GARCH Volatility Models

Returns have almost no forecastable mean, but their volatility is highly forecastable, calm days cluster with calm days, storms with storms. GARCH models are the standard machinery for that fact. They take the one robust empirical regularity of financial time series (volatility clustering) and turn it into a parametric, estimable, forecastable process. Engle's ARCH and Bollerslev's GARCH won a Nobel Prize for good reason: they are the backbone of volatility forecasting, risk models, and option-pricing calibration.

The stylized facts they target

Three features recur across essentially every asset and frequency:

1. Volatility clustering, large moves follow large moves. The ACF of returns is near zero, but the ACF of squared returns is strongly positive and slowly decaying.
2. Fat tails, the unconditional return distribution is leptokurtic (kurtosis > 3), with more extreme events than a normal.
3. Leverage effect, negative returns raise future volatility more than equal positive returns (in equities).

GARCH is engineered to reproduce the first two automatically, and its asymmetric variants capture the third.

From ARCH to GARCH

Model the return as $r_t = \mu + \varepsilon_t$ with $\varepsilon_t = \sigma_t z_t$, where $z_t$ is i.i.d. mean-zero unit-variance ("standardized innovation") and $\sigma_t^2$ is the conditional variance given the past. Engle's ARCH($q$) makes today's variance a function of recent squared shocks:

$\sigma_t^2 = \omega + \sum_{i=1}^q \alpha_i\,\varepsilon_{t-i}^2.$

A big shock yesterday raises variance today, clustering, by construction. But capturing persistent clustering needs many lags. Bollerslev's GARCH($p,q$) adds lagged variance terms, giving an ARMA-like parsimony to the variance:

$\boxed{\ \sigma_t^2 = \omega + \sum_{i=1}^q \alpha_i\,\varepsilon_{t-i}^2 + \sum_{j=1}^p \beta_j\,\sigma_{t-j}^2\ }$

The everyday workhorse is GARCH(1,1):

$\sigma_t^2 = \omega + \alpha\,\varepsilon_{t-1}^2 + \beta\,\sigma_{t-1}^2.$

Three parameters and it fits most return series remarkably well. The $\alpha$ term is the reaction to news (the size of the last shock); the $\beta$ term is the memory (how much of yesterday's variance carries over).

Persistence, stationarity, and the leverage of $\alpha+\beta$

Take expectations of the GARCH(1,1) equation. Since $\mathbb{E}[\varepsilon_{t-1}^2] = \mathbb{E}[\sigma_{t-1}^2] = \bar\sigma^2$ (the unconditional variance) in the stationary case,

$\bar\sigma^2 = \omega + \alpha\bar\sigma^2 + \beta\bar\sigma^2 \ \Longrightarrow\ \bar\sigma^2 = \frac{\omega}{1 - \alpha - \beta}.$

For this to be positive and finite the stationarity condition is $\alpha + \beta < 1$ (with $\omega, \alpha, \beta \ge 0$). The sum $\alpha + \beta$ is the persistence: it governs how slowly a volatility shock decays back to $\bar\sigma^2$. Empirically it is close to 1 for equities (often 0.97–0.99), meaning volatility shocks are extremely persistent, a spike today still elevates variance months out. In the limit $\alpha + \beta = 1$ you get IGARCH (integrated GARCH): the unconditional variance is undefined, shocks are permanent, and RiskMetrics' exponentially-weighted moving average is exactly the $\omega = 0$ special case. GARCH also generates fat tails automatically: even with Gaussian $z_t$, mixing over time-varying $\sigma_t^2$ produces unconditional kurtosis above 3.

Forecasting the variance term structure

The multi-step forecast reverts geometrically toward $\bar\sigma^2$ at rate $(\alpha+\beta)$:

$\mathbb{E}_t[\sigma_{t+h}^2] = \bar\sigma^2 + (\alpha+\beta)^{h-1}\big(\sigma_{t+1}^2 - \bar\sigma^2\big).$

This is the volatility term structure: when current vol is above its long-run level, near-dated forecasts are high and the curve slopes down to $\bar\sigma^2$; when below, it slopes up. The mean-reversion speed, hence the whole shape, is set by the persistence $\alpha+\beta$, which is exactly why the estimate of that sum matters so much for pricing longer-dated options and for scaling VaR across horizons.

Estimation and the worked example

GARCH is fit by maximum likelihood. Under a Gaussian innovation the log-likelihood (dropping constants) is

$\ell(\theta) = -\frac12\sum_{t}\Big(\log\sigma_t^2(\theta) + \frac{\varepsilon_t^2}{\sigma_t^2(\theta)}\Big),$

with $\sigma_t^2$ recursively built from the data given $(\omega,\alpha,\beta)$. Numerically maximizing gives the estimates and information-matrix standard errors. Suppose a daily equity GARCH(1,1) fit returns $\omega = 2\times10^{-6}$, $\alpha = 0.08$, $\beta = 0.90$. Persistence is $0.98$, very sticky. The unconditional daily variance is $\bar\sigma^2 = 2\times10^{-6}/0.02 = 10^{-4}$, i.e. a daily vol of $1\%$, about $16\%$ annualized ($1\%\times\sqrt{252}$). A shock has a half-life of $\log(0.5)/\log(0.98)\approx 34$ trading days, a vol spike takes roughly seven weeks to half-decay. These numbers are typical and are exactly what a risk desk reads off the fit.

The extensions desks actually use

• GARCH-$t$, replace Gaussian $z_t$ with Student-$t$ innovations to match the fat tails that Gaussian GARCH still understates; standard for VaR.
• GJR-GARCH / TARCH, add an asymmetry term $\gamma\,\varepsilon_{t-1}^2\mathbf 1\{\varepsilon_{t-1}<0\}$ so negative shocks raise vol more: the leverage effect.
• EGARCH, models $\log\sigma_t^2$, guaranteeing positivity without parameter constraints and allowing asymmetry.

Failure modes in financial data

• Spurious IGARCH from structural breaks. A regime shift in the level of volatility inflates estimated persistence toward 1; you can mistake a break for genuine long memory and over-forecast persistence. See Stationarity and Regime Detection.
• Gaussian innovations understate tails. Even fitted GARCH with normal $z_t$ underestimates extreme-quantile risk; use $t$ or empirical innovations for VaR/ES.
• Jumps and overnight gaps. GARCH assumes a smooth diffusion of variance; discrete jumps (earnings, macro prints, gaps) are not well captured and can dominate risk.
• It forecasts variance, not direction. GARCH says nothing about the sign of returns, useful for sizing and risk, not for a directional signal. Do not confuse volatility forecastability with return forecastability.

In interviews

Write the GARCH(1,1) recursion from memory and state the two headline results: unconditional variance $\bar\sigma^2 = \omega/(1-\alpha-\beta)$ and stationarity requires $\alpha+\beta<1$. Explain persistence $\alpha+\beta$ as the volatility mean-reversion speed, why it's near 1 for equities, and what IGARCH / EWMA mean in the limit. Be able to say why GARCH exists, returns' mean is unforecastable but squared returns are strongly autocorrelated (clustering), connecting it to the ARMA discussion, and that GARCH generates fat tails even from Gaussian shocks. Naming GJR/EGARCH for the leverage effect and GARCH-$t$ for the tails signals desk-level familiarity. A common numeric ask is the vol-shock half-life, $\log(0.5)/\log(\alpha+\beta)$.