SV · Volatility estimators
GARCH
Also called: GARCH(1,1), generalized autoregressive conditional heteroskedasticity, volatility clustering model
GARCH models volatility that clusters: today's variance depends on yesterday's squared shock and yesterday's variance, reverting to a long-run level.
Definition
GARCH, generalized autoregressive conditional heteroskedasticity, is a family of time-series models in which the variance of returns changes over time and depends on past shocks and past variance. Bollerslev (1986) generalized Engle's 1982 ARCH model; GARCH(1,1) is by far the most used specification.
It captures volatility clustering: large moves tend to be followed by large moves, and calm periods by calm periods.
Intuition
Markets do not have one volatility. After a shock, volatility jumps and then decays back towards a normal level over days or weeks. GARCH turns that pattern into a recursion: tomorrow's variance is a weighted blend of a long-run average, the latest squared surprise, and today's variance.
That makes it useful for volatility forecasts, risk limits and position sizing that adapt to conditions without the lag of a long rolling window.
How it's calculated
For returns rt = μ + εt with εt = σt zt and zt independent with unit variance:
σt² = ω + α · εt−1² + β · σt−1²
with ω > 0, α ≥ 0, β ≥ 0. The process is covariance-stationary when α + β < 1, and the long-run variance is ω / (1 − α − β). The sum α + β is the persistence; the half-life of a variance shock is ln(0.5) / ln(α + β) periods.
Parameters are estimated by maximum likelihood, often with a Student-t distribution for zt to allow fat tails. Variants add asymmetry, such as GJR-GARCH and EGARCH, so negative shocks raise volatility more than positive ones.
How to read it
Typical daily fits on equity indices show α small and β large, with α + β close to one: shocks have a modest immediate effect but decay slowly. A k-step-ahead forecast converges geometrically towards the long-run variance.
Compare the GARCH forecast with implied volatility for the same horizon to frame whether options look rich or cheap relative to a model of recent behavior.
Common mistakes
- Fitting on too short a sample; persistence estimates are unstable.
- Ignoring asymmetry in equities, where down moves raise volatility more than up moves.
- Treating α + β very close to one as harmless; forecasts then barely mean-revert and long-run variance is poorly identified.
- Using a normal distribution for zt when returns are clearly fat-tailed.
- Reading the long-run variance as today's variance; the conditional forecast can sit far above or below it.
Go deeper
Saqrid Institute lesson: Volatility Clustering & GARCH(1,1). Lesson text is open to read without an account.