Volatility & GARCH Models

Modeling the changing volatility of financial returns.

Volatility is Not Constant

A key observation in financial markets is that volatility is not constant over time. Periods of high volatility (large price swings) tend to be followed by more high volatility, and periods of low volatility tend to be followed by more low volatility. This phenomenon is known as **volatility clustering**.

Standard time series models assume constant variance (homoscedasticity), which is a poor fit for financial data. Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models were developed to address this by explicitly modeling the variance as a time-varying process.

The GARCH(1,1) Model

The most common GARCH model, GARCH(1,1), models the next period's variance (σt2\sigma^2_{t}) as a combination of a long-term average variance, the previous period's squared return (the ARCH term), and the previous period's variance (the GARCH term).

σt2=ω+αat−12+βσt−12\sigma^2_t = \omega + \alpha a^2_{t-1} + \beta\sigma^2_{t-1}
  • ω\omega is a constant term.
  • α\alpha (alpha) governs the reaction to past shocks. A large alpha means volatility reacts intensely to market movements.
  • β\beta (beta) governs the persistence of volatility. A large beta means shocks to volatility take a long time to die out.
Interactive GARCH Simulation
Adjust the ARCH (α\alpha) and GARCH (β\beta) parameters to see their effect on the simulated returns and the modeled volatility.