## Thesis Format

Monograph

## Degree

Doctor of Philosophy

## Program

Statistics and Actuarial Sciences

## Supervisor

Escobar-Anel, Marcos

## Abstract

Financial markets and instruments are continuously evolving, displaying new and more refined stylized facts. This requires regular reviews and empirical evaluations of advanced models. There is evidence in literature that supports stochastic volatility models over constant volatility models in capturing stylized facts such as "smile" and "skew" presented in implied volatility surfaces. In this thesis, we target commodity and volatility index markets, and develop a novel stochastic volatility model that incorporates mean-reverting property and 4/2 stochastic volatility process. Commodities and volatility indexes have been proved to be mean-reverting, which means their prices tend to revert to their long term mean over time. The 4/2 stochastic volatility process integrates two processes that have contrary behaviors. As a result, not only is the 4/2 stochastic volatility process able to reproduce "smile" and "skew", but also model the asset price time series very well even in the most extreme situation like financial crisis where the assets' prices become highly volatile. In the one-dimensional study, we derive a semi-closed form conditional characteristic function (c.f.) for our model and propose two feasible approximation approaches. Numerical study shows that the approximations are able to approximate the c.f. well in a large region of the parametric space. With the approximations, we are able to price options using c.f. based pricing algorithms which outperform Monte Carlo simulation in speed. We are also the first to study two estimation methods for the 4/2 stochastic volatility process. We show numerically that the estimation methods produce consistent estimators. By applying our model to empirical data, especially volatility indexes data, we find evidence for an embedded 4/2 stochastic volatility process, which also can be seen by observing drastic spikes from data. In option pricing applications, we realize there can be 20% difference on option prices if the underlying model is specified by a 4/2 stochastic volatility process as opposed to a 1/2 stochastic volatility process. We further test our approximation approaches by comparing the option prices generated by Monte Carlo simulation and those obtained from Fast Fourier Transform using the approximated c.f., the error turns out to be negligible. We next consider a generalized multivariate model based on our one-dimensional mean-reverting 4/2 stochastic volatility model and principal component stochastic volatility framework to capture the behavior of multiple commodities or volatility indexes. The model structure enables us to express the model in terms of a linear combination of independent one-dimensional mean-reverting 4/2 stochastic volatility processes. We find a quasi-closed form c.f. for the generalized model and analytic approximations of the c.f. under certain model assumptions. We propose a scaling factor to connect empirical variance series to theoretical variances and estimate the parameters with the methodology developed for our one-dimensional mean-reverting 4/2 stochastic volatility model. The effectiveness of our approximation approaches is supported by comparing the Value-at-Risk (VaR) values of a portfolio of two risky assets and a cash account using Monte Carlo simulations and the approximated distributions.

## Summary for Lay Audience

Financial markets and instruments are continuously evolving, displaying new and more refined stylized facts. This requires regular reviews and empirical evaluations of advanced models. There is evidence in literature that supports stochastic volatility models over constant volatility models in capturing stylized facts such as ``smile' and ``skew" presented in implied volatility surfaces. In this thesis, we target commodity and volatility index markets, and develop a novel stochastic volatility model that incorporates mean-reverting property and 4/2 stochastic volatility process. Commodities and volatility indexes have been proved to be mean-reverting, which means their prices tend to revert to their long term mean over time; the 4/2 stochastic volatility process is able to reproduce "smile" and "skew", but also model the asset price time series very well even in situations like financial crisis where the assets' prices become unstable. In the study of single asset, we study theoretical properties of our model and propose two approximation approaches. With the approximations, we are able to price options using fast pricing algorithms instead of simulations. We are also the first to study two estimation methods for the 4/2 stochastic volatility process. Our estimation methods can produce accurate estimators when sample size is large enough. By applying our model to empirical data, especially volatility indexes data, we find evidence for an embedded 4/2 stochastic volatility process. In option pricing applications, we realize there can be 20% difference on option prices if the underlying model is incorrectly specified. We compare the option prices generated by simulation and those obtained from approximation approaches, the error turns out to be negligible. We next consider a multi-asset setting as an extension of our single asset study aiming to capture the behavior of multiple commodities or volatility indexes. The model structure enables us to express one asset as a linear combination of independent artificial "assets". We propose a scaling factor to connect empirical variance series to theoretical variances and estimate the parameters. As an application, we further construct a portfolio that consists of two risky assets and a cash account and calculate Value-at-Risk using simulations and the approximated distributions.

## Recommended Citation

Gong, Zhenxian, "The Mean-Reverting 4/2 Stochastic Volatility Model: Properties And Financial Applications" (2021). *Electronic Thesis and Dissertation Repository*. 7686.

https://ir.lib.uwo.ca/etd/7686

#### Included in

Applied Statistics Commons, Other Applied Mathematics Commons, Probability Commons, Statistical Models Commons