Lesson 04

GARCH

Volatility is not constant. Markets alternate between calm and turbulent periods — a property called volatility clustering. GARCH models this directly by letting today's variance depend on yesterday's variance and yesterday's shock.

Volatility clustering

Mandelbrot observed in 1963 that large price changes tend to be followed by large changes (of either sign), and small by small. This violates the constant-variance assumption of ARIMA and is one of the most robust stylised facts in finance.

Simulated returns with alternating low-vol and high-vol regimes

■ Low vol ■ High vol

The key insight. Volatility is autocorrelated. Yesterday's volatility is the best predictor of today's volatility. GARCH formalises this observation into a tractable model with maximum likelihood estimation.

ARCH effects

Before GARCH came ARCH (Autoregressive Conditional Heteroskedasticity), introduced by Engle (Nobel 2003). ARCH(q) lets conditional variance depend on the last q squared residuals.

ARCH(q): σ²_t = ω + α₁ε²_t−1 + α₂ε²_t−2 + ... + α_qε²_t−q where ε_t = σ_t·z_t, z_t ~ N(0,1) Constraints: ω > 0, α_k ≥ 0, Σα_k < 1 (stationarity) Engle's ARCH-LM test: Regress ε²_t on its own lags → F-test H₀: no ARCH effects (α₁ = ... = α_q = 0)

ARCH test in practice. Fit an AR model to the mean (even AR(0) = demeaning suffices). Square the residuals. Run an OLS regression of ε²_t on ε²_t−1, ..., ε²_t−q. A significant F-statistic means ARCH effects are present and a GARCH model is warranted.

GARCH(1,1)

GARCH extends ARCH by also including lagged conditional variance. GARCH(1,1) is the workhorse of financial volatility modelling — one extra parameter (β) that captures persistence far more parsimoniously than high-order ARCH.

GARCH(1,1): Return: r_t = μ + ε_t Innovation: ε_t = σ_t·z_t, z_t ~ N(0,1) Variance: σ²_t = ω + α·ε²_t−1 + β·σ²_t−1 ω > 0 : long-run average variance weight α ≥ 0 : ARCH term — how strongly new shocks raise volatility β ≥ 0 : GARCH term — how long volatility persists α + β < 1 : stationarity condition Long-run variance: σ²_∞ = ω / (1 − α − β)

Typical equity estimates: α ≈ 0.10, β ≈ 0.85. The high β means volatility is highly persistent — it decays slowly after a shock. α + β ≈ 0.95 is common, meaning the unconditional variance barely exceeds the current level. In a crisis, elevated volatility can persist for months.

Interactive GARCH simulation

Adjust α and β to see how each parameter shapes the volatility dynamics. Watch the conditional σ_t (orange) respond to shocks in the returns (above).

α — ARCH term (shock sensitivity) 0.10

β — GARCH term (persistence) 0.85

Returns (top) and conditional volatility σ_t (bottom)

Try extreme values. Set α = 0.28, β = 0.70: each shock produces a big spike but volatility decays quickly. Set α = 0.05, β = 0.93: smaller spikes but volatility stays elevated for many bars after a shock — matching the slow decay seen in equity crises.

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