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Home / Financial Tools / GARCH(1,1) Volatility Clustering Estimator

Graduate-Level Modeling · Time Series Econometrics

GARCH(1,1) Volatility Clustering Estimator

Volatility isn't constant, and it isn't random either, it clusters. A big move today makes another big move tomorrow more likely. GARCH(1,1) is the workhorse model that actually captures this, and forecasts exactly how fast it fades.

How To Use This Model

Reading This Tool

Set α and β, the long-run volatility level, and a shock return to simulate.

The chart shows the correct multi-step-ahead GARCH forecast, volatility spikes on the shock day, then mean-reverts back toward the long-run level at a pace set by persistence (α+β).

GARCH Parameters

α+β must stay below 1.0 for the process to be stationary (mean-reverting). If your sliders push the sum to 1.0 or above, β is automatically capped.

Volatility Response To The Shock

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Persistence (α+β)

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Post-Shock Annualized Vol

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Half-Life Of The Shock

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Forecasted Annualized Volatility Path

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What The Half-Life Actually Tells You

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Why Alpha And Beta Trade Off Differently

A high α with low β produces a sharp, short-lived volatility spike, the model reacts strongly to news but forgets it quickly. A low α with high β produces a smaller immediate reaction but far more persistent elevated volatility. Equity index returns typically show low α (0.05-0.10) and high β (0.85-0.92), a slow-fading, highly persistent pattern.

Reading The Forecast Path

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Conditional Heteroskedasticity Modeling

The Core Formulas

σ²t = ω + α·ε²t−1 + β·σ²t−1
Long-run variance: σ²LR = ω / (1−α−β)
h-step forecast: E[σ²t+h] = σ²LR + (α+β)h·(σ²t+1 − σ²LR)

This tool fixes ω by targeting your specified long-run volatility, then applies the correct multi-step-ahead forecast formula, which decays geometrically at rate (α+β) back toward the long-run variance. A naive simulation that simply assumes zero future shocks converges to the wrong asymptote, ω/(1−β) rather than ω/(1−α−β), this tool avoids that error.

When To Actually Use This Model

  • Forecasting near-term volatility for option pricing or risk management after a large market move.
  • Teaching conditional heteroskedasticity and volatility clustering in an econometrics or empirical finance course.
  • Setting time-varying VaR or margin requirements that should widen immediately after a shock and normalize gradually.
  • Illustrating why historical (unconditional) volatility understates near-term risk right after a large market move.

Key Assumptions & Limitations

  • This tool takes α and β as given rather than estimating them by maximum likelihood from an actual return series.
  • GARCH(1,1) assumes symmetric response to positive and negative shocks; EGARCH and GJR-GARCH relax this to capture the leverage effect.
  • Assumes normally distributed innovations; fat-tailed alternatives (Student-t GARCH) are common in practice.
  • A single isolated shock is modeled here; a real return series has continuous shocks arriving every period.

Foundational Reference

Bollerslev, T. (1986). Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics, 31(3), 307-327.

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