Advanced✓ Mathematical
◆ The PatternModeling volatility that clusters
Financial returns exhibit volatility clustering — calm periods and turbulent periods tend to persist. GARCH(1,1) models the conditional variance as a function of past squared returns and past variance, capturing this self-exciting behaviour.
σ2t = ω + αε2t−1 + βσ2t−1
α captures the reaction to recent shocks. β captures persistence of volatility. α + β close to 1 means high persistence.
// Interactive — GARCH volatility clustering
α (reaction)0.10
β (persistence)0.85
# Python — GARCH(1,1) from arch import arch_model model = arch_model(returns, vol='Garch', p=1, q=1) result = model.fit(disp='off') print(result.summary()) # Forecast next 5 periods of volatility fcast = result.forecast(horizon=5)
Pattern bridge: GARCH is the quantitative foundation behind Value-at-Risk calculations and Bollinger Bands. The same volatility clustering appears in heteroscedastic data across ML problems.