Doctor of Philosophy
Statistics and Actuarial Sciences
Over the last two decades, trading of financial derivatives has increased significantly along with richer and more complex behaviour/traits on the underlying assets. The need for more advanced models to capture traits and behaviour of risky assets is crucial. In this spirit, the state-of-the-art 4/2 stochastic volatility model was recently proposed by Grasselli in 2017 and has gained great attention ever since. The 4/2 model is a superposition of a Heston (1/2) component and a 3/2 component, which is shown to be able to eliminate the limitations of these two individual models, bringing the best out of each other. Based on its success in describing stock dynamics and pricing options, the 4/2 stochastic volatility model is an ideal candidate for portfolio optimization. Hence, in this thesis, we focus on portfolio optimization problems under the 4/2 stochastic volatility class of models.
To highlight the 4/2 stochastic volatility model in portfolio optimization problems, five related and self-contained projects are conducted. We firstly investigate, in Chapter 2, portfolio optimization problems under the 4/2 stochastic volatility model within the framework of expected utility theory for a constant-relative-risk-averse (CRRA) investor in incomplete and complete markets. We postulate the market prices of risk are proportional to variance's driver. By employing a dynamic programming approach, we formulate the corresponding Hamilton-Jacobi-Bellman (HJB) equations and solve them via an exponential-affine ansatz. Verification theorems are provided to ensure optimality. We find that the optimal strategy recommended by the 4/2 model depends on the levels of current volatility, a reasonable feature not reported in the existing literature. To present a meaningful empirical study, a full estimation is performed for the 4/2 model along with its embedded popular models (i.e. the 3/2 and 1/2). We compare the optimal recommendations from various models and illustrate the wealth-equivalent losses from classical sub-optimal strategies (i.e. as those produced by 1/2, 3/2 and Geometric Brownian motion). Given the fact that investors are not only risk-averse but also ambiguity-averse, we further take ambiguity-aversion into account and examine, in Chapter 3, a robust portfolio optimization problem under the setting described before. We determine the robust optimal strategy and the worst-case measure by allowing separate levels of uncertainty for variance and stock drivers. The impact of ambiguity aversion on the optimal strategy is studied under a realistic parametric set and viable ambiguity-aversion levels following a detection error analysis. The theoretical and numerical analyses confirm an inverse relation between absolute risky exposure and the level of ambiguity aversion. In particular, exposures to assets could decrease by 50\% for reasonable ambiguity-aversion values. In Chapter 4, we incorporate a consumption decision into the most complete portfolio optimization problem described before, and employ the preferable proportion-to-volatility market price of risk in a new analysis of the 4/2 model. Due to the non-affine nature, the solution for the value function involves confluent hypergeometric functions similar to those needed for option pricing within the 4/2 model. The chapter explores all cases where closed-form solutions are available, in all combinations of settings: complete-incomplete, consumption or its absence, ambiguity-averse or its absence, leverage or its absence. In Chapter 5, we propose a multivariate 4/2 stochastic volatility model to capture advanced stylized facts in the behaviour of multiple assets, such as co-volatility movements and stochastic correlations among assets. The model is built as a linear combination of independent one-dimensional 4/2 processes, which keeps the number of parameters parsimonious. In this new model, the conditional characteristic functions (c.f.) under historical and risk-neutral measures are derived in closed-form, which would allow for derivative pricing and risk management analysis (not conducted here). A rich portfolio optimization problem in a multivariate setting is presented which included risk-aversion and incomplete markets, conditions for verification and proper solutions are provided; we also study the co-volatility movements and highlight the importance of the 4/2 underlying structure as part of a numerical analysis. In Chapter 6, we combine all we have learned about stochastic volatility and expected utility to reveal the largest class of stochastic volatility processes solvable in closed form within a larger family of utility functions, the HARA (hyperbolic absolute risk aversion). This chapter lists all solvable cases reported in the literature, adding many others thanks to a change of control technique. The extension to ambiguity-aversion analyses is also considered.
Summary for Lay Audience
Over the last two decades, the trading of financial products has increased significantly along with evidence of more complex behaviour in assets. The need for more advanced models to capture this complexity is crucial. In this spirit, the state-of-the-art 4/2 stochastic volatility model was recently proposed by Grasselli in 2017 and has gained great success in capturing various stylized facts of stock prices. Hence, in this thesis, we apply the 4/2 stochastic volatility class of models to the problem of finding the best allocation of wealth between a risky asset and a cash account. We find that the optimal strategy recommended by the 4/2 model depends on the levels of current volatility, an intuitive feature that investors may expect but not reported in the existing literature. Investors are uncertain about the probability distribution of the stock, so they would consider a set of alternative models when making investment decisions, this is called a robust portfolio optimization. Based on the optimal portfolio problems above, we extend the literature in two directions. On the one hand, the optimal allocations hinge upon the choices of market price of risk, i.e., the return in excess of the risk-free rate that investors demand as a compensation for taking/bearing risk, but there is no consensus in the existing literature. Hence, we consider a conventional choice of market price of risk, namely, proportional to volatility, solving the corresponding (robust) portfolio optimization problem. On the other hand, due to the importance of modelling the stochastic correlation implied by multivariate volatility, instead of constant correlation or single-factor stochastic volatility, we further consider a multivariate model with 4/2 structured counterparts in an optimal portfolio choice problem. Lastly, we combine all we have learned about stochastic volatility and expected utility to reveal the largest class of stochastic volatility processes solvable in closed form within a larger family of utility functions, the hyperbolic absolute risk aversion, which is more flexible in assessing risk aversion of investors and can be connected with yet another area of portfolio problems known as mean variance theory.
Cheng, Yuyang, "Portfolio Optimization Analysis in the Family of 4/2 Stochastic Volatility Models" (2022). Electronic Thesis and Dissertation Repository. 8952.