Malliavin Calculus for Stochastic Strings with Applications To Barrier Options and Optimal Portfolios

Abstract

We apply the Malliavin calculus to the stochastic string framework and obtain a Clark-Ocone-like formula. This result allows us to rewrite the hedging portfolio explicitly in terms of the Malliavin derivative of the discounted payoff. We illustrate this new result with two applications. Firstly, we obtain a closed-form expression for the hedging portfolio of a barrier option on a bond. Secondly, we solve explicitly the optimal portfolio problem in our framework. Related to this issue, we also find a mutual fund theorem and provide an example with HARA utility functions.