Questions: Modeling Time-Varying Volatility with GARCH

Question 1

Following a major market crash, a GARCH(1,1) model estimates today's conditional volatility as very high. With α = 0.09 and β = 0.90, how will volatility behave over the following weeks?

• A. Volatility will immediately return to the long-run average since markets are efficient
• B. Volatility will remain elevated for weeks, decaying gradually, because α + β = 0.99 indicates very high persistence
• C. Volatility will continue rising indefinitely as past shocks compound
• D. The model cannot forecast future volatility — GARCH only describes current variance

Answer: (B)

The parameter sum α + β = 0.99 controls how quickly conditional variance reverts to its long-run mean ω/(1 − α − β). With the sum close to 1, shocks decay very slowly — each period retains 99% of the previous period's elevated variance. This captures the empirical reality of volatility clustering: after a major shock, elevated volatility persists for days or weeks, not just one period. Markets can be informationally efficient and still exhibit this second-moment persistence.

Question 2

In the GARCH(1,1) equation σ²ₜ = ω + αε²ₜ₋₁ + βσ²ₜ₋₁, what does the term αε²ₜ₋₁ represent?

• A. The long-run average variance level that the process mean-reverts toward
• B. The persistence of yesterday's variance estimate carried into today's forecast
• C. The impact of yesterday's unexpected return shock on today's conditional variance
• D. The leverage effect from negative returns exceeding positive returns of the same size

Answer: (C)

ε²ₜ₋₁ is the squared residual from yesterday — the unexpected return, large when there was a surprise move (positive or negative). The α coefficient scales how much this 'news' updates the current variance estimate. A large shock yesterday (ε² large) pushes today's estimated variance upward. This is the 'news impact' or ARCH component. The β term separately captures persistence — carrying forward the previous variance estimate regardless of what new shock arrived.

Question 3

In a GARCH(1,1) model, volatility is assumed constant across time, with shocks causing primarily temporary, single-period deviations before immediately reverting.

• T. True
• F. False

Answer: False

GARCH is explicitly designed to model time-varying volatility. The β term (typically 0.85–0.90 in equity markets) ensures shocks persist across multiple periods: a large ε²ₜ today raises σ²ₜ₊₁, which carries over into σ²ₜ₊₂, and so on. A constant-variance model (ARCH(0)) would assume all shocks die after one period. GARCH's improvement over constant variance is precisely capturing this multi-period clustering of volatility.

Question 4

The closer α + β is to 1 in a GARCH(1,1) model, the more persistent volatility is and the slower it reverts to its long-run average.

• T. True
• F. False

Answer: True

The long-run variance is ω/(1 − α − β), and the speed of reversion toward it after a shock is governed by (α + β). When α + β = 1 (IGARCH), shocks are permanent — volatility never mean-reverts. When α + β = 0.99 (typical for equity indices), reversion is extremely slow. When α + β = 0.80, reversion is much faster. This parameter sum is therefore the key indicator of how long-lived volatility episodes will be.

Question 5

What is volatility clustering, and why does it make the constant-variance assumption inadequate for modeling financial returns?

Model answer: Volatility clustering is the empirical pattern where large price changes tend to be followed by more large price changes, and calm periods by more calm periods — regardless of sign. A constant-variance model assigns the same σ² to every time period, which is contradicted by this autocorrelation in the magnitude of returns. In calm periods it overestimates risk; in turbulent periods it underestimates it. GARCH addresses this by making σ²ₜ a function of past shocks and past variance, allowing the model to track changing risk levels dynamically.

The practical consequence of ignoring volatility clustering is severe: risk management models (VaR) become unreliable, option prices are mispriced (constant-σ Black-Scholes prices are wrong during high-volatility regimes), and portfolio allocations based on a fixed σ are stale. GARCH's time-varying conditional variance gives a day-specific risk estimate, which is why it became standard in financial risk management.