GARCH Model

The GARCH model is one of the most widely used tools in quantitative finance for modelling and forecasting volatility — how much asset returns swing around over time. Because volatility isn't constant, GARCH gives a way to capture its changing behaviour. This guide explains what the GARCH model is, the intuition behind it, the formula, and its uses — in clear, plain language. It complements our guide to the exponentially weighted moving average and is relevant to anyone studying quantitative finance, econometrics or risk.

What is the GARCH model?

GARCH stands for Generalised Autoregressive Conditional Heteroskedasticity — a mouthful, but each part has meaning. "Heteroskedasticity" means non-constant variance (volatility that changes over time); "conditional" means the variance depends on past information; and "autoregressive" means it depends on its own past values. Put together, a GARCH model is a way of describing how the variance (volatility) of a time series changes over time, based on its recent history. It was introduced by Tim Bollerslev in 1986, generalising Robert Engle's earlier ARCH model from 1982 — work for which Engle later won the Nobel Prize in Economics.

The intuition: volatility clustering

The key observation GARCH captures is volatility clustering: in financial markets, calm periods tend to be followed by calm periods, and turbulent periods by more turbulence. Big price moves cluster together; small ones cluster together. In other words, today's volatility depends partly on how volatile things were recently. A model with constant volatility can't capture this, but GARCH can — it lets the variance respond to recent shocks and recent volatility, producing the realistic pattern of quiet and stormy spells seen in real markets.

The GARCH(1,1) formula

The most common version is GARCH(1,1), which models the conditional variance as:

σ²_t = ω + α · ε²_t−1 + β · σ²_t−1

Here σ²_t is today's variance, ε²_t−1 is last period's squared "shock" (return surprise), and σ²_t−1 is last period's variance. The parameter ω is a constant, α weights the recent shock, and β weights the recent variance. So today's volatility is a blend of a baseline level, how big the last move was, and how volatile things already were. The "(1,1)" simply means one lag of each term; higher-order versions like GARCH(p,q) use more lags, but GARCH(1,1) is usually enough to fit financial returns well.

Persistence and the long-run level

A useful feature is that the sum α + β measures the persistence of volatility. When this sum is close to 1, shocks to volatility fade only slowly — volatility stays elevated for a long time after a big move. The model also implies a long-run (unconditional) variance of ω ÷ (1 − α − β), toward which volatility tends to mean-revert over time. This mean reversion is realistic: after a spike, volatility eventually settles back toward its long-run average rather than staying high or low forever.

How GARCH relates to EWMA

The exponentially weighted moving average (EWMA) model is actually a special case of GARCH(1,1) — the case where ω = 0 and α + β = 1. EWMA gives a simple, single-parameter way to update volatility, but it has no mean reversion (no long-run level). GARCH is more flexible: by including ω and allowing α + β < 1, it adds a long-run variance and mean reversion, which often fits financial data better.

What the GARCH model is used for

GARCH models are used wherever changing volatility matters. They forecast volatility for risk management; they feed into Value at Risk (VaR) calculations; they help with option pricing, where volatility is a key input; and they're used in portfolio and trading decisions that depend on expected risk. They're a workhorse of financial econometrics precisely because they capture, simply and effectively, the volatility clustering and mean reversion seen throughout financial markets.

Frequently asked questions

What is the GARCH model?

A model of how the variance (volatility) of a time series changes over time based on its recent history — Generalised Autoregressive Conditional Heteroskedasticity, introduced by Bollerslev in 1986.

What is the GARCH(1,1) formula?

σ²_t = ω + αε²_t−1 + βσ²_t−1: today's variance depends on a constant, last period's squared shock, and last period's variance.

What does GARCH capture?

Volatility clustering (calm and turbulent periods cluster together) and mean reversion of volatility toward a long-run level — both of which constant-volatility models miss.

What is GARCH used for?

Forecasting volatility, Value at Risk, option pricing, and risk and portfolio decisions — anywhere changing volatility is important.

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