Option pricing and hedging with execution costs and market impact.

Summary In this article we consider the pricing and (partial) hedging of a call option when liquidity matters, that is either for a large nominal or for an illiquid underlying. In practice, as opposed to the classical assumptions of a price-taker agent in a frictionless market, traders cannot be perfectly hedged because of execution costs and market impact. They face indeed a trade-off between mishedge errors and hedging costs that can be solved using stochastic optimal control. Our framework is inspired from the recent literature on optimal execution and permits to account for both execution costs and the lasting market impact of our trades. Prices are obtained through the indifference pricing approach and not through super-replication. Numerical examples are provided using PDEs, along with comparison with the Black model.