Hedging with Transient Price Impact for Non-Covered and Covered Options

We solve the superhedging problem for European options in a market with finite liquidity where trading has transient impact on prices, and possibly a permanent one in addition. Impact is multiplicative to ensure positive asset prices. Hedges and option prices depend on the physical and cash delivery specifications of the option settlement. For non-covered options, where impact at the inception and maturity dates matters, we characterize the superhedging price as a viscosity solution of a degenerate semilinear pde that can have gradient constraints. The non-linearity of the pde is governed by the transient nature of impact through a resilience function. For covered options, the pricing pde involves gamma constraints but is not affected by transience of impact. We use stochastic target techniques and geometric dynamic programming in reduced coordinates