Robust Optimization of Order Execution

Order execution for algorithmic trading has been studied in the literature to determine the optimal strategy by minimizing a trade-off between expected execution cost and risk. Usually, the variance of the execution cost is taken as a proxy of risk due to mathematical tractability. However, the variance has been recognized not to be practical since it is a symmetric measure of risk and, hence, penalizes the low-cost events. In this paper, we propose the use of the conditional value-at-risk (CVaR) of the execution cost as risk measure, which allows to take into consideration only the unfavorable part of the return distribution, or, equivalently, unwanted high cost. In addition, due to the parameter estimation errors in the price model, the naive strategies given by the nominal problem may perform badly in the real market, and hence it is extremely important to take such parameters estimation errors into consideration. To deal with this, we extend both the traditional mean-variance approach and our proposed CVaR approach to their robust design counterparts.