Modeling Volatility and GARCH Family › Modeling Volatility and GARCH Family

In our study of the stylized facts of financial returns (Part III, Chapter 1), we observed a striking phenomenon. Large price movements tend to cluster together, followed by periods of relative calm. This volatility clustering violates the constant volatility assumption in the Black-Scholes-Merton framework we developed in Chapters 5 through 8. When we examined implied volatility and the volatility smile, we found that markets reject constant volatility, pricing options at different implied volatilities across strikes and maturities.

This chapter addresses a fundamental question: how do we model volatility that changes over time? The answer came from Robert Engle's groundbreaking ARCH model in 1982, which earned him the Nobel Prize in Economics. Tim Bollerslev extended this work with GARCH in 1986, creating what became the most widely used volatility model in finance. These models capture the empirical reality that today's volatility depends on yesterday's shocks and yesterday's volatility, creating the persistence and clustering we observe in markets.

Time-varying volatility affects several financial decisions. You need accurate volatility forecasts for Value at Risk calculations, exploit the gap between implied and forecasted volatility, and adjust position sizes based on volatility regimes. By the end of this chapter, you will be able to specify, estimate, and use GARCH models to capture the dynamic nature of financial volatility.