Optimal Trade Execution Under Stochastic Volatility and Liquidity

Abstract

We study the problem of optimally liquidating a financial position in a discrete-time model with stochastic volatility and liquidity. We consider the three cases where the objective is to minimize the expectation, an expected exponential or a mean-variance criterion of the implementation cost. In the first case, the optimal solution can be fully characterized by a forward-backward system of stochastic equations depending on conditional expectations of future liquidity. In the other two cases, we derive Bellman equations from which the optimal solutions can be obtained numerically by discretizing the control space. In all three cases, we compute optimal strategies for different simulated realizations of prices, volatility and liquidity and compare the outcomes to the ones produced by the deterministic strategies of Bertsimas and Lo (1998; Optimal control of execution costs. Journal of Financial Markets, 1, 1–50) and Almgren and Chriss (2001; Optimal execution of portfolio transactions. Journal of Risk, 3, 5–33).

Key Words:

• Optimal trade execution
• implementation cost
• discrete-time stochastic control
• Bellman equation
• stochastic volatility
• stochastic liquidity

Acknowledgements

We thank an anonymous referee for helpful comments.

Funding

The first author gratefully acknowledges financial support from NSF Grant DMS-0642361.