Hedging in Lévy models and the time step equivalent of jumps

Abstract

We show that the option hedging risk of an optimal, continuously rebalanced hedging strategy in an exponential Lévy model is well approximated by the risk taken from the discrete-time Black-Scholes model whose time step equals half of the excess kurtosis rate of the real-world stock returns. The result is obtained by asymptotic analysis considering the Lévy model as a perturbation of the Black-Scholes model. The resulting approximation depends 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 formulae work well for a variety of Lévy models suggested in the literature. The analysis is additionally performed for some popular suboptimal hedging strategies, with the same conclusion.

Keywords: excess kurtosis rate, quadratic hedging, Lévy processes, Black-Scholes sensitivities (Greeks)

JEL Classification: C61, C63, G11, G13