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

Question

AcadiFi · Accepted Answer

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.

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

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