carlo method. As we have seen in Section 12.4.2, the simplest way to compute sensitivities by the Monte Carlo method consists in approximating the derivatives by incremental ratios obtained by simulating the payoffs corresponding to close values of the underlying asset. If the payoff function is not regular (for example, in the case of a digital option with strike K and payoff function 1[K,+∞[) this technique is not efficient since the incremental ratio has typically a very large variance. In Section 12.4.2 we have seen that the problem can be solved by integrating by parts and differentiating the density function of the underlying asset, provided it is sufficiently regular: if the underlying asset follows a geometric Brownian motion, this is possible since the explicit expression of the density is known.
KeywordsBrownian Motion Duality Relation Geometric Brownian Motion Underlying Asset Asian Option
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