Optimal Portfolio Liquidation with Execution Cost and Risk

Abstract

We study the optimal portfolio liquidation problem over a finite horizon in a limit order book with bid-ask spread and temporary market price impact penalizing speedy execution trades. We use a continuous-time modeling framework, but in contrast with previous related papers (see, e.g., [L. C. G. Rogers and S. Singh, Math. Finance, 20 (2010), pp. 597–615] and [A. Schied and T. Schöneborn, Finance Stoch., 13 (2009), pp. 181–204]), we do not assume continuous-time trading strategies. We consider instead real trading that occur in discrete time, and this is formulated as an impulse control problem under a solvency constraint, including the lag variable tracking the time interval between trades. A first important result of our paper is to prove rigorously that nearly optimal execution strategies in this context actually lead to a finite number of trades with strictly increasing trading times, and this holds true without assuming ad hoc any fixed transaction fee. Next, we derive the dynamic programming quasi-variational inequality satisfied by the value function in the sense of constrained viscosity solutions. We also introduce a family of value functions which converges to our value function and is characterized as the unique constrained viscosity solutions of an approximation of our dynamic programming equation. This convergence result is useful for numerical purpose but is postponed until a companion paper [F. Guilbaud, M. Mnif, and H. Pham, Numerical Methods for an Optimal Order Execution Problem, preprint, 2010].