How do ARCH and GARCH models capture volatility clustering, and how do you estimate them?

GARCH (Generalized ARCH) models are the industry standard for modeling time-varying volatility. The GARCH(1,1) specification captures volatility clustering with just three parameters.

GARCH(1,1) Model:

Mean equation: r_t = mu + epsilon_t, where epsilon_t = sigma_t x z_t, z_t ~ N(0,1)

Variance equation: sigma_t^2 = omega + alpha x epsilon_{t-1}^2 + beta x sigma_{t-1}^2

Where:

• omega > 0: baseline variance level
• alpha >= 0: reaction to new shocks (ARCH term)
• beta >= 0: persistence of past variance (GARCH term)
• alpha + beta < 1: required for stationarity

Parameter Interpretation:

• alpha measures how strongly today's volatility reacts to yesterday's surprise. High alpha = 'jumpy' volatility.
• beta measures how persistent volatility is. High beta = volatility decays slowly after a shock.
• alpha + beta = total persistence. Closer to 1 means shocks to volatility last longer.

Long-Run Variance:

sigma_LR^2 = omega / (1 - alpha - beta)

Estimation via MLE:

GARCH cannot be estimated by OLS. You must use Maximum Likelihood:

1. Assume z_t ~ N(0,1)
2. Log-likelihood: ln L = Sum of [-0.5 x ln(2*pi) - 0.5 x ln(sigma_t^2) - 0.5 x epsilon_t^2/sigma_t^2]
3. Numerically optimize over (omega, alpha, beta)

Example — Havenbrook Capital, S&P 500 daily returns (2020-2025):

Volatility Forecasting:

Multi-step ahead forecast reverts toward long-run variance:

sigma_{t+h}^2 = sigma_LR^2 + (alpha + beta)^h x (sigma_t^2 - sigma_LR^2)

FRM Key Points:

• GARCH(1,1) captures ~90% of volatility dynamics for most assets
• RiskMetrics EWMA is a special case with alpha + beta = 1 and omega = 0 (no mean reversion)
• EGARCH and GJR-GARCH add asymmetry (leverage effect — negative returns increase vol more than positive)
• Always check alpha + beta < 1 for stationarity; if >= 1, the model is IGARCH (integrated)

Master GARCH modeling in our FRM Part I Quantitative module.