In a financial market consisting of a nonrisky asset and a risky one, we study the minimal initial capital needed in order to superreplicate a given contingent claim under a gamma constraint. This is a constraint on the unbounded variation part of the hedging portfolio. We first consider the case in which the prices are given as general Markov diffusion processes and prove a verification theorem which characterizes the superreplication 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.
All Science Journal Classification (ASJC) codes
- Control and Optimization
- Applied Mathematics