Optimal Execution of Limit and Market Orders with Trade Director, Speed Limiter, and Fill Uncertainty

Abstract

We study the optimal execution of market and limit orders with permanent and temporary price impacts under continuous time. Our model also features uncertainty in the filling of limit orders, a trade speed limiter, and a trader director. We formulate an optimal stochastic control problem to determine the dynamic execution strategy, with a quadratic terminal penalty to ensure complete liquidation. For comparison, we also solve the schedule-following optimal execution problem that penalizes deviations from an order schedule. Numerical results are provided to illustrate the optimal market and limit orders over time. Our model is able to encourage non-negativity of the trading rates. In addition, we identify conditions on the model parameters to ensure optimality of the controls and boundedness of the associated stochastic control problem.