The multi-dimensional super-replication problem under gamma constraints

The classical Black-Scholes hedging strategy of a European contingent claim may require rapid changes in the replicating portfolio. One approach to avoid this is to impose a priori bounds on the variations of the allowed trading strategies, called gamma constraints. Under such a restriction, it is in general no longer possible to replicate a European contingent claim, and super-replication is a commonly used alternative. This paper characterizes the infimum of the initial capitals that allow an investor to super-replicate the contingent claim by carefully choosing an investment strategy obeying a gamma constraint. This infimum is shown to be the unique viscosity solution of a nonstandard partial differential equation. Due to the lower gamma bound, the "intuitive" partial differential equation is not parabolic and the actual equation satisfied by the infimum is the parabolic majorant of this equation. The derivation of the viscosity property is based on new results on the small time behavior of double stochastic integrals.