1.9.1 GARCH from Zero

Basic Prerequisites: Econometrics & Time Series (variance, stationarity) and Probability & Measure Theory (conditional expectation).

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1. Intuition & Practical Objective

This page builds the why of volatility modeling with no prior time-series knowledge needed. The objective is one idea: while you cannot predict whether a return will be positive or negative, you can predict how big the next return is likely to be - and that prediction is what risk management is made of.

Start with the dumbest question: what is volatility? Colloquially it is “how jumpy the market is”. Precisely, it is the standard deviation of returns over a period - the size of the wiggles, not their direction. It matters because every risk number you have ever seen (VaR, margin, an option’s price, a Sharpe ratio, a position size) is some function of a volatility estimate.

Three “aha”s:

1. Returns are uncorrelated but not independent. Compute the autocorrelation of daily returns: it is essentially zero. You cannot use yesterday’s return to predict today’s sign. Now compute the autocorrelation of squared returns: it is large and positive and decays slowly. Yesterday’s magnitude predicts today’s magnitude. This is volatility clustering - Mandelbrot’s observation (1963) that “large changes tend to be followed by large changes, of either sign”.

2. Volatility is latent. You never observe ${\sigma }_t$; you observe one realized return, which is ${\sigma }_t$ times a random draw. So all volatility modeling is inference about a hidden quantity from noisy data. This is exactly why we need a model - and why the model is a statement of belief, not a measurement.

3. Clustering is forecastable, so it is tradable and hedgeable. Because ${\sigma }_t^2$ depends on the past, a bad shock today means a higher risk estimate tomorrow. That single fact powers vol targeting, dynamic VaR, and the entire GARCH family.

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2. Mathematical Ground Truth & Derivations

The decomposition. Write any return as a predictable mean plus an unpredictable shock:

$$r_t={\mu }_t+a_t,{\mu }_t=\mathbb{E}[r_t\mid {\mathcal{F}}_{t-1}],a_t=r_t-{\mu }_t,\mathrm{Var}(r_t\mid {\mathcal{F}}_{t-1})=\mathbb{E}[a_t^2\mid {\mathcal{F}}_{t-1}]={\sigma }_t^2.$$

The shocks $a_t$ form a martingale difference sequence: $\mathbb{E}[a_t\mid {\mathcal{F}}_{t-1}]=0$, so they are uncorrelated ($\mathrm{Cov}(a_t,a_{t-s})=0$) - hence ACF(returns)$\approx 0$. But they are not independent, because their conditional variance varies:

$$a_t={\sigma }_t{\epsilon }_t,{\epsilon }_t\stackrel{iid}{\sim }N(0,1).$$

Unconditional vs conditional. The unconditional (long-run) variance is a single fixed number. The conditional variance ${\sigma }_t^2$ moves. The whole game is the trading-off between them: if ${\sigma }_t^2$ reverts to a constant mean ${\bar{\sigma }}^2$, then forecasting is possible but the forecasts are transient. This mean reversion is precisely what makes volatility “bounded and stationary” (stylized fact 3 of Tsay §3.1).

The ARCH seed (Engle 1982). The minimal way to make ${\sigma }_t^2$ depend on the past is to let it be an autoregression on the squared shocks:

$${\sigma }_t^2={\alpha }_0+{\alpha }_1{a}_{t-1}^2+{\alpha }_2{a}_{t-2}^2+\cdots +{\alpha }_q{a}_{t-q}^2,{\alpha }_0>0,{\alpha }_i\ge 0.$$

“Squaring” is what turns a signed shock into a magnitude; the ${\alpha }_i\ge 0$ constraint keeps variance positive. A big $a_{t-1}^2$ (either sign) raises ${\sigma }_t^2$, which in turn makes $|a_t|$ likely large - that is clustering, generated endogenously. GARCH (Bollerslev 1986) adds autoregressive terms in the variance itself, ${\beta }_j{\sigma }_{t-j}^2$; the derivation and estimation live in 02 · ARCH & GARCH.

The four stylized facts of volatility (Tsay §3.1, Cont 2001):

1. Clustering - calm and turbulent periods persist.
2. Continuous evolution with rare jumps - ${\sigma }_t$ wanders smoothly, occasionally gaps.
3. Bounded / mean-reverting - it does not run off to infinity (so a stationarity condition must hold).
4. Asymmetry (leverage) - big negative returns raise volatility more than big positive ones (Black 1976).

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3. Computational Implementation - seeing clustering with no model at all

Standard library only. No GARCH here - just a two-state “regime” toy that switches between a calm and a turbulent volatility. It shows the two signatures of clustering: near-zero ACF in levels, strongly positive ACF in squares, and big shocks arriving in bunches.

Read the two columns: ACF of returns hovers around zero (no predictability of sign - consistent with weak-form efficiency), while ACF of squared returns is positive and decays slowly (magnitude is highly predictable). Nearly a quarter of the largest shocks share a day with another large shock - clustering, made visible without writing a single GARCH equation.

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4. Failure Modes & First-Principles Breakdowns

1. Volatility is latent - every estimate is noisy. One day of returns gives you a noisy draw of a hidden ${\sigma }_t$. Any statement about “today’s volatility” is an inference, and the noise is large at daily frequency. This is why range estimators and realized volatility (04) exist: to sample the same latent quantity more efficiently.
2. Clustering is not a trend in price. Volatility clustering holds for magnitude only. A common beginner error is to think “high vol $\Rightarrow$ down”. High vol is direction-agnostic; the asymmetry (leverage) in fact makes high vol weakly bearish in equities, but that is a separate fact, not clustering.
3. The regime label is unobservable in real time. In the toy above we know the state; in real markets you must estimate it - and mislabeling a regime is the seed of Regime Detection and of the structural-break failure in 05 · Failure Modes.
4. Mean reversion has a half-life. If ${\sigma }_t^2$ reverts to a constant, a vol spike is temporary - overreacting to one shock (assuming it persists forever) is as wrong as ignoring it. The speed of that reversion is the persistence parameter, central to 02.

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5. References

• Mandelbrot, Benoît (1963): The Variation of Certain Speculative Prices, J. Business 36(4)
• Engle, Robert F. (1982): Autoregressive Conditional Heteroscedasticity…, Econometrica 50(4)
• Cont, Rama (2001): Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues, Quantitative Finance 1(2)
• Tsay, Ruey S.: Analysis of Financial Time Series (3rd ed., 2010)

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6. Connected Graph Bridges

• Base: Econometrics & Time Series · Probability & Measure Theory
• Continue: 02 · ARCH & GARCH · Index Hub
• Related: Regime Detection (volatility as a hidden state) · Momentum (vol as a scaling factor)