Optimal trade execution aims at balancing price impact and timing risk. With respect to the mathematical formulation of the optimization problem, we primarily focus on Mean Variance (MV) optimization, in which the two conflicting objectives are maximizing expected revenue (the flip side of trading impact) and minimizing variance of revenue (a measure of timing risk). We also consider the use of expected quadratic variation of the portfolio value process as an alternative measure of timing risk, which leads to Mean Quadratic Variation (MQV) optimization. We demonstrate that MV-optimal strategies are quite different from MQV-optimal strategies in many aspects. These differences are in stark contrast to the common belief that MQV-optimal strategies are similar to, or even the same as, MV-optimal strategies. These differences should be of interest to practitioners since we prove that the classic Almgren-Chriss strategies (industry standard) are MQV-optimal, in contrary to the common belief that they are MV-optimal. From a computational point of view, we extend theoretical results in the literature to prove that the mean variance efficient frontier computed using our method is indeed the complete Pareto-efficient frontier. First, we generalize the result in Li (2000) on the embedding technique and develop a post-processing algorithm that guarantees Pareto-optimality of numerically computed efficient frontier. Second, we extend the convergence result in Barles (1990) to viscosity solution of a system of nonlinear Hamilton Jacobi Bellman partial differential equations (HJB PDEs). On the numerical aspect, we combine the techniques of similarity reduction, non-standard interpolation, and careful grid construction to significantly improve the efficiency of our numerical methods for solving nonlinear HJB PDEs.

Optimal trade execution aims at balancing price impact and timing risk. With respect to the mathematical formulation of the optimization problem, we primarily focus on Mean Variance (MV) optimization, in which the two conflicting objectives are maximizing expected revenue (the flip side of trading impact) and minimizing variance of revenue (a measure of timing risk). We also consider the use of expected quadratic variation of the portfolio value process as an alternative measure of timing risk, which leads to Mean Quadratic Variation (MQV) optimization. We demonstrate that MV-optimal strategies are quite different from MQV-optimal strategies in many aspects. These differences are in stark contrast to the common belief that MQV-optimal strategies are similar to, or even the same as, MV-optimal strategies. These differences should be of interest to practitioners since we prove that the classic Almgren-Chriss strategies (industry standard) are MQV-optimal, in contrary to the common belief that they are MV-optimal. From a computational point of view, we extend theoretical results in the literature to prove that the mean variance efficient frontier computed using our method is indeed the complete Pareto-efficient frontier. First, we generalize the result in Li (2000) on the embedding technique and develop a post-processing algorithm that guarantees Pareto-optimality of numerically computed efficient frontier. Second, we extend the convergence result in Barles (1990) to viscosity solution of a system of nonlinear Hamilton Jacobi Bellman partial differential equations (HJB PDEs). On the numerical aspect, we combine the techniques of similarity reduction, non-standard interpolation, and careful grid construction to significantly improve the efficiency of our numerical methods for solving nonlinear HJB PDEs.