GARCH Volatility Clustering Lab (2D)

GARCH(1,1) Parameters

Long-run level ω 0.05

Shock reaction α 0.12

Volatility persistence β 0.75

Display

Tick speed 8/s

Visible bars 90

Drag left/right inside the chart to scrub through the history buffer (up to 500 ticks). Drag back to the right edge — or hit "Jump to live" — to resume auto-scroll.

Live Readouts

Conditional volatility σₜ

0.00%

Persistence α+β

0.87

Long-run variance

0.38

ACF of εₜ² at lag 1 (clustering signal)

0.00

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.
Top panel: bars are the simulated daily return εₜ (colour = current σₜ, blue=calm, red=turbulent); the two curves trace the ±σₜ envelope. Middle panel: σₜ itself over time, against the long-run level. Bottom panel: the sample autocorrelation function (ACF) of εₜ², computed live over the full 500-tick history — the textbook signature of volatility clustering is that this stays positive for many lags instead of vanishing at lag 1 like it would for a plain white-noise return series.

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)

live