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

Assuming geometric Brownian motion as unaffected price process <inline-formula> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ramf_a_683963_o_ilm0001.gif"/> </inline-formula>, Gatheral and Schied (2011; Optimal trade execution under geometric Brownian motion in the Almgren and Chriss framework, <italic>International Journal of Theoretical and Applied Finance,</italic> 14, pp. 353--368) 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 <inline-formula> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ramf_a_683963_o_ilm0002.gif"/> </inline-formula>. We prove the surprising result that the strategy remains optimal whenever <inline-formula> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ramf_a_683963_o_ilm0003.gif"/> </inline-formula> is a square-integrable martingale. We then analyse the optimization criterion of Gatheral and Schied (2011) in the case in which <inline-formula> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ramf_a_683963_o_ilm0004.gif"/> </inline-formula> 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.