Explicit F\"ollmer--Schweizer decomposition and discretization with jump correction in exponential L\'evy models

Abstract

We investigate two hedging problems in exponential L\'evy models. First, we provide an explicit representation for the F\"ollmer--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 L\'evy process using a jump correction 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 $L_p$-estimates, $p \in (2, \infty)$. Moreover, the effect of a change of measure satisfying a reverse H\"older inequality is addressed. As an application, the error caused by discretizing the local risk-minimizing strategies is investigated in dependence of properties of the L\'evy measure, the regularity of the payoff function and the chosen random discretization times.