FINC6023 Chap.7 Forecasting Volatility & Correlations
Forecasting Volatility & Correlations
Forecasting Volatility and Correlations supplies the changing volatility input used by market-risk models. An equally weighted estimator treats observations in a window alike, so a large shock remains fully represented until it leaves the window and then disappears abruptly. Exponentially weighted moving average responds more smoothly by assigning geometrically declining weight to older squared returns.
The decay parameter controls persistence and the weight on the newest shock.
GARCH adds a constant and separates the response to the latest squared innovation from persistence in the previous conditional variance. The shock coefficient measures reaction; the lagged-variance coefficient measures persistence. Their sum controls the speed of mean reversion.
When that sum is below one, the process has a finite long-run variance. A current variance above the long-run level is forecast to decline toward it, while a current variance below the long-run level rises toward it.
Forecasting several steps ahead requires care because the model predicts variance, not volatility.
Apply the persistence power to the variance gap, add the long-run variance, and take a square root only at the end if volatility is requested.
EWMA can be understood as the limiting case with no constant and persistence equal to one, so its forecasts do not revert to a distinct long-run mean.
Exam questions can ask for a one-day update, the long-run variance, a multi-step forecast, or an interpretation of parameter changes. Begin by checking that coefficients are non-negative and that the persistence condition is satisfied.
Keep returns and variances in decimal form, distinguish a percentage volatility from a squared decimal variance, and state whether the forecast moves toward or away from its long-run level.
What this chapter covers
- 01
Volatility as the SD of log returns; annualise by √252
- 02
Implied volatility and the VIX (risk-neutral)
- 03
Equally-weighted estimator σ_n² = (1/m)Σ u_{n−i}²
- 04
EWMA σ_n² = λσ_{n−1}² + (1−λ)u_{n−1}² (λ = 0.94)
- 05
ARCH(m) and GARCH(1,1) σ_n² = ω + αu_{n−1}² + βσ_{n−1}²
- 06
Long-run variance V_L = ω/(1−α−β); α+β < 1 gives mean reversion
- 07
N-step forecast E[σ_{n+t}²] = V_L + (α+β)ᵗ(σ_n² − V_L); EWMA = GARCH with ω = 0
Worked example · free
GARCH(1,1): long-run vol, a one-day update and a forecast
- 2Long-run variance V_L = ω/(1 − α − β) = 0.000002/(1 − 0.08 − 0.90) = 0.000002/0.02 = 0.0001.
- 1Long-run daily vol σ_L = √0.0001 = 0.01 = 1.00%/day.
- 2Update: σ_n² = 0.000002 + 0.08(0.010)² + 0.90(0.013)² = 0.000002 + 0.0000080 + 0.0001521 = 0.0001621.
- 1Today's vol = √0.0001621 = 0.012732 = 1.273%/day.
- 2Persistence α + β = 0.98. 10-step variance E[σ²₊₁₀] = V_L + (α+β)¹⁰(σ_n² − V_L) = 0.0001 + 0.98¹⁰(0.0001621 − 0.0001) = 0.0001 + 0.8171(0.0000621) = 0.00015074.
- 110-day-ahead vol = √0.00015074 = 0.012278 = 1.228%/day (reverting toward 1.00%).
Key terms
- EWMA volatility
- An exponentially-weighted estimator σ_n² = λσ_{n−1}² + (1−λ)u_{n−1}², giving recent shocks more weight (RiskMetrics uses λ = 0.94). It is GARCH(1,1) with ω = 0, so it has no mean reversion.
- GARCH(1,1)
- σ_n² = ω + αu_{n−1}² + βσ_{n−1}², where ω = γV_L. It blends a long-run variance, the latest squared shock (weight α) and yesterday's variance (weight β), and mean-reverts when α + β < 1.
- Long-run variance (V_L)
- The unconditional variance a GARCH process reverts to, V_L = ω/(1 − α − β). Rearranging from ω to V_L is OFF the provided formula sheet, so memorise it.
- Persistence (α + β)
- The speed of mean reversion in GARCH: closer to 1 means shocks decay slowly and volatility stays elevated longer. α is the reaction to new shocks; β is the carry-over of yesterday's variance.
- Volatility clustering
- Volatility clustering is the empirical tendency for large absolute returns to be followed by large absolute returns and quiet periods to be followed by quiet periods.
- Mean reversion
- Mean reversion in a variance model is the tendency of conditional variance forecasts to move back toward a finite long-run variance after a shock.
Forecasting Volatility & Correlations FAQ
Is the GARCH long-run-variance formula on the sheet?
The basic GARCH relation V_L = ω/(1−α−β) appears, but the unit flags that you must be able to REARRANGE it (recover V_L from ω, or ω = γV_L) quickly under pressure — practise both directions, since the intercept ω = γV_L trips many students.
How is EWMA related to GARCH?
EWMA is exactly GARCH(1,1) with ω = 0 and α + β = 1. Because there is no long-run-variance term, EWMA never mean-reverts — its forecast for every future horizon is just today's variance. GARCH, with α + β < 1, pulls forecasts back toward V_L.
What do high α and high β tell me?
α (typically ~0.03–0.10) is how strongly volatility reacts to the latest shock; β (typically ~0.85–0.95) is how persistent yesterday's volatility is. A high-α, low-β process is 'spiky' (jumps up then fades fast); a high-β process keeps volatility elevated for a long time.
Why do I forecast variance before volatility?
EWMA and GARCH are written as variance recursions, so shocks, coefficients, and the long-run target all operate on squared returns or variances. Keep the calculation in variance units through the update and multi-step forecast. Take a square root only when the question asks for volatility, otherwise the scale and persistence calculation become inconsistent.
How can I check a multi-step GARCH forecast?
Compare today's conditional variance with the long-run variance. A stable model should move the forecast gap toward zero at a rate governed by persistence. If current variance is above the long-run value, later forecasts should decline but remain above it; if current variance is below, they should rise toward it.
Exam move
Use one consistent GARCH worksheet. Record the constant, shock coefficient, lagged-variance coefficient, latest squared return, and previous variance in separate boxes. Compute persistence first and verify that it supports a stable long-run variance. Next calculate the long-run variance, perform the one-period update, and project the variance gap several steps ahead.
Delay every square root until a volatility rather than a variance is actually requested. After each answer, state the direction of mean reversion and compare the forecast with both today's variance and the long-run value. For EWMA, practise the same update while recognising that the weights sum to one and there is no separate long-run anchor.
Use quick parameter comparisons to build intuition: a larger shock coefficient creates a sharper immediate reaction, a larger lagged-variance coefficient makes shocks last longer, and persistence near one slows reversion. Treat percentage-to-decimal conversion as a separate written step so squared values do not acquire a scale error.
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