Optimality of payoffs in Lévy models

In this paper we determine the lowest cost strategy for a given payoff in Lévy markets where the pricing is based on the Esscher martingale measure. In particular, we consider Lévy models where the price process is driven by an NIGand a VG-process. Explicit solutions for cost-efficient strategies are derived for a variety of vanilla options, spreads, and forwards. Determination of efficient put prices based on estimated parameters from German stock prices reveals that the potential savings that optimal payoffs provide can be quite substantial. The empirical findings are supplemented by a result that relates the magnitude of these savings to the strength of the market trend. Moreover, we consider the problem of hedging efficient claims, derive explicit formulas for the deltas of efficient calls and puts and apply the results to German stock market data. As a main result we find that cost-efficient options also show an improved behaviour concerning delta hedging compared to their classical counterparts. AMS subject : 60G51; 60E15