# GARCH

> GARCH models capture volatility clustering, where big moves follow big moves. Learn the GARCH(1,1) formula, persistence, forecasting and uses in risk and options.

Source: https://learn.tradelabsai.com/math/garch/  
Track: Math and Statistics · Level: Advanced · Updated: 2026-10-03  
Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice.  
Cite as: TradeLabs Learn, "GARCH", https://learn.tradelabsai.com/math/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](https://learn.tradelabsai.com/math/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
```

| Parameter | Meaning |
|---|---|
| ω (omega) | Baseline level |
| α (alpha) | Reaction to yesterday's shock |
| β (beta) | Persistence of yesterday's variance |
| α + β | Overall persistence; close to 1 means shocks fade slowly |

Long run (unconditional) variance:

```
long run variance = ω / (1 - α - β)
```

Parameters are estimated by maximum likelihood. See [Maximum Likelihood](https://learn.tradelabsai.com/math/maximum-likelihood/).

## Worked example

**Example: A volatility forecast**
Fitted daily GARCH(1,1) parameters for an index: ω = 0.000002, α = 0.09, β = 0.89. Persistence α + β = 0.98.

Long run daily variance = 0.000002 / 0.02 = 0.0001, so long run daily volatility = 1%, or about 15.9% annualised.

Yesterday's variance estimate was 0.0002 (1.41% daily) and yesterday's return was minus 3% (ε² = 0.0009).

Today's variance = 0.000002 + 0.09 × 0.0009 + 0.89 × 0.0002 = 0.000002 + 0.000081 + 0.000178 = 0.000261, so today's volatility ≈ 1.62%, about 25.6% annualised. The large shock pushed volatility up; with persistence of 0.98, it will decay back toward 1% only gradually, with a half life of about ln(0.5) / ln(0.98) ≈ 34 days.

## Variations

| Model | Adds | Why |
|---|---|---|
| EGARCH (Nelson, 1991) | Log variance and asymmetric effects | Falls raise volatility more than rises |
| GJR GARCH (1993) | Extra term for negative shocks | Captures the leverage effect in stocks |
| GARCH with Student t errors | Fat tailed shocks | More realistic tails. See [Student's t-Distribution](https://learn.tradelabsai.com/math/students-t-distribution/) |
| Multivariate GARCH (DCC) | Time varying correlations | Portfolio risk. See [Covariance and Correlation](https://learn.tradelabsai.com/math/covariance-and-correlation/) |
| EWMA (RiskMetrics) | A simple special case with fixed weights | Easy, widely used. See [Rolling and Expanding Windows](https://learn.tradelabsai.com/math/rolling-and-expanding-windows/) |

## Uses in trading

| Use | Lesson |
|---|---|
| Volatility forecasts for position sizing | [Volatility and ATR-Based Sizing](https://learn.tradelabsai.com/risk/volatility-and-atr-based-sizing/) |
| Value at risk and expected shortfall | [Value at Risk (VaR)](https://learn.tradelabsai.com/portfolio/value-at-risk/), [Expected Shortfall (CVaR)](https://learn.tradelabsai.com/portfolio/expected-shortfall/) |
| Comparing implied and forecast volatility | [Volatility Arbitrage](https://learn.tradelabsai.com/volatility/volatility-arbitrage/) |
| Volatility targeting portfolios | [Risk Budgeting and Risk Parity](https://learn.tradelabsai.com/portfolio/risk-budgeting-and-risk-parity/) |
| Simulating realistic return paths | [Monte Carlo Simulation](https://learn.tradelabsai.com/research/monte-carlo-simulation/) |

## Limits

- **Volatility forecasts are not direction forecasts.**
- **Sudden jumps** from news are not anticipated.
- **Parameters change** across regimes. See [Structural Breaks and Regime Changes](https://learn.tradelabsai.com/math/regime-changes/).
- **Implied volatility** often contains information beyond GARCH, such as known upcoming events. See [Implied Volatility (IV)](https://learn.tradelabsai.com/volatility/implied-volatility/).
- **Daily data limits:** high frequency realised volatility models can forecast better in some settings. See [Historical and Realized Volatility](https://learn.tradelabsai.com/volatility/historical-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](https://learn.tradelabsai.com/programming/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](https://learn.tradelabsai.com/math/compounding/).

## Continue learning

- Next lesson: [Compounding and Geometric vs Arithmetic Returns](https://learn.tradelabsai.com/math/compounding/)
- Previous lesson: [ARIMA](https://learn.tradelabsai.com/math/arima/)
- Related: [ARIMA](https://learn.tradelabsai.com/math/arima/): ARIMA models forecast a time series from its own past values and errors. Learn the AR, I and MA terms, how to choose orders and why returns are hard to predict.
- Related: [Historical and Realized Volatility](https://learn.tradelabsai.com/volatility/historical-volatility/): Historical volatility measures how much a price actually moved, using past returns. Learn the standard formula, range based estimators and how traders use it.
- Related: [Implied Volatility (IV)](https://learn.tradelabsai.com/volatility/implied-volatility/): Implied volatility is the market's forecast of future movement, backed out from option prices. Learn how to read it, convert it to expected moves and use it.
- Related: [Autocorrelation and Partial Autocorrelation](https://learn.tradelabsai.com/math/autocorrelation/): Autocorrelation measures how a series relates to its own past values. Learn the formula, the ACF, what positive and negative autocorrelation mean and why it matters.
- Related: [Value at Risk (VaR)](https://learn.tradelabsai.com/portfolio/value-at-risk/): Value at risk estimates the loss a portfolio should not exceed with a given confidence over a set period. Learn the three methods, an example and the limits.
- Related: [Volatility Trading](https://learn.tradelabsai.com/volatility/volatility-trading/): Volatility trading profits from the size of price moves, not their direction. Learn implied vs realised bets, the main instruments and how to manage risk.