Optimal liquidation under stochastic price impact

TL;DR

This paper develops approximate optimal trading strategies for liquidating assets in a stochastic price impact environment using coefficient expansion methods, providing practical solutions with numerical validation.

Contribution

It introduces a novel approach to solve the stochastic price impact liquidation problem via coefficient expansion of the Hamilton-Jacobi-Bellman equation, resulting in closed-form approximations.

Findings

Closed-form approximations for the value function and strategies

Effective numerical validation of the approximations

Insights into special cases with financial interpretations

Abstract

We assume a continuous-time price impact model similar to Almgren-Chriss but with the added assumption that the price impact parameters are stochastic processes modeled as correlated scalar Markov diffusions. In this setting, we develop trading strategies for a trader who desires to liquidate his inventory but faces price impact as a result of his trading. For a fixed trading horizon, we perform coefficient expansion on the Hamilton-Jacobi-Bellman equation associated with the trader's value function. The coefficient expansion yields a sequence of partial differential equations that we solve to give closed-form approximations to the value function and optimal liquidation strategy. We examine some special cases of the optimal liquidation problem and give financial interpretations of the approximate liquidation strategies in these cases. Finally, we provide numerical examples to demonstrate the effectiveness of the approximations.