M.Sc. Thesis Defence (DSAS) - Xize Ye

"On the Estimation of Heston-Nandi GARCH Using Returns and/or Options: A Simulation-based Approach"

The development of option pricing models has been a productive research area ever since the first Nobel prize-winning proposal of the Black-Scholes-Merton model in the 1970s. In 2000, Heston and Nandi proposed a particular GARCH$(p,q)$ conditional volatility model with a closed-form option pricing formula for European option prices. The filtering and estimation of conditional volatility of this model can be completed solely from daily observables. Yet, many questions remain unanswered. For instance, is the model calibration process robust and reliable? How accurate and valid are the parameter estimates? In this thesis, we simulated market data based on the Heston Nandi-GARCH(1,1) (henceforth, HN-GARCH) model, and then examined and compared the 4 maximum likelihood-based estimation and calibration approaches using returns and/or options. We first followed Bollerslev (1986) and Heston and Nandi (2000) to investigate the fundamental returns-only estimation on GARCH models, during which we found that the price of risk parameter, $\lambda$, is particularly difficult to estimate from returns data only, and its estimator is highly influenced by the average level of simulated daily noises. Hence, we simulated option prices with a pre-defined noise structure, calibrated the model jointly with options data, and compared its performance with the benchmark returns-only MLE method. We conjectured beforehand that bringing in option data shall help calibrate all parameters, with the potential of capturing $\lambda$ more precisely. From our empirical studies, with the additional option sample, we can improve the efficiency of the estimates for parameters $\alpha$, $\beta$, and $\gamma$. Nonetheless, the improvements for $\lambda$, both from empirical standard errors, and sample RMSEs, are insignificant. This seems largely due to the fact that, from the configuration of HN-GARCH, options data can only capture $\gamma^* = \gamma + \lambda$, but fail to distinguish $\lambda$ from $\gamma$ (also, note that the relative size of $\gamma$ is hundred times bigger than $\lambda$). In addition, we simulated the option sample with different noise levels, to demonstrate the consequence when we have a noisy option sample versus a less noisy one. The result shows that, with added option samples the RMSEs for estimated GARCH parameters are reduced dramatically, even with a very noisy option data set. This suggests that calibrating GARCH option pricing models with a relatively short return series of around 6 years, plus an option sample is more ideal than using a long return series of 20 years alone. Finally, as a by-product, we studied which type of options leads to the larger calibration improvements. Our controlled experiment confirms that out-of-the-money, short-maturity options are the best choices.

Keywords: Heston-Nandi GARCH, Simulation, Joint Calibration-Estimation, Maximum Likelihood Estimation