Continuous-Time and Distributionally Robust Mean-Variance Models

de Lin Chen

This thesis contains three works in both continuous-time and distributionally robust mean-variance Markowitz models. In the first work, we study naive strategies in the continuous-time mean-variance model. We propose a new type of agent to approximate the dynamic of the naive agent by partitioning the time line into numerous small equal length time intervals. Then, we prove that, the wealth process of the proposed agent converges to that of the naive agent and derive the explicit formula for the limiting wealth process and its corresponding portfolio process. In the end, we compare the naive strategies with two equilibrium strategies in the Black-Scholes market. The second work contributes to the mean-variance model by considering its distributionally robust counterpart, where the region of distributional uncertainty is around the empirical measure and the discrepancy between probability measures is dictated by the Wasserstein distance. We reduce this problem to an empirical variance minimization problem with an additional regularization term. Moreover, we extend the recently developed inference methodology to our setting in order to select the size of the distributional uncertainty as well as the associated robust target return rate in a data-driven way. Finally, we report extensive backtesting results on the S&P 500 that compares the performance of our model with those of several well-known models, including the Fama--French model and the Black--Litterman model. In the last part, we develop a distributionally robust model based on the Sharpe ratio optimization problem. We transform the problem into an equivalent convex optimization problem that can be solved numerically. In this model, we do not need to choose the target return parameter, which has to be decided by subjective judgement in previous distributionally robust mean-variance models. As a result, the distributionally robust Sharpe ratio model is completely data-driven. We also provide guidance on the choice of ambiguity set size by using a much simpler scheme than that employed in the second work. In the end, we compare the performance of this model to that of the second work and some other well-known models on S&P500.