ETF5952 · Quantitative Methods for Risk Analysis
Volatility Models: EWMA, GARCH and GJR-GARCH
In ETF5952 Quantitative Methods for Risk Analysis at Monash University, this chapter models the conditional variance σt2 = Var(Rt | It−1) — the fact that the size of return swings changes over time and clusters. It builds three workhorse models: EWMA, an exponentially weighted average of past squared shocks with the RiskMetrics rule λ = 0.94; GARCH(1,1), which adds a constant so volatility mean-reverts to a finite long-run level σ̄2 = ω/(1−α−β); and GJR-GARCH, which adds a leverage term so negative shocks raise volatility more than equal-size positive ones. These volatility forecasts feed directly into the unit's VaR and Expected Shortfall machinery.
What this chapter covers
- 01Volatility clustering: time-varying conditional variance vs a roughly constant conditional mean
- 02The constant-mean return model R_t = μ + σ_t Z_t and the squared mean-adjusted shock
- 03Rolling-window (moving-average) volatility: equal weights, sharp cut-off, short vs long window
- 04EWMA: recursive form σ_t² = (1−λ)(R_{t−1}−μ)² + λ σ_{t−1}², geometric weights, no cut-off
- 05The RiskMetrics rule λ = 0.94 for daily data (set, not estimated)
- 06GARCH(1,1): ω, α (reaction), β (persistence); stationarity α + β < 1
- 07Long-run variance σ̄² = ω/(1−α−β) and the weighted-average rewrite (mean reversion)
- 08EWMA as a special GARCH(1,1) with ω = 0, α = 1−λ, β = λ (α + β = 1, no anchor)
- 09GJR-GARCH: the indicator 1{R_{t−1} < μ}, the leverage effect γ > 0, persistence α + γ/2 + β
- 10Exam technique: square the shock, check α + β < 1, read γ's sign, report σ with units
Worked example: GARCH(1,1) long-run volatility and a one-step update
- +1Stationarity check. α + β = 0.10 + 0.85 = 0.95 < 1 ✓ — a finite long-run variance exists, and persistence 0.95 means volatility is highly persistent (reverts slowly).
- +1(a) Long-run variance. σ̄2 = ω/(1−α−β) = 0.000004 / (1 − 0.95) = 0.000004 / 0.05 = 0.00008.
- +1(a) Long-run volatility. σ̄ = √0.00008 = 0.00894 (0.894% per day).
- +1(b) One-step update. σt2 = ω + α(Rt−1−μ)2 + βσt−12 = 0.000004 + 0.10·(−0.03)2 + 0.85·0.000225 = 0.000004 + 0.00009 + 0.00019125 = 0.00028525.
- +1(b) Volatility and direction. σt = √0.00028525 = 0.01689 (1.689% per day). This is well above the 0.894% long-run level, so the forecast will drift down toward σ̄ over the coming days (mean reversion).
Key terms
- Volatility clustering
- The empirical pattern that large returns (of either sign) tend to be followed by large returns and small by small, so the conditional variance is strongly time-varying while the conditional mean is roughly constant. It is the motivation for every model in the chapter.
- Conditional variance
- σt2 = Var(Rt | It−1), the variance of the next return given the information set It−1 = {r1, …, rt−1}. Its square root σt is the conditional volatility that daily VaR and ES require.
- EWMA (exponentially weighted moving average)
- A variance estimate σt2 = (1−λ)(Rt−1−μ)2 + λσt−12 that weights past squared shocks geometrically with decay λ. No sharp cut-off; higher λ = longer memory, lower λ = more reactive.
- RiskMetrics λ = 0.94
- The RiskMetrics rule of thumb that fixes the EWMA decay at λ = 0.94 for daily data (0.97 for monthly). It is imposed rather than estimated from the sample, which makes EWMA a single-parameter, no-optimisation model.
- GARCH(1,1)
- σt2 = ω + α(Rt−1−μ)2 + βσt−12 with ω > 0, α, β ≥ 0. α is the reaction to the last shock, β the persistence of past variance; α + β is the total persistence.
- Long-run (unconditional) variance
- σ̄2 = ω/(1−α−β), defined only when α + β < 1. GARCH volatility mean-reverts toward this anchor; EWMA (α + β = 1) has no finite long-run variance.
- GJR-GARCH / leverage effect
- GJR-GARCH adds γ(Rt−1−μ)2·1{Rt−1 < μ} to GARCH. With γ > 0 a negative shock (indicator = 1) raises next-period volatility more than an equal-size positive shock — the leverage effect. Persistence is α + γ/2 + β.
- Persistence
- How slowly volatility shocks decay: α + β for GARCH(1,1), α + γ/2 + β for GJR-GARCH. Close to 1 means very persistent volatility; it must be strictly below 1 for a finite long-run variance.
Volatility Models: EWMA, GARCH and GJR-GARCH FAQ
When is the GARCH long-run variance formula σ̄² = ω/(1−α−β) actually valid?
Only when the persistence α + β is strictly less than 1. That condition guarantees the process is stationary and the geometric series behind the unconditional variance converges. If α + β ≥ 1 — as in EWMA, where α + β = 1 exactly — there is no finite long-run variance to revert to, and quoting a number for σ̄ would be wrong. Always state the check before you compute the anchor.
What is the leverage effect and how does GJR-GARCH capture it?
The leverage effect is the empirical tendency for a fall in price (a negative shock) to raise future volatility more than a rise of the same size. Standard GARCH cannot see it because returns enter only through their square, so +2% and −2% are identical. GJR-GARCH adds a term γ(R_{t−1}−μ)²·1{R_{t−1}<μ}: the indicator switches on only after a loss, so a negative shock loads at α + γ while a positive one loads at just α. A positive γ means bad news bites harder.
Can AI help me with volatility models like EWMA and GARCH in ETF5952?
Yes — Sia can explain the method step by step. Ask it to walk you through substituting into an EWMA or GARCH(1,1) update, why the long-run variance needs α + β < 1, or how the GJR indicator switches the leverage term on and off, and it will show the reasoning so you can reproduce it under closed-book exam conditions. Sia coaches the technique and checks your working; it does not sit the assessment for you or guarantee a mark.
Exam move
Put the three recursions on one page and drill the update chain until it is automatic: square the mean-adjusted shock, multiply by the reaction weight, add the persistence-weighted old variance, then take the square root to report a volatility in per cent. Burn in three checks — EWMA weights (1−λ) and λ sum to 1 with no ω; GARCH needs α + β < 1 for a finite long-run variance σ̄² = ω/(1−α−β); and the GJR indicator fires on the sign of the shock, not its square. Practise reading direction words aloud (a negative shock raises volatility more; the forecast reverts toward the long-run anchor), because a correct number with the wrong direction earns nothing. The final exam is worth 40% of the unit, is closed book, and its duration is not published in the unit materials, so plan your time in proportion to the marks and confirm the exact date and length on Moodle.
Working through Volatility Models: EWMA, GARCH and GJR-GARCH in ETF5952? Sia is AskSia’s AI Finance tutor — ask any ETF5952 Volatility Models: EWMA, GARCH and GJR-GARCH question and get a clear, step-by-step explanation grounded in how ETF5952 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
