Malliavin Calculus and the Optimal Weighting Function in a Pure Jump Lévy Setting

Abstract

The paper is devoted to the problem of obtaining weighting functions for the Greeks of an option price written on a stock whose dynamics are of pure jump type. The problem is motivated by the work of Fourni\'e et al. [8, 9], who considered the price sensitivities of a frictionless market and proved that Greeks can be computed as the expectation of the product of the discounted payoff $\Phi$ and a suitable weighted function, i.e.Greek = E[Φ(XT)weight]. Since the weighting functions are random variables that need to be explicitly computed on each specific case, we establish necessary and sufficient conditions to be satisfied. The method used relied on the Malliavin calculus for Levy processes.