Hedging in L\'evy Models and the Time Step Equivalent of Jumps

Abstract

We consider option hedging in a model where the underlying follows an exponential L\'evy 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 L\'evy model as a perturbation of the Black-Scholes model. The approximations depend on the first four moments of logarithmic stock returns in the L\'evy model and option price sensitivities (greeks) in the limiting Black-Scholes model. We illustrate numerically that our formulas work well for a variety of L\'evy models suggested in the literature. From a theoretical point of view, it turns out that jumps have a similar effect on hedging errors as discrete-time hedging in the Black-Scholes model.