Doctoral Public Lecture | Xize Ye

Student Name: Xize Ye
Program: Statistics

Supervisors: Dr. Marcos Escobar-Anel & Dr. Lars Stentoft
Time: November 5th, 2025, 9:00 AM - 10:00 AM

Location: Western Science Centre 248

Title:  Affine Generalized Autoregressive Conditionally Stochastic Heteroskedasticity: Theory and Applications

Abstract: 

Conventional GARCH models consider perfect causality between returns and variance, which contradicts real market data and fails to match historical returns with forward-looking volatility filtered from option prices. This thesis proposes the novel Generalized Autoregressive Conditionally Stochastic Heteroskedasticity (GARCSH) framework, which adds a second innovation in the variance process, independent of return shocks, to capture joint dynamics of return and volatility. The structure is motivated by continuous-time stochastic volatility models where correlated Brownian motions simultaneously drive return and variance.

Combined with the celebrated Heston-Nandi GARCH model, our framework yields the affine HN-GARCSH model. This thesis first develops theoretical results and applications of the baseline model, including a pricing kernel from the local risk-neutral valuation relationship (LRNVR), maximum likelihood estimation on asset–VIX data, and closed-form pricing formulas for return options, VIX futures, and VIX options, as well as convergence in continuous time.

The second part applies the baseline model to study which (maturity of) VIX should be used in option pricing models. Our empirical results from extensive studies and robustness checks demonstrate that the short-term (1-month) VIX is preferred over long-term VIXs. 

The third part explores robust estimation of HN-GARCSH and the distributional parameter under generalized measurement error structures, which is crucial in the modelling of the VIX. Using an extended dataset with return–VIX pairs and joint estimation techniques with option data, the results confirm the superior performance of HN-GARCSH in VIX fitting and derivative pricing, showing robustness under improved specifications and additional data.

The final part combines our methodology with recent affine GARCH advances, proposing three new affine GARCSH models with formulas for moment generating functions and implied VIXs, thereby providing a foundation for future research. Overall, our proposed framework has interesting theoretical features, is shown to be robust when considering different assets, time periods, and estimation methods, leading to significantly improved performance in VIX fitting and the pricing of return and volatility derivatives. The extraordinary empirical performance, combined with convenient applicability, demonstrates the importance of our methodology in the field of discrete-time option pricing.

Summary for Lay Audience:

Returns on financial assets are random, and one of the critical areas of research is to model uncertainty and how returns and their volatility (how much prices fluctuate) are connected. The most widely used model, known as GARCH, assumes a very tight relationship between the two. Such an assumption is contrary to the real-world data. GARCH models (particularly standard ones) also struggle to explain the VIX index, a real-time financial measure of volatility, demonstrating potential room for improvement.

My thesis introduces a new methodology that directly improves on the GARCH models by assuming a more generalized dynamics of asset volatility. Our framework implies a much looser connection between asset return and volatility. As a consequence, our model is able to explain the VIX index much more precisely. It is also shown in the thesis that this model not only works well in theory but is also very practical for computing and forecasting the price of various financial products. We have picked two financial indices and two stocks, and different time periods. The improved performance of our model persists -- our new model consistently does a better job than older models at matching the real market and explaining the behavior of volatility.

Finally, I extend the model further, connecting it with recent advances in finance research, and show how it can be used in many new directions. In conclusion, this thesis provides a more generalized and realistic way of modelling volatility that better explains the market behaviour along many important dimensions and greatly improves the accuracy of pricing financial products. Due to its flexible nature, many areas of potential research are possible, marking a significant milestone and achievement in financial research.

Please contact the Graduate Assistant in the program for further information: https://grad.uwo.ca/about_us/program_contacts.cfm.