How do GARCH models capture volatility clustering, and when are they better than historical volatility?

GARCH (Generalized Autoregressive Conditional Heteroskedasticity) models are the industry standard for modeling time-varying volatility. The key insight they capture is **volatility clustering** — the empirical observation that large returns tend to be followed by large returns, and small returns by small returns. **GARCH(1,1) Model:** **σ²_t = ω + α × r²_{t-1} + β × σ²_{t-1}** Where: - **σ²_t** = today's conditional variance forecast - **ω** (omega) = long-run variance weight (constant) - **α** (alpha) = weight on yesterday's squared return (the "shock" or "news" coefficient) - **β** (beta) = weight on yesterday's variance (the "persistence" coefficient) - **r²_{t-1}** = yesterday's squared return - **σ²_{t-1}** = yesterday's conditional variance **Constraint:** α + β < 1 for the process to be stationary (mean-reverting). **Intuitive Interpretation:** Think of it as: "Today's volatility is a weighted average of the long-run average volatility, yesterday's surprise, and yesterday's volatility level." **Worked Example:** Lancaster Risk Analytics estimates GARCH(1,1) parameters for the S&P 500: - ω = 0.000002 - α = 0.08 - β = 0.90 Yesterday's return was -3.2% (a large negative shock), and yesterday's volatility estimate was σ_{t-1} = 1.2% daily. **Step 1: Calculate components** - r²_{t-1} = (-0.032)² = 0.001024 - σ²_{t-1} = (0.012)² = 0.000144 **Step 2: Apply GARCH formula** - σ²_t = 0.000002 + 0.08 × 0.001024 + 0.90 × 0.000144 - σ²_t = 0.000002 + 0.0000819 + 0.0001296 - σ²_t = 0.0002135 - **σ_t = √0.0002135 = 1.46% daily** The volatility jumped from 1.2% to 1.46% because of the large negative return yesterday. This is volatility clustering in action. **Long-Run Variance:** - V_L = ω / (1 - α - β) = 0.000002 / (1 - 0.08 - 0.90) = 0.000002 / 0.02 = 0.0001 - Long-run daily volatility = √0.0001 = **1.0%** (or ~15.8% annualized) - Current forecast (1.46%) is above long-run, so it will gradually mean-revert **Persistence:** - α + β = 0.98 → very high persistence, meaning volatility shocks decay slowly - Half-life of a volatility shock ≈ ln(2) / ln(α + β) ≈ 0.693 / 0.0202 ≈ **34 days** **GARCH vs. Simple Historical Volatility:** | Feature | Historical Vol | GARCH(1,1) | |---|---|---| | Weights on observations | Equal | Exponentially decaying | | Captures clustering | No | Yes | | Responds to shocks | Slowly (window-dependent) | Immediately | | Mean reversion | Implicit | Explicit (ω parameter) | | VaR accuracy | Underestimates during crises | Better tail estimates | **When to Use GARCH:** - Risk management and VaR calculations (regulatory requirement under Basel) - Options pricing when implied volatility is unavailable - Any situation where volatility is clearly time-varying **EWMA as a Special Case:** EWMA is GARCH with ω = 0 and α + β = 1 (integrated GARCH). This means no mean reversion — once volatility spikes, the model doesn't pull it back toward a long-run average. RiskMetrics uses λ = 0.94 (so α = 0.06, β = 0.94). **Exam Tip:** FRM Part I frequently asks you to calculate the next-day variance using GARCH(1,1) given the parameters and yesterday's data. Practice the arithmetic — it's straightforward but easy to botch under time pressure. Explore more quantitative analysis topics in our FRM Part I course.