Volatility Models: ARCH and GARCH
Why returns are unpredictable in the mean but not in the variance, how GARCH captures volatility clustering, and what its parameters imply.
Leads to: GARCH volatility feeds option pricing (Phase 15) and risk models throughout.
Learning Objectives
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- Explain volatility clustering and why it violates the constant-variance assumption of ARMA.
- Define the ARCH(\(q\)) and GARCH(\(p,q\)) models and interpret their parameters.
- Derive the stationarity condition and unconditional variance of GARCH(1,1).
- Relate GARCH persistence \(\alpha+\beta\) to volatility half-life and the IGARCH boundary.
- Simulate a GARCH process and detect clustering via the ACF of squared returns.
Key Vocabulary
- Volatility clustering
- The empirical tendency of large moves to follow large moves: \(|r_t|\) is autocorrelated even when \(r_t\) is not.
- Conditional variance
- \(\sigma_t^2=\Var(r_t\mid\mathcal F_{t-1})\), the variance given the past; time-varying in ARCH/GARCH.
- ARCH(\(q\)
- \(\sigma_t^2=\omega+\sum_{i=1}^q\alpha_i r_{t-i}^2\); conditional variance driven by recent squared returns.
- GARCH(\(p,q\)
- Adds lagged variances: \(\sigma_t^2=\omega+\sum\alpha_i r_{t-i}^2+\sum\beta_j\sigma_{t-j}^2\).
- Persistence
- \(\alpha+\beta\) in GARCH(1,1); how slowly volatility shocks decay; near 1 means very persistent.
- Unconditional variance
- \(\omega/(1-\alpha-\beta)\) for stationary GARCH(1,1); the long-run average variance.
- IGARCH
- The boundary \(\alpha+\beta=1\): shocks to variance never die out (integrated volatility).
Intuition & Motivation
ARCH makes today’s variance a function of recent squared shocks: a big surprise raises tomorrow’s expected variance. GARCH adds a smoothing term - today’s variance also remembers yesterday’s variance - which captures the long, gentle decay of real volatility with very few parameters. The single number \(\alpha+\beta\) summarizes how persistent turbulence is: close to 1 (as it usually is for equities) means a volatility shock takes many weeks to fade, and exactly 1 means it never does.
Why ARMA is not enough
Write a return as \(r_t=\sigma_t z_t\) with i.i.d. standardized \(z_t\) (\(\E z_t=0,\ \Var z_t=1\)) and a time-varying conditional standard deviation \(\sigma_t\). Then \(r_t\) is serially uncorrelated (so its ACF is flat, matching returns) but \(r_t^2\) is autocorrelated (matching the clustering). The mean model and the variance model are separate: ARMA handles the mean, ARCH/GARCH the conditional variance \(\sigma_t^2\).
ARCH and GARCH defined
with \(\omega\gt 0,\ \alpha_i,\beta_j\ge0\) to keep \(\sigma_t^2\gt 0\). GARCH(1,1) - just three parameters - is the workhorse and fits most return series remarkably well. Parameters are estimated by maximum likelihood: assuming \(z_t\sim\Normal(0,1)\), maximize \(\ell=-\tfrac12\sum_t\big(\log\sigma_t^2+r_t^2/\sigma_t^2\big)\) over \((\omega,\alpha,\beta)\) (a numerical optimization, since \(\sigma_t^2\) is recursive).
Persistence \(\alpha+\beta\) controls the decay of a volatility shock: the conditional variance mean-reverts toward \(\bar\sigma^2\) at rate \(\alpha+\beta\) per step, giving a half-life \(\log(1/2)/\log(\alpha+\beta)\). For daily equity returns \(\alpha+\beta\) is commonly \(\approx0.98\text{--}0.99\), a half-life of weeks. GARCH also generates fat tails in the unconditional return distribution even with Gaussian \(z_t\), matching a key stylized fact; a Student-\(t\) innovation fits the tails even better.
Interactive: simulate GARCH and see the clustering
- Applying ARMA to returns and assuming constant variance - it misses clustering entirely and mis-states risk.
- Reading GARCH as a mean-return forecast; it forecasts variance, not direction.
- Estimating GARCH without the constraints \(\omega\gt 0,\ \alpha,\beta\ge0,\ \alpha+\beta\lt 1\); violating them gives negative or explosive variances.
- Assuming Gaussian innovations when tails are heavy; use Student-\(t\) and check standardized-residual diagnostics.
- Ignoring leverage (down moves raise vol more than up moves); symmetric GARCH misses it - use EGARCH/GJR.
- Fit the mean (often just a constant, or an AR(1)) and the variance jointly, or fit GARCH to the mean residuals.
- Report persistence \(\alpha+\beta\) and the implied half-life; near-1 persistence warns that shocks fade slowly.
- Check standardized residuals \(r_t/\hat\sigma_t\): they should be i.i.d. with no remaining ACF in their squares.
- For asymmetry, prefer GJR-GARCH or EGARCH; for tails, Student-\(t\) innovations - both matter for equity risk.
Knowledge Check
Practical Exercise
A GARCH(1,1) fit to daily equity returns yields \(\omega=0.00001,\ \alpha=0.08,\ \beta=0.90\) (returns in decimals). (a) Is it stationary, and what is the unconditional daily volatility? (b) What is the persistence and the volatility half-life? (c) Interpret what happens to forecast volatility after a large market shock.
(a) \(\alpha+\beta=0.98\lt 1\), so it is covariance-stationary. Unconditional variance \(\bar\sigma^2=\omega/(1-\alpha-\beta)=0.00001/0.02=0.0005\), so daily volatility \(\bar\sigma=\sqrt{0.0005}\approx0.0224\), about 2.24% per day (roughly \(0.0224\sqrt{252}\approx35\%\) annualized).
(b) Persistence \(\alpha+\beta=0.98\). Half-life \(=\log(0.5)/\log(0.98)\approx34\) trading days: a volatility shock takes about a month and a half to halve.
(c) A large \(r_{t-1}^2\) spikes \(\sigma_t^2\) through the \(\alpha\) term; because \(\beta=0.90\) is large, the elevated variance then feeds forward and decays only slowly toward \(\bar\sigma^2\). So after a shock the model forecasts a persistent regime of elevated volatility that mean-reverts over roughly the half-life - capturing the clustering seen in crises.
Lesson Summary
Formula Sheet Additions
- Did I enforce \(\omega\gt 0,\ \alpha,\beta\ge0,\ \alpha+\beta\lt 1\) in estimation?
- Did I model the mean separately and fit GARCH to residuals?
- Are standardized residuals i.i.d. with no ACF in their squares?
- Did I consider fat-tailed (Student-\(t\)) innovations and leverage (EGARCH/GJR)?
Retrieval Practice
Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.
A: ARMA assumes constant variance but returns show volatility clustering (autocorrelated squares); ARCH/GARCH model the time-varying conditional variance \(\sigma_t^2\).
A: Covariance-stationary iff \(\alpha+\beta\lt 1\); unconditional variance \(\omega/(1-\alpha-\beta)\).
A: Volatility shocks decay very slowly (long half-life); at exactly 1 (IGARCH) they never die out.
Flashcards
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Completion Checklist
- I can explain the core ideas in my own words
- I worked the derivations/examples by hand
- I completed the interactive workbench(es)
- I passed the knowledge check
Source References
This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.
- Time Series Analysis (James Hamilton, 1994) foundational - Hamilton 21 - Ch. 21: ARCH/GARCH models, conditional heteroscedasticity, and maximum-likelihood estimation.
- The Volatility Surface (Jim Gatheral, 2006) foundational - Gatheral 1 - Volatility as a persistent, forecastable process underlying option pricing.