Robust Strategies for Optimal Order Execution in the Almgren-Chriss Framework
20 Pages Posted: 25 Jan 2012 Last revised: 25 Sep 2012
Date Written: January 24, 2012
Abstract
Assuming geometric Brownian motion as unaffected price process S0, Gathertal and 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 S0. We prove the surprising result that the strategy remains optimal whenever S0 is a square-integrable martingale. We then analyze the optimization criterion of Gatheral and Schied (2011) in the case in which S0 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.
Keywords: market impact, optimal order execution, Almgren-Chriss model, robustness, model uncertainty
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