Volatility Clustering & GARCH

ARIMA modeled the mean of returns and assumed the variance was constant. That assumption is spectacularly false. Stare at any return chart and you’ll see it: stretches of placid little wiggles, then bursts of violent swings, then calm again. Big moves cluster with big moves; quiet clusters with quiet. This is volatility clustering, and it’s one of the most robust empirical facts in all of finance — and one the ARCH/GARCH family was built to capture. Modeling it is how you forecast tomorrow’s risk, price options, and size positions. If ARIMA is the grammar of the mean, GARCH is the grammar of the variance.

Volatility clustering: the stylized fact

Analogy. Volatility behaves like weather, not climate. You can’t predict whether tomorrow brings rain or sun (the direction), but you absolutely can predict that a storm season is upon you — turbulent days come in runs, and calm days come in runs. Markets are the same: a panicky day is usually followed by more panicky days, and a sleepy day by more sleepy days.

The evidence. Recall from the autocorrelation lesson the tell-tale signature: the ACF of returns is nearly flat (direction unforecastable), but the ACF of squared returns (or absolute returns) is strongly positive and decays slowly over many lags. That slow decay in squared returns is the fingerprint of volatility clustering — today’s squared return helps predict tomorrow’s.

Why it matters. A risk model that assumes constant variance will badly understate risk in turbulent regimes (and overstate it in calm ones). During a crisis, “average” volatility is useless; you need the current, elevated volatility. Volatility clustering means recent volatility is your best guide to near-term volatility — and that’s forecastable structure worth real money in risk management and options pricing.

ARCH: variance that depends on recent shocks

The breakthrough (Robert Engle, 1982, eventually a Nobel) was simple and radical: let the variance itself be a function of recent data. The ARCH model — AutoRegressive Conditional Heteroskedasticity — says today’s variance depends on the size of recent shocks.

Unpacking the name: Heteroskedasticity = non-constant variance. Conditional = given the recent past. Autoregressive = the variance feeds on its own recent history. Put together: the variance, conditional on recent returns, varies over time and depends on its own recent inputs.

Definition. An ARCH(1) model sets the conditional variance to a baseline plus a piece of the last squared shock: $\sigma_t^2 = \omega + \alpha\,\varepsilon_{t-1}^2,$ where $\varepsilon_{t-1}$ is the previous return shock, $\omega > 0$ is a baseline, and $\alpha \geq 0$. The mechanism: a big move last period ($\varepsilon_{t-1}^2$ large) pushes up this period’s variance — clustering, built right in.

The limitation. Pure ARCH needs many lagged squared terms (ARCH(q) with large q) to capture the slow, persistent decay real volatility shows. That’s clunky and over-parameterized — which is exactly the problem GARCH solves.

GARCH(1,1): the workhorse

The fix — Tim Bollerslev, 1986 — was to add a term that lets variance feed on its own previous value, not just past shocks. That’s GARCH (Generalized ARCH), and the version that dominates practice is GARCH(1,1).

Definition. $\sigma_t^2 = \omega + \alpha\,\varepsilon_{t-1}^2 + \beta\,\sigma_{t-1}^2.$ Three forces set tomorrow’s variance:

• $\omega$ — a constant baseline (the long-run anchor).
• $\alpha\,\varepsilon_{t-1}^2$ — the reaction to the last shock: how much a fresh surprise jolts volatility (the “news” term).
• $\beta\,\sigma_{t-1}^2$ — the persistence of yesterday’s variance: how much elevated volatility carries over (the “memory” term).

This recursion is enormously efficient. Because $\sigma_{t-1}^2$ itself depends on $\sigma_{t-2}^2$, and so on, GARCH(1,1) implicitly contains an infinite weighted sum of all past squared shocks — capturing the slow decay that pure ARCH needed dozens of terms for, with just three parameters. That parsimony is why GARCH(1,1) became the default.

Worked example — one update. Suppose $\omega = 0.00001$, $\alpha = 0.08$, $\beta = 0.90$. Yesterday’s variance was $\sigma_{t-1}^2 = 0.0004$ (a daily vol of $\sqrt{0.0004} = 0.02 = 2\%$), and yesterday’s return shock was a large $\varepsilon_{t-1} = -0.03$ (a $-3\%$ day, so $\varepsilon_{t-1}^2 = 0.0009$). Then: $\sigma_t^2 = 0.00001 + 0.08(0.0009) + 0.90(0.0004) = 0.00001 + 0.000072 + 0.00036 = 0.000442.$ Today’s vol forecast is $\sqrt{0.000442} \approx 0.021 = 2.1\%$ — the big move bumped expected volatility up. A calm day ($\varepsilon_{t-1} = 0$) would instead give $\sigma_t^2 = 0.00001 + 0 + 0.00036 = 0.00037$, vol $\approx 1.92\%$ — volatility drifting back down. That up-on-shock, decay-on-calm behavior is clustering.

Persistence and the long-run variance

The single most important quantity in a fitted GARCH is the sum $\alpha + \beta$ — the persistence.

What it controls. $\alpha + \beta$ measures how slowly a volatility shock dies out. For typical daily equity data it’s often around $0.95$–$0.99$ — very persistent: a volatility spike takes weeks to fade. The closer to 1, the longer turbulence lingers.

The unconditional (long-run) variance. Provided $\alpha + \beta < 1$, the variance is mean-reverting and settles toward a long-run level: $\bar{\sigma}^2 = \frac{\omega}{1 - \alpha - \beta}.$ This is the “climate” volatility the process reverts to between storms. Notice the parallel with the AR(1) long-run mean $c/(1-\phi)$ — same structure, applied to variance.

Worked example — the long-run anchor. With $\omega = 0.00001$, $\alpha = 0.08$, $\beta = 0.90$: persistence $= 0.98$, and $\bar{\sigma}^2 = \frac{0.00001}{1 - 0.98} = \frac{0.00001}{0.02} = 0.0005,$ so long-run daily vol $= \sqrt{0.0005} \approx 0.0224 = 2.24\%$ (annualized, $\times\sqrt{252} \approx 35.5\%$). Today’s $2.1\%$ forecast sits just below this anchor and will drift back toward it as calm returns.

Forecasting future variance

GARCH’s payoff is a term structure of volatility — a forecast not just for tomorrow but for any horizon.

One step ahead is direct from the equation: $\sigma_{t+1}^2 = \omega + \alpha\varepsilon_t^2 + \beta\sigma_t^2$, using today’s known shock and variance.

Multi-step ahead is where mean reversion shows up. The $h$-step forecast pulls geometrically from today’s variance toward the long-run anchor: $E[\sigma_{t+h}^2] = \bar{\sigma}^2 + (\alpha + \beta)^{h-1}\big(\sigma_{t+1}^2 - \bar{\sigma}^2\big).$ The gap between current and long-run variance shrinks by a factor of $(\alpha + \beta)$ each step. High persistence → slow convergence → today’s elevated vol matters far into the future. Low persistence → fast convergence → the forecast snaps back to the anchor quickly.

Worked example — the volatility term structure. Using $\bar{\sigma}^2 = 0.0005$, $\alpha + \beta = 0.98$, and a current elevated $\sigma_{t+1}^2 = 0.0009$ (vol $\approx 3\%$, mid-crisis):

• 1 day: $0.0009$ → vol $3.00\%$.
• 10 days: $0.0005 + 0.98^{9}(0.0009 - 0.0005) = 0.0005 + 0.834 \times 0.0004 = 0.000834$ → vol $\approx 2.89\%$.
• 50 days: $0.0005 + 0.98^{49}(0.0004) = 0.0005 + 0.372 \times 0.0004 = 0.000649$ → vol $\approx 2.55\%$.
• 250 days: $0.0005 + 0.98^{249}(0.0004) \approx 0.0005 + 0.0066\times 0.0004 \approx 0.000503$ → vol $\approx 2.24\%$ (back at the anchor).

The forecast slides smoothly from today’s crisis level back toward the long-run climate — exactly the shape that lets a risk desk say “elevated now, normalizing over the quarter.”

Putting it together

Volatility clustering — turbulent and calm stretches bunching into runs — is the robust stylized fact that returns have a flat level ACF but a strongly persistent squared-return ACF: magnitude is forecastable even when direction isn’t. ARCH captured this by letting today’s variance react to the last squared shock ($\sigma_t^2 = \omega + \alpha\varepsilon_{t-1}^2$), but needed too many lags. GARCH(1,1) fixed it by adding a persistence term, $\sigma_t^2 = \omega + \alpha\varepsilon_{t-1}^2 + \beta\sigma_{t-1}^2$: $\omega$ anchors the baseline, $\alpha$ is the reaction to fresh news, $\beta$ is the carryover of yesterday’s variance, and three parameters implicitly weight all past shocks. The key diagnostic is persistence $\alpha + \beta$ — near 1 for daily equities, so shocks linger — and, provided it’s below 1, the variance mean-reverts to the long-run level $\omega/(1-\alpha-\beta)$. Forecasts form a volatility term structure that decays geometrically by $(\alpha + \beta)$ per step from today’s level back toward that anchor. GARCH-normal already manufactures some fat tails through clustering; Student-t errors and asymmetric variants (GJR/EGARCH, the leverage effect) close the rest of the gap to real returns.

Next up — EWMA & RiskMetrics — we meet GARCH’s famous simplified cousin: the exponentially weighted moving average. It’s the $\alpha + \beta = 1$ special case, it powers the classic RiskMetrics risk system, and its single decay parameter $\lambda$ makes it the pragmatic volatility estimator that desks reach for first.