Superreplication Under Gamma Constraints

We consider a nancial market consisting of a nonrisky asset and a risky one. We study the minimal initial capital needed in order to super-replicate some given contingent claim under Gamma constraint, i.e. constraint on the unbounded variation part of the hedging portfolio. In the general Markov diiusion case, we prove a veriication Theorem which characterizes the super-replication cost as the solution of some varia-tional inequality. In the context of the Black-Scholes model, our veriication Theorem allows to derive an explicit solution of the problem. The results are based on an original dynamic programming principle which is well-suited with \stochastic target" problems. We gratefully acknowledge conversations with Professor F. Delbaen; in particular he has provided us with the key idea in the proof of Lemma 8.1.