A Malliavin Calculus Approach to Minimal Variance Hedging

Abstract

In this paper, we investigate the following problem: How can a financial institution, which has sold an option to a client, optimally hedge the payoff of this option by investing into a stock and into the option itself? Optimality is measured in terms of minimal variance and the associated optimal hedging portfolio is derived by a stochastic maximum principle. Moreover, we deduce the time dynamics of the stochastic option price process by Malliavin calculus methods, particularly by an application of the Clark-Ocone formula. We finally apply our theoretical results to several examples.