Local risk-minimization in exponential Lévy models: Explicit representation and jump-adapted discretization

Abstract

The focus of this article is two-fold. First, we provide an explicit representation for the Föllmer–Schweizer decomposition of European type options under mild conditions, which implies a closed-form expression of the corresponding local risk-minimizing strategies. Secondly, we discretize stochastic integrals driven by an exponential Lévy process using a jump-adapted method. The convergence rate of the resulting discretization error as the expected number of discretization times increases is measured in weighted BMO spaces, implying also -estimates, . Moreover, the effect of a change of measure satisfying a reverse Hölder inequality is addressed. As an application, the error caused by discretizing the local risk-minimizing strategies is investigated in dependence on properties of the Lévy measure, the regularity of the payoff function and the chosen random discretization times