Optimal Portfolio Execution Strategies: Uncertainty and Robustness
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Optimal investment decisions often rely on assumptions about the models and their associated parameter values. Therefore, it is essential to assess suitability of these assumptions and to understand sensitivity of outcomes when they are altered. More importantly, appropriate approaches should be developed to achieve a robust decision. In this thesis, we carry out a sensitivity analysis on parameter values as well as model speci cation of an important problem in portfolio management, namely the optimal portfolio execution problem. We then propose more robust solution techniques and models to achieve greater reliability on the performance of an optimal execution strategy. The optimal portfolio execution problem yields an execution strategy to liquidate large blocks of assets over a given execution horizon to minimize the mean of the execution cost and risk in execution. For large-volume trades, a major component of the execution cost comes from price impact. The optimal execution strategy then depends on the market price dynamics, the execution price model, the price impact model, as well as the choice of the risk measure. In this study, rst, sensitivity of the optimal execution strategy to estimation errors in the price impact parameters is analyzed, when a deterministic strategy is sought to minimize the mean and variance of the execution cost. An upper bound on the size of change in the solution is provided, which indicates the contributing factors to sensitivity of an optimal execution strategy. Our results show that the optimal execution strategy and the e cient frontier may be quite sensitive to perturbations in the price impact parameters. Motivated by our sensitivity results, a regularized robust optimization approach is devised when the price impact parameters belong to some uncertainty set. We rst illustrate that the classical robust optimization might be unstable to variation in the uncertainty set. To achieve greater stability, the proposed approach imposes a regularization constraint on the uncertainty set before being used in the minimax optimization formulation. Improvement in the stability of the robust solution is discussed and some implications of the regularization on the robust solution are studied. Sensitivity of the optimal execution strategy to market price dynamics is then investigated. We provide arguments that jump di usion models using compound poisson processes naturally model uncertain price impact of other large trades. Using stochastic dynamic programming, we derive analytical solutions for minimizing the expected execution cost under jump di usion models and compare them with the optimal execution strategies obtained from a di usion process. A jump di usion model for the market price dynamics suggests the use of Conditional Value-at-Risk (CVaR) as the risk measure. Using Monte Carlo simulations, a smoothing technique, and a parametric representation of a stochastic strategy, we investigate an approach to minimize the mean and CVaR of the execution cost. The devised approach can further handle constraints using a smoothed exact penalty function.