Explicit Representation of the Minimal Variance Portfolio in Markets driven by Lévy processes.

Abstract

In a market driven by a Lévy martingale, we consider a claim x. We study the problem of minimal variance hedging and we give an explicit formula for the minimal variance portfolio in terms of Malliavin derivatives. We discuss two types of stochastic (Malliavin) derivatives for x: one based on the chaos expansion in terms of iterated integrals with respect to the power jump processes and one based on the chaos expansion in terms of iterated integrals with respect to the Wiener process and the Poisson random measure components. We study the relation between these two expansions, the corresponding two derivatives and the corresponding versions of the Clark-Haussmann-Ocone theorem.