Optimal Hedging in Discrete and Continuous Time

Abstract

In this article we find the optimal solution of the hedging problem in discrete time by minimizing the mean square hedging error, when the underlying assets are multidimensional, extending the results of Schweizer (1995). We also find explicit expressions for the optimal hedging problem in continuous time when the underlying assets are modeled by a regime-switching geometric L'evy process. It is also shown that the continuous time solution can be approximated by discrete time Hidden Markov models processes. In addition, in the case of the regime-switching geometric Brownian motion, the optimal prices are the same as the prices under an equivalent martingale measure, making that measure a natural choice. However, the optimal hedging strategy is not the usual delta hedging but it can be easily computed by Monte Carlo methods. These results presented here are different from those of Follmer & Schweizer (1991) on hedging and Prigent (2003) on weak convergence, who both considered the problem of local quadratic risk minimizing strategies instead of the global quadratic risk minimizing strategies. Even in the discrete case, these two settings were shown to lead, in general, to different solutions by Schweizer (1995).