GARCH Volatility Forecasting Fundamentals

Why it matters

Financial markets exhibit volatility clustering—periods of high volatility tend to follow high volatility, and calm periods follow calm periods. GARCH (Generalized Autoregressive Conditional Heteroskedasticity) models capture this stylized fact, making them essential for risk management, option pricing, and portfolio allocation. Understanding how volatility persists after shocks helps you forecast Value-at-Risk and size positions appropriately.

Core intuition

Think of volatility like weather patterns: a stormy day makes another stormy day more likely, but eventually the storm passes. GARCH models formalize this by making today's volatility forecast depend on:

Yesterday's volatility forecast (persistence)
Yesterday's actual shock (reactivity to news)

The model captures how quickly volatility returns to its long-run average after a shock—this decay rate determines the "half-life" of volatility spikes.

Main content

The GARCH(1,1) Model

The standard GARCH(1,1) specification for conditional variance $\sigma_t^2$ is:

\sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2

where:

$\omega$ = long-run variance baseline (constant term)
$\alpha$ = weight on yesterday's squared shock (ARCH term)
$\beta$ = weight on yesterday's variance forecast (GARCH term)
$\epsilon_t$ = return shock at time $t$

Key constraint:

\alpha + \beta < 1

ensures stationarity (volatility doesn't explode)

Volatility Clustering

When $\alpha + \beta$ is close to 1 (typically 0.95-0.99 for daily financial returns), volatility shocks persist for many periods. This high persistence creates the clustering effect we observe in markets.

Half-Life of Volatility

The half-life measures how long it takes for a volatility shock to decay to half its initial impact:

\text{Half-life} = \frac{\ln(0.5)}{\ln(\alpha + \beta)}

Example: If

\alpha + \beta = 0.95

\text{Half-life} = \frac{\ln(0.5)}{\ln(0.95)} \approx 13.5 \text{ days}

A shock to volatility takes roughly two weeks to halve in impact.

Multi-Step Forecasting

To forecast variance $h$ steps ahead, GARCH produces:

\sigma_{t+h|t}^2 = \bar{\sigma}^2 + (\alpha + \beta)^h (\sigma_{t+1|t}^2 - \bar{\sigma}^2)

where $\bar{\sigma}^2 = \omega / (1 - \alpha - \beta)$ is the unconditional variance.

As $h$ increases, the forecast converges to the long-run average at rate $(\alpha + \beta)^h$ .

In quant finance

Value-at-Risk Estimation

The 99% daily VaR for a position with exposure $X$ is commonly estimated as:

\text{VaR}_{99\%} = 2.33 \times \sigma_t \times X

where

\sigma_t

comes from the GARCH forecast and 2.33 is the 99th percentile of the standard normal distribution.

Why GARCH matters here: During volatile periods,

\sigma_t

increases, automatically raising VaR limits and reducing position sizes—exactly when risk is highest.

Practical Applications

Risk budgeting: Scale position sizes by $1/\sigma_t$ to target constant volatility exposure
Option trading: GARCH forecasts help identify mispriced implied volatility
Stop-loss placement: Widen stops when GARCH forecasts elevated volatility
Regime detection: Persistent high $\sigma_t$ signals a volatility regime shift

Model Limitations

Assumes symmetric response to positive/negative shocks (leverage effects ignored)
Relies on normality assumption for return innovations
Parameters may be unstable across long samples
Doesn't capture jumps or structural breaks

Summary

GARCH models provide a systematic framework for forecasting time-varying volatility by capturing:

Clustering: high (low) volatility persists through the $\alpha + \beta$ terms
Mean reversion: volatility eventually returns to long-run average $\bar{\sigma}^2$
Shock persistence: measured by half-life $\ln(0.5)/\ln(\alpha + \beta)$

The model's