Robust Strategies for Optimal Order Execution in the Almgren-Chriss Framework

TL;DR

This paper investigates the robustness of a known optimal order execution strategy under model misspecification, proving its optimality for a broad class of price processes and providing explicit solutions for minimizing liquidation costs.

Contribution

It extends the optimal execution framework to general square-integrable semimartingales, showing the strategy's robustness and deriving closed-form solutions without Markovian assumptions.

Findings

Strategy remains optimal for all square-integrable martingale price processes.

Provides explicit solutions for minimizing expected liquidation costs.

Extends the model to non-Markovian semimartingale price processes.

Abstract

Assuming geometric Brownian motion as unaffected price process $S^0$, Gatheral & Schied (2011) derived a strategy for optimal order execution that reacts in a sensible manner on market changes but can still be computed in closed form. Here we will investigate the robustness of this strategy with respect to misspecification of the law of $S^0$. We prove the surprising result that the strategy remains optimal whenever $S^0$ is a square-integrable martingale. We then analyze the optimization criterion of Gatheral & Schied (2011) in the case in which $S^0$ is any square-integrable semimartingale and we give a closed-form solution to this problem. As a corollary, we find an explicit solution to the problem of minimizing the expected liquidation costs when the unaffected price process is a square-integrable semimartingale. The solutions to our problems are found by stochastically solving a finite-fuel control problem without assumptions of Markovianity.