1. Introduction to Volatility Models From Ruey. S. Tsay’s slides

2. Characteristics of Volatility

3. Characteristics of Volatility • Not directly observable • Existence of volatility clusters (volatility maybe high for certain time periods and low for other periods) • Evolving over time in a continuous manner • Volatility does not diverge to infinity, i.e. volatility is stationary • Volatility reacts differently to big price increase/drop

4. Structure of Volatility Models Basic idea: Shocks of asset returns are NOT serially correlated, but dependent.

5. Model Building Steps • Specify a mean equation by testing for serial dependence in the data. • Use the residuals of the mean equation to test for ARCH effects. • Specify a volatility model if ARCH effects are statistically significant and perform a joint estimation of the mean and volatility equation. • Check the fitted model and refine it if necessary.

6. ARCH Model

7. ARCH Model Properties

8. Pro and Con of ARCH Model • Pro: • Simplicity • Generates Volatility Clustering • Heavy Tails (outlier study) • Con: • Symmetric btw positive & negative prior returns • Restrictive • Provides no explanation • Not sufficiently adaptive in prediction

9. Building an ARCH Model • Modeling the mean effect and testing • Use Q-statistics of squared residuals; McLeod and Li (1983) &Engle (1982) • Order determination • Use PACF of the squared residuals • Estimation • Conditional MLE • Model checking • Q-stat of standardized residuals and squared standardized residuals. Skewness & Kurtosis of standardized residuals.

10. GARCH Model

11. ARCH Model Properties