Hedging in Lévy models and the time step equivalent of jumps

Abstract

We consider option hedging in a model where the underlying follows an exponential Lévy process. We derive approximations to the variance-optimal and to some suboptimal strategies as well as to their mean squared hedging errors. The results are obtained by considering the Lévy model as a perturbation of the Black–Scholes model. The approximations depend on the first four moments of logarithmic stock returns in the Lévy model and option price sensitivities (Greeks) in the limiting Black–Scholes model. We illustrate numerically that our formulas work well for a variety of Lévy models suggested in the literature. From a practical point of view, it turns out that jumps have an effect on hedging errors similar to discrete-time hedging in the Black–Scholes model.