Ph.D. Thesis Defence Public Lecture (DSAS) - Yichen Zhu

“Application of a polynomial affine method in dynamic portfolio choice”.

Abstract: 

This thesis develops numerical approaches to attain optimal multi-period portfolio strategies in the context of advanced stochastic models within expected utility and mean-variance theories.  Unlike common buy-and-hold portfolio strategies, dynamic asset allocation reflects the investment philosophy of a portfolio manager that benefits from the most recent market conditions to rebalance the portfolio accordingly. This enables managers to capture fleeting opportunities in the markets thereby enhancing the portfolio performance. However, the solvability of the dynamic asset allocation problem is often non-analytical, especially when considering a high-dimensional portfolio with advanced models mimicking practical asset's return. To overcome this issue, this thesis presents a competitive methodology to approximate optimal  dynamic portfolio strategies.   

The thesis can be categorized into two large sections. The development, algorithmic description, testing and extension of the methodology are presented in detail in the first section. Specifically, the main method, named PAMC, is originally developed for constant relative risk aversion investors. In a comparison with two existing well-known benchmark methods, our approach demonstrates superior efficiency and accuracy, this is not only for cases with no known solution but also for models where the analytical solution is available. We consequently extend the method into the wider hyperbolic absolute risk aversion utility family which is more flexible in capturing the risk aversion of investors. This extension permits the applicability of our method to both expected utility theory and mean-variance theory. Furthermore, the quality of portfolio allocation is directly linked to the quality of the portfolio value function approximation. This generates another important extension: the replacement of the polynomial regression in the original method by neural networks. Besides, we successfully implement the method on two important but closed-form unsolvable models: the Ornstein-Uhlenbeck 4/2 model and the Heston model with a stochastic interest rate, which further confirms the practicality and effectiveness of our novel methodology. 

The second part of the thesis addresses the application of our numerical method to investments involving financial derivatives. In addition to portfolio performance maximization and given the infinitely many choices of derivatives, we propose another criterion, namely, risk exposure minimization, to help investors meet regulatory constraints and protect their capital in the case of a market crash. The complexity of derivatives’ price dynamics leads to new challenges on the solvability of the optimal allocation for a derivative-based portfolio. With proper modifications, our method is applicable to this type of problem. We then consider a portfolio construction with equity options and volatility index (VIX) options in the presence of volatility risk, providing insight into best investment practices with derivatives.