On Optimal Options Book Execution Strategies with Market Impact

We consider the optimal execution of a book of options when market impact is a driver of the option price. We aim at minimizing the mean-variance risk criterion for a given market impact function. First, we develop a framework to justify the choice of our market impact function. Our model is inspired from Leland's option replication with transaction costs where the market impact is directly part of the implied volatility function. The option price is then expressed through a Black-Scholes-like PDE with a modified implied volatility directly dependent on the market impact. We set up a stochastic control framework and solve an Hamilton-Jacobi-Bellman equation using finite differences methods. The expected cost problem suggests that the optimal execution strategy is characterized by a convex increasing trading speed, in contrast to the equity case where the optimal execution strategy results in a rather constant trading speed. However, in such mean valuation framework, the underlying spot price does not seem to affect the agent's decision. By taking the agent risk aversion into account through a mean-variance approach, the strategy becomes more sensitive to the underlying price evolution, urging the agent to trade faster at the beginning of the strategy.