Price-Sensitive Liquidation In Continuous-Time

Abstract

We consider the stochastic control problem of how to optimally close a large asset position in an illiquid market with price impact. We assume that the risk attributed to an open position depends on the price evolvement since the beginning of the trading period. Within a continuous-time model with a linear temporary price impact we show how to obtain an optimal trade off between liquidity costs and price risk. The optimal trading rates, turning out to be price sensitive, can be characterized in terms of a PDE describing by how much they differ from a linear trading rate. The PDE, in general, does not possess a closed-form solution. We provide a uniqueness result for solutions in the viscosity sense, allowing in the following to identify the value function and optimal trading rates. Finally we show that optimal strategies from discrete model approximations converge to the continuous-time optimal trading rates.

Keywords: optimal liquidation, price impact, stochastic control of trading speed, skewness, price-sensitive preferences

JEL Classification: C61, C63, G12