Doctor of Philosophy
Statistics and Actuarial Sciences
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.
Summary for Lay Audience
Financial markets, i.e. markets for financial securities, are important mediums for economic development. There are two typical players in financial markets. The demand side refers to participants who seek funds for daily operational cash flows, bridge financing or long-term special projects. In contrast, the supply side refers to those who have excess money that can be used in favor of demand side thereby earning appreciation of their capital. This “dedication” of assets is often called investment, which is investigated in a scientific way in this thesis. In practice, investment is a systematic process for large institutional investors including defining objectives, security selection, ongoing monitoring, etc. In this thesis, we focus on one of the most important steps, i.e. portfolio construction. A portfolio is a collection of financial investments like stocks, bonds, commodities, that take advantage of the wisdom of diversification—which simply means never put all your eggs in one basket. Moreover, we consider the portfolio in a continuous-time framework which allows investors to adjust their portfolio allocation in real time in response to new information that can appear at any time. Thus, Chapters 2, 3 and 4 are mainly concerned on the methodologies for approximating the optimal portfolio strategy. Financial derivatives, defined as contracts that derive value from the performance of underlying assets, are widely used for hedging, speculation and taking arbitrage opportunities. There are numerous studies showing that derivatives may effectively enhance portfolio performance. This generates interest in the theme of the last two chapters: investments across traditional financial markets and derivatives markets. Specifically, Chapter 5 develops a new framework for derivatives-based portfolios. Chapter 6 illustrates an application of derivatives-based portfolios in presence of volatility risk.
Zhu, Yichen, "Application Of A Polynomial Affine Method In Dynamic Portfolio Choice" (2022). Electronic Thesis and Dissertation Repository. 8467.