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

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

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

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.

ARCH Model

ARCH Model Properties

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

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.

GARCH Model

ARCH Model Properties