Super - replication under Gamma constraints 1

In a nancial market consisting of a non risky asset and a risky one, we study the minimal initial capital needed in order to super-replicate a given contingent claim under a Gamma constraint. This is a constraint on the unbounded variation part of the hedging portfolio. We rst consider the case in which the prices are given as general Markov di usion processes and prove a veri cation theorem which characterizes the super-replication cost as the unique solution of a quasi variational inequality. In the context of the Black-Scholes model (i.e., when volatility is constant), this theorem allows us to derive an explicit solution of the problem. These results are based on a new dynamic programming principle for general \stochastic target" problems.