Ph.D. Thesis Proposal Public Lecture (DSAS) - Yuyang Cheng

Portfolio optimization analysis in the family of 4/2 stochastic volatility models

Over the last two decades, financial trading of derivatives has increased significantly along with a richer and more complex behaviour/traits of the underlyings rapidly emerging. The need for more advanced model to capture those traits and behaviour of risky assets is crucial. In this spirit, the state-of-art 4/2 stochastic 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 component and a 3/2 component, which is shown to be able to eliminate the limitations of these two individual components, 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, four 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 an incomplete and complete market. 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. Given the fact that investors are not only risk-averse but also ambiguity-averse, we further take ambiguity-averse 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 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. Furthermore, 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 function (c.f.) is derived in closed-form, which allows for derivative pricing,  and a rich portfolio optimization problem in a multivariate setting is presented which included risk-aversion and incomplete markets.