GARCH

Market volatility comes in clusters: calm periods tend to stay calm, and turbulent periods tend to stay turbulent. GARCH models, short for generalised autoregressive conditional heteroskedasticity, capture this pattern by letting today's volatility depend on recent shocks and recent volatility. Robert Engle introduced the ARCH model in 1982, work that earned him a share of the 2003 Nobel Memorial Prize in Economic Sciences, and Tim Bollerslev generalised it to GARCH in 1986. GARCH models are widely used for volatility forecasting, risk management and option analysis.

Volatility clustering#

Daily returns themselves show little autocorrelation, but squared or absolute returns show strong positive autocorrelation. That means the size of moves is predictable even if their direction is not. GARCH models exploit exactly this. See Autocorrelation and Partial Autocorrelation.

The GARCH(1,1) model#

r_t = μ + ε_t,   ε_t = σ_t × z_t
σ_t² = ω + α × ε_(t-1)² + β × σ_(t-1)²

Long run (unconditional) variance:

long run variance = ω / (1 - α - β)

Parameters are estimated by maximum likelihood. See Maximum Likelihood.

Worked example#

Variations#

Uses in trading#

Limits#

• Volatility forecasts are not direction forecasts.
• Sudden jumps from news are not anticipated.
• Parameters change across regimes. See Structural Breaks and Regime Changes.
• Implied volatility often contains information beyond GARCH, such as known upcoming events. See Implied Volatility (IV).
• Daily data limits: high frequency realised volatility models can forecast better in some settings. See Historical and Realized Volatility.

Fitting GARCH in practice#

In Python, the arch package fits GARCH models in a few lines. Use daily returns in percent, which helps the optimiser, and compare a normal and a Student t error distribution. Check that α + β is below 1, that the standardised residuals show no remaining volatility clustering, and that out of sample forecasts beat a simple rolling volatility estimate. If they do not, the simpler method is usually the better choice. See Python for Trading.

GARCH and volatility targeting#

Some funds target a constant level of portfolio volatility, such as 10% a year. They use a forecast, often GARCH or EWMA based, and scale positions down when forecast volatility rises and up when it falls. Research has found that volatility scaling can improve risk adjusted returns for some assets, because high volatility periods have often coincided with poor returns, although results vary by asset and period.

Frequently asked questions#

What is a GARCH model?#

A volatility model in which today's variance depends on yesterday's squared return and yesterday's variance, capturing volatility clustering.

What does persistence mean in GARCH?#

The sum α + β, showing how slowly volatility shocks fade; values close to 1 mean high volatility lasts a long time.

Can GARCH predict market direction?#

No. GARCH forecasts the size of moves (volatility), not whether prices will rise or fall.

Next, learn how money grows over time in Compounding and Geometric vs Arithmetic Returns.