Optimal Portfolio Liquidation with Limit Orders

Abstract

This paper addresses portfolio liquidation using a new angle. Instead of focusing only on the scheduling aspect like Almgren and Chriss in [J. Risk, 3 (2000), pp. 5--39], or only on the liquidity-consuming orders like Obizhaeva and Wang in [Optimal Trading Strategy and Supply/Demand Dynamics, SSRN eLibrary, 2005], we link the optimal trade schedule to the price of the limit orders that have to be sent to the limit order book to optimally liquidate a portfolio. Most practitioners address these two issues separately: they compute an optimal trading curve, and they then send orders to the markets to try to follow it. The results obtained in this paper can be interpreted and used in two ways: (i) we solve simultaneously the two problems and provide a strategy to liquidate a portfolio over a few hours, and (ii) we provide a tactic for following a trading curve over slices of a few minutes. As far as the model is concerned, the interactions of limit orders with the market are modeled via a point process pegged to a diffusive “fair price.” A Hamilton--Jacobi--Bellman equation is used to solve the control problem involving both nonexecution risk and price risk. Backtests are carried out to exemplify the use of our results, both on long periods of time (for the entire liquidation process) and on slices of 5 minutes (to follow a given trading curve).