GARCH Volatility Clustering Lab

GARCH(1,1) Parameters

Long-run level ω 0.05

Shock reaction α 0.12

Volatility persistence β 0.75

Live Readouts

Conditional volatility σₜ

0.00%

Persistence α+β

0.87

Long-run variance

0.38

How it works

GARCH(1,1) — Generalized AutoRegressive Conditional Heteroskedasticity — models a return series whose variance itself changes over time, the mechanism behind real financial "volatility clustering": calm periods and turbulent periods each tend to persist.

εₜ = σₜ · zₜ,        zₜ ~ N(0,1)
σₜ² = ω + α·εₜ₋₁² + β·σₜ₋₁²
long-run variance = ω / (1 − α − β)   [requires α+β < 1]

ω (omega) — the base variance the process reverts to when nothing unusual is happening.
α (alpha) — how strongly yesterday's squared shock εₜ₋₁² feeds into today's variance. Higher α = sharper, more reactive volatility spikes.
β (beta) — how much of yesterday's variance carries over. Higher β = volatility decays slowly, producing long clustered runs.
Inject shock — forces one very large |z| draw, so you can watch σₜ jump and then relax back toward the long-run level at a rate set by α+β.
Bars are the simulated daily return εₜ (height = ± return, colour + depth = current σₜ); the two curves trace the ±σₜ envelope, visibly widening during clustered turbulence and narrowing during calm stretches.

This is the model at the core of real risk engines (option pricing, Value-at-Risk, margin calculators) — the same equation the article's "Finance" application section points to when it mentions forecasting volatile markets.

Calm (low σₜ) Turbulent (high σₜ)

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