GARCH(1,1) — Volatility Clustering

GARCH(1,1):
r_t = σ_t ε_t
σ²_t = ω + α ε²_{t-1}σ²_{t-1} + β σ²_{t-1}

α+β < 1 for stationarity

ω (baseline var.) 0.001 α (ARCH effect) 0.10 β (GARCH effect) 0.85 Shock magnitude 3.0σ N simulation steps 500

α + β persistence = —

GARCH(1,1) (Bollerslev 1986) models conditional heteroskedasticity: the variance σ²_t depends on past squared returns and past variance. This produces volatility clustering — calm periods followed by turbulent ones — as seen in real financial markets. The unconditional variance is ω/(1−α−β). The distribution of returns has fat tails (excess kurtosis > 0), meaning extreme events are far more probable than a Gaussian model predicts. Value at Risk (VaR) at 95% confidence = −1.645σ_t gives the loss not exceeded 95% of the time. The return distribution plot shows GARCH (orange) vs standard normal (green) for comparison.

GARCHVolatility clusteringFat tailsVaRFinancial econometrics

GARCH(1,1) — Volatility Clustering & Fat Tails