M.6.4 Volatility Modeling

Basic Prerequisites: 02 · Stationarity & ARMA and 03 · Forecasting & Unit Roots.

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1. Intuition & Practical Objective

Returns are not iid: even when the mean has no linear memory, the squared (or absolute) returns are strongly autocorrelated - calm periods follow calm periods and storms follow storms. This is volatility clustering, and it is the most tradeable fact in all of quantitative finance: it is what option markets quote, what risk engines forecast, and what stops a naive “assume constant risk” position from blowing up. This page’s objective is to model the conditional variance ${\sigma }_t^2=\mathrm{Var}(r_t\mid {\mathcal{F}}_{t-1})$ and to forecast it.

Tsay (Ch 3) isolates four stylized facts of volatility:

1. Clustering - big moves follow big moves, small follow small;
2. Continuous evolution with rare jumps - volatility wanders, occasionally jumps;
3. Bounded / stationary - it does not diverge to infinity;
4. Leverage effect - volatility responds asymmetrically: a negative return shock raises volatility more than an equal positive one.

The discipline: fit a mean model first (remove linear dependence → residuals $a_t$), then test for ARCH effects on $a_t^2$, then specify a volatility model for ${\sigma }_t^2$ and estimate jointly (Tsay §3.3 four-step loop). The first-order workhorse is GARCH(1,1).

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2. Mathematical Ground Truth & Derivations

Setup (Tsay eq. 3.2–3.4). With $r_t={\mu }_t+a_t$, ${\mu }_t=\mathbb{E}[r_t\mid {\mathcal{F}}_{t-1}]$, the conditional variance is ${\sigma }_t^2=\mathrm{Var}(r_t\mid {\mathcal{F}}_{t-1})=\mathrm{Var}(a_t\mid {\mathcal{F}}_{t-1})$.

ARCH effect test. Ljung–Box $Q(m)$ on $a_t^2$ (McLeod–Li) or Engle’s LM test - regress $a_t^2$ on $m$ lags, $LM=T{R}^2\sim {\chi }_m^2$. Rejecting ⇒ volatility is time-varying.

ARCH(m) (Engle 1982): $a_t={\sigma }_t{\epsilon }_t$, ${\sigma }_t^2={\alpha }_0+{\sum }_{i=1}^m{\alpha }_i{a}_{t-i}^2$, with ${\alpha }_0>0,{\alpha }_i\ge 0$. Unconditional variance ${\alpha }_0/(1-\sum {\alpha }_i)$ requires $0\le \sum {\alpha }_i<1$. ARCH(1) unconditional kurtosis $3(1-{\alpha }_1^2)/(1-3{\alpha }_1^2)>3$ - heavy tails come for free from a conditional-Gaussian model (finite 4th moment needs ${\alpha }_1^2<1/3$).

GARCH(m,s) (Bollerslev 1986): ${\sigma }_t^2={\alpha }_0+{\sum }_{i=1}^m{\alpha }_i{a}_{t-i}^2+{\sum }_{j=1}^s{\beta }_j{\sigma }_{t-j}^2$, nonnegative, with stationarity ${\sum }_{i=1}^{\max (m,s)}({\alpha }_i+{\beta }_i)<1$ (Tsay eq. 3.14). $a_t^2$ itself is an ARMA (eq. 3.15) with martingale-difference noise - not iid. GARCH(1,1):

$${\sigma }_t^2=\omega +\alpha a_{t-1}^2+\beta {\sigma }_{t-1}^2.$$

• persistence $=\alpha +\beta$; stationary iff $\alpha +\beta <1$;
• unconditional variance ${\bar{\sigma }}^2=\frac{\omega }{1-\alpha -\beta }$;
• heavy tails if $1-2{\alpha }^2-(\alpha +\beta {)}^2>0$;
• multi-step forecast ${\sigma }_h^2(\ell )={\alpha }_0+({\alpha }_1+{\beta }_1){\sigma }_h^2(\ell -1)$ → converges to ${\bar{\sigma }}^2$ (eq. 3.17).

IGARCH(1,1) (Tsay §3.6): $\alpha +\beta =1$, ${\sigma }_t^2={\alpha }_0+\beta {\sigma }_{t-1}^2+(1-\beta )a_{t-1}^2$; unconditional variance is undefined; the multi-step forecast is a straight line ${\sigma }_h^2(\ell )={\sigma }_h^2(1)+(\ell -1){\alpha }_0$. The special case ${\alpha }_0=0$ is RiskMetrics EWMA ${\sigma }_t^2=(1-\beta )a_{t-1}^2+\beta {\sigma }_{t-1}^2$, $\beta \approx 0.94$ - exponential smoothing. (The square-root-of-time VaR rule relies on exactly this form; see 03 · Unit Roots.)

Leverage / asymmetry - ARCH and GARCH respond equally to $+$ and $-$ shocks. Fix:

• EGARCH (Nelson 1991): $\ln {\sigma }_t^2={\alpha }_0+\frac{1+{\beta }_{1}B}{1-{\alpha }_{1}B}g({\epsilon }_{t-1})$ with weighted innovation $g({\epsilon }_t)=\theta {\epsilon }_t+\gamma [|{\epsilon }_t|-\mathbb{E}|{\epsilon }_t|]$, $\mathbb{E}|\epsilon |=\sqrt{2/\pi }$. The log form relaxes positivity; asymmetry is carried by $\theta$ (slopes $\theta +\gamma$ for $\epsilon \ge 0$, $\theta -\gamma$ for $\epsilon <0$; expect $\theta <0$). Tsay’s IBM example: a $-2\sigma$ shock raises volatility ~37.4% more than a $+2\sigma$ shock.
• TGARCH/GJR (Glosten–Jagannathan–Runkle 1993, Zakoian 1994): ${\sigma }_t^2={\alpha }_0+\sum ({\alpha }_i+{\gamma }_i{N}_{t-i})a_{t-i}^2+\sum {\beta }_j{\sigma }_{t-j}^2$, where $N_{t-i}=1$ if $a_{t-i}<0$ else 0 - a negative shock gets ${\alpha }_i+{\gamma }_i$, a positive only ${\alpha }_i$.

GARCH-M (Tsay eq. 3.23): $r_t=\mu +c{\sigma }_t^2+a_t$ - the conditional variance enters the mean (risk premium $c$), inducing serial correlation in returns. GARCH excess kurtosis (Tsay §3.16): Gaussian case $K_a^{(g)}=6{\alpha }_1^2/[1-2{\alpha }_1^2-({\alpha }_1+{\beta }_1{)}^2]$.

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3. Computational Implementation - simulate, test, and fit GARCH(1,1)

Simulate a GARCH(1,1) with known $(\omega ,\alpha ,\beta )=(0.05,0.10,0.85)$, run the ARCH-LM test on the squared returns, and recover the parameters by conditional Gaussian maximum likelihood (via a derivative-free pattern search - stdlib only, no scipy).

The ARCH-LM test screams rejection ($2817\gg 11.07$) - the squared returns are unmistakably autocorrelated even though the returns themselves look white. The Gaussian MLE recovers $(\alpha ,\beta )\approx (0.118,0.826)$ against truth $(0.10,0.85)$, and the persistence $\alpha +\beta =0.944\approx 0.95$ (so unconditional variance $1.04\approx 1.0$) - the fitted model reproduces the level of risk almost exactly even with modest parameter error.

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4. Failure Modes & First-Principles Breakdowns

1. Symmetric response (ARCH/GARCH). They treat a $+2\sigma$ and a $-2\sigma$ shock identically, so they systematically mis-state risk after negative news. Use EGARCH (leverage in $\theta <0$) or TGARCH/GJR when the data shows asymmetry - Tsay’s IBM example quantifies it: a $-2\sigma$ shock raises vol ~37% more than a $+2\sigma$ one.
2. ARCH’s restrictive kurtosis bound. ARCH(1) needs ${\alpha }_1^2<1/3$ for a finite 4th moment - easily violated, and ARCH tends to overpredict volatility after large isolated shocks. GARCH’s two-parameter persistence handles this better.
3. Persistence $\approx 1$ ⇒ undefined long-run variance. IGARCH/RiskMetrics ($\alpha +\beta =1$) have no stationary unconditional variance and forecast volatility along a straight line. If your data is near this boundary, “mean reversion to a long-run vol” is an assumption, not a finding - and the square-root-of-time VaR rule breaks (see 03 · Unit Roots).
4. GARCH is ARMA on $a_t^2$ - do not treat ${\eta }_t=a_t^2-{\sigma }_t^2$ as iid. It is a martingale difference sequence; applying iid-based diagnostics to the model’s noise is invalid.
5. Positivity & stationarity constraints. Fitted GARCH can wander into $\alpha +\beta \ge 1$ or negative variance; enforce $\alpha ,\beta \ge 0$ and $\alpha +\beta <1$ during estimation (as the pattern-search reparameterization above does via the softmax transform).

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5. References

• Tsay, Analysis of Financial Time Series
• Engle, R.F. (1982), “Autoregressive Conditional Heteroscedasticity,” Econometrica
• Bollerslev, T. (1986), “Generalized Autoregressive Conditional Heteroskedasticity,” J. Econometrics
• Nelson, D. (1991), “Conditional Heteroskedasticity in Asset Returns: A New Approach,” Econometrica
• RiskMetrics, Technical Document (1996)

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6. Connected Graph Bridges

• Back: 03 · Forecasting & Unit Roots · Index Hub
• Forward: 05 · Cointegration & Multivariate · 06 · Advanced Extensions (multivariate GARCH/BEKK/DCC)
• Applied: Implied Volatility Surfaces · Heston & SABR · VaR & Expected Shortfall