1.9 GARCH & Volatility Modeling

Basic Prerequisites: Econometrics & Time Series (stationarity, ARMA, forecasting, conditional moments) and Probability & Measure Theory (conditional expectation, filtrations, martingale-difference sequences). (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

Financial returns are uncorrelated in level but strongly dependent in magnitude. Look at any equity index: today’s return tells you almost nothing about tomorrow’s sign, but a big move today makes a big move tomorrow far more likely. This is volatility clustering - the single most robust stylized fact in finance (Mandelbrot 1963) - and it means the unconditional variance $\mathrm{Var}(r_t)$ is the wrong risk number: risk is time-varying and forecastable.

GARCH-family models make volatility a deterministic function of the past. The core declaration is that the return decomposes into a predictable mean and a conditionally Gaussian shock whose variance itself follows an ARMA-type recursion:

Everything below is a choice about how ${\sigma }_t^2$ depends on the filtered past. The point of the whole discipline: volatility is the one genuinely forecastable quantity in returns, so it drives risk management (VaR/ES), position sizing (vol targeting), option pricing, and hedging.

This folder is the topic-hub for GARCH & volatility modeling in Kwant-Atlas. It (a) gives the fast formula lookup below - job #1 of a hub - and (b) routes to six sub-pages walking from raw intuition, through ARCH/GARCH estimation, asymmetric models (leverage), realized volatility and HAR, failure modes, and multivariate/forecasting extensions.

The one-sentence essence. “Returns are unpredictable in direction but their conditional variance is highly persistent and forecastable; model ${\sigma }_t^2$ as a positive, mean-reverting recursion on past squared shocks so that one-day-ahead risk is a computable number - but remember the model is only as good as its stationarity and its symmetry assumptions.”

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). All formulas below are transcribed from the verified Tsay (2010) corpus chapters (§3.4–3.16 ARCH/GARCH/EGARCH/TGARCH, §3.15 realized vol; §10.4 DCC) and the canonical papers (Engle 1982, Bollerslev 1986, Nelson 1991, Glosten–Jagannathan–Runkle 1993, Corsi 2009, Engle 2002). The numbers in the check column were re-executed and reproduced exactly from the working Python in §3 and the sub-pages.

Notation: $r_t$ return, $a_t=r_t-{\mu }_t$ shock/innovation, ${\sigma }_t^2=\mathrm{Var}(a_t\mid {\mathcal{F}}_{t-1})$ conditional variance, ${\epsilon }_t$ standardized shock; $\omega \equiv {\alpha }_0$, ARCH coefficients ${\alpha }_i$, GARCH coefficients ${\beta }_j$; $N(\cdot )$ is irrelevant here, all recursions are deterministic given data.

Critical caveat (the model’s blind spot). Plain ARCH/GARCH responds symmetrically to $+$ and $-$ shocks - it deliberately cannot see the leverage effect. Empirics say negative shocks raise future volatility more (Black 1976). Any GARCH used for equity risk without a GJR/EGARCH asymmetry term is mis-specified on the downside - see 03 · Asymmetric Models.

3. Computational Implementation - the formula engine

Standard library only (math + random). Simulates a GARCH(1,1), then verifies the two identities that define the model: the unconditional-variance match and the serial dependence of squared returns (the clustering signature). Reproduces the §2 check numbers.

The simulated variance lands within ~12% of the analytic ${\alpha }_0/(1-{\alpha }_1-{\beta }_1)$ - the residual gap is finite-sample noise over just 5000 draws. The slow, still-elevated ACF of $r_t^2$ at lags 1–5 is the clustering: squared shocks are positively, persistently autocorrelated even though the shocks themselves are not.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the full analysis lives in 05 · Failure Modes & Practice. In one line each:

1. Persistence vs. structural break. A genuine regime shift makes the estimated ${\alpha }_1+{\beta }_1$ creep toward the IGARCH boundary ($1$) - a unit-root artefact, not true infinite persistence. (Verified: break-sample persistence $0.9913$ vs calm subsample $0.9384$.)
2. Symmetry / leverage misspecification. ARCH and GARCH treat $+2\sigma$ and $-2\sigma$ shocks identically; in equities the negative one raises future vol far more, so symmetric models understate downside risk.
3. High-order estimation curse. ARCH($q$) and GARCH($p,q$) add parameters fast; likelihood surfaces are flat and estimates unstable. GARCH(1,1) is famously hard to beat - parsimony is a feature, not a limitation.
4. Non-stationarity and tail estimation error. The $4$th-moment condition can fail, making the “unconditional variance” a fiction, and the estimate of the right tail (which drives VaR) rests on a handful of shocks.

5. References

• Engle, Robert F. (1982): Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica 50(4), 987–1007
• Bollerslev, Tim (1986): Generalized Autoregressive Conditional Heteroskedasticity, J. Econometrics 31(3), 307–327
• Nelson, Daniel B. (1991): Conditional Heteroskedasticity in Asset Returns: A New Approach, Econometrica 59(2), 347–370
• Glosten, Jagannathan & Runkle (1993): On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks, J. Finance 48(5), 1779–1801
• Corsi, Fulvio (2009): A Simple Approximate Long-Memory Model of Realized Volatility, J. Financial Econometrics 7(2), 174–196
• Tsay, Ruey S.: Analysis of Financial Time Series (3rd ed., 2010)

6. Connected Graph Bridges

• Foundational base: Econometrics & Time Series (stationarity, ARMA, forecasting) · Probability & Measure Theory (martingale differences) · Statistics & Inference (MLE)
• Sibling topics (this pillar): Regime Detection (vol regimes as states) · Momentum (vol scaling) · Signal Processing & Kalman (SV as state space)
• Risk & portfolio: VaR & Expected Shortfall (uses the vol forecast) · Risk Parity (vol targeting) · Black–Scholes–Merton (vol as the pricing parameter)
• Sub-pages (in-folder): 01 From Zero · 02 ARCH & GARCH · 03 Asymmetric Models · 04 Realized Vol & HAR · 05 Failure Modes & Practice · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05