FINC6023 · Financial Risk Management
Forecasting Volatility & Correlations
Forecasting Volatility & Correlations is the engine room that feeds σ into every VaR. You move from the equally-weighted estimator to EWMA (which decays old shocks geometrically, RiskMetrics λ = 0.94) and then to GARCH(1,1), σ_n² = ω + αu_{n−1}² + βσ_{n−1}², which adds mean reversion toward a long-run variance V_L = ω/(1−α−β). The off-sheet trap the unit flags lives here: you must be able to rearrange to V_L under exam pressure, and to project σ forward t steps as it reverts toward V_L.
What this chapter covers
- 01Volatility as the SD of log returns; annualise by √252
- 02Implied volatility and the VIX (risk-neutral)
- 03Equally-weighted estimator σ_n² = (1/m)Σ u_{n−i}²
- 04EWMA σ_n² = λσ_{n−1}² + (1−λ)u_{n−1}² (λ = 0.94)
- 05ARCH(m) and GARCH(1,1) σ_n² = ω + αu_{n−1}² + βσ_{n−1}²
- 06Long-run variance V_L = ω/(1−α−β) [rearrangement is OFF-SHEET to apply]
- 07α = reaction, β = persistence; need α+β < 1 for mean reversion
- 08n-step forecast E[σ_{n+t}²] = V_L + (α+β)ᵗ(σ_n² − V_L); EWMA = GARCH with ω = 0
GARCH(1,1): long-run vol, a one-day update and a forecast
- 2 marksLong-run variance V_L = ω/(1 − α − β) = 0.000002/(1 − 0.08 − 0.90) = 0.000002/0.02 = 0.0001.
- 1 markLong-run daily vol σ_L = √0.0001 = 0.01 = 1.00%/day.
- 2 marksUpdate: σ_n² = 0.000002 + 0.08(0.010)² + 0.90(0.013)² = 0.000002 + 0.0000080 + 0.0001521 = 0.0001621.
- 1 markToday's vol = √0.0001621 = 0.012732 = 1.273%/day.
- 2 marksPersistence 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.
- 1 mark10-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.
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.
Exam move
Rehearse the three GARCH moves as one routine: compute V_L, do a one-day update, then project t steps with (α+β)ᵗ. Memorise the V_L rearrangement because it is off-sheet, and be ready to state EWMA = GARCH with ω = 0.
