On Optimal Options Book Execution Strategies with Market Impact

Abstract

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.

Managing investment and liquidity risks for derivatives within a market impact perspective

“…Then we can solve the problem numerically using finite differences methods as in Kalife and Mouti (2017) In our numerical experiment, we present results for a long position on ATM put options with the parameters as follows: This chart illustrates the optimal execution strategy through the rate of trading as a function of the underlying price S and time t. The strategy does not depend on the trader inventory position. However, as time increases the trading rate increases (convex in time): as the maturity approaches the agent must acquire faster as time passes.…”