Hedging of covered options with linear market impact and gamma constraint

Within a financial model with linear price impact, we study the problem of hedging a covered European option under gamma constraint. Using stochastic target and partial differential equation smoothing techniques, we prove that the super-replication price is the viscosity solution of a fully non-linear parabolic equation. As a by-product, we show how $\epsilon$-optimal strategies can be constructed. Finally, a numerical resolution scheme is proposed.