GARCH doesn't just fit volatility — it forecasts it. Multi-step variance forecasts converge to the long-run level. Value at Risk adapts dynamically to current conditions. Asymmetric extensions capture the leverage effect seen in equity markets.
Multi-step variance forecasts
From the GARCH(1,1) recursion, h-step ahead forecasts can be derived analytically. They converge to the long-run variance exponentially at rate (α + β) per step.
h-step ahead forecast from time t:
σ²t+1|t = ω + α·ε²t + β·σ²t (1-step)
σ²t+h|t = σ²∞ + (α+β)^{h−1} · (σ²t+1|t − σ²∞)
σ²∞ = ω / (1 − α − β) ← long-run variance
As h → ∞: σ²t+h|t → σ²∞ (mean reversion in variance)
Conditional variance history → 30-step forecast converging to σ²∞ (dashed)
Half-life of a variance shock. With α = 0.10, β = 0.85 (α+β = 0.95), the half-life of a variance shock is log(0.5) / log(0.95) ≈ 14 bars. A spike in volatility takes about 14 trading days to decay halfway back to normal. A crash in 2008 that tripled volatility took months to fully normalise.
Value at Risk
VaRα is the loss not exceeded with probability 1−α over a given horizon. Under GARCH, VaR adapts to current volatility — it's tighter in calm periods and wider during turbulence.
1-day VaR at 95% confidence under GARCH:
VaR95%(t+1) = μ + σt+1|t · Φ⁻¹(0.05)
= μ − 1.645 · σt+1|t
Φ⁻¹(0.05) = −1.645 is the 5th percentile of N(0,1)
10-day VaR (proper GARCH calculation):
VaR10d = μ·10 + √(Σ σ²t+k|t) · Φ⁻¹(0.05) k=1..10
(√10 scaling underestimates in GARCH — volatility clusters)
Return distributions: normal-vol day vs high-vol day — VaR 95% marked
Dynamic VaR in the 2008 crisis. Static VaR models (using historical average volatility) catastrophically underestimated risk — because they used the calm-period average σ. GARCH-based dynamic VaR adapts: as market turmoil begins σt+1|t rises and VaR warnings increase before losses compound. Basel III now requires dynamic models.
Asymmetric GARCH — the leverage effect
Equity markets show a persistent asymmetry: negative returns raise volatility more than equivalent positive returns. This "leverage effect" (Black, 1976) is missed by symmetric GARCH(1,1).
News impact curve: σ² response to a shock ε_{t-1} — symmetric GARCH vs GJR-GARCH
The leverage effect in practice. For major equity indices (S&P 500, FTSE, DAX), γ is typically 0.08–0.15. A −2% return raises volatility about twice as much as a +2% return. For FX and commodities the effect is weaker or absent — there is no "short position" acting as leverage.