Question:

Assume current annualized volatility is 30%. Under a daily model, the GARCH(1,1) parameter estimates are:

omega = 0.00018
alpha = 0.10
beta = 0.70

(i) What is the implied long-run average daily volatility?
(ii) What is the persistence of the specification?
(iii) What is the daily volatilty forecast in four days (n+4)?

Answers:

(i)
L.R. average variance = Omega / [1 - alpha - beta]
L.R. average variance = 0.00018 / [1 - 0.1 - 0.7] = 0.0009
L.R. average volatility = SQRT() = 3.0%

(ii)
Persistence = (alpha + beta)
If it helps, think of persistence as the weights that are NOT mean-reverting.
In this case, persistence = 0.01 + 0.7 = 0.8

(iii)
Note, if the current annualized volatility is 30%, then the current daily variance is given by:
Daily Variance = 30%^2/250 = 0.00036

Forecast is given by:
E[variance in k days] = Long-run variance + [(alpha + beta)^k][current variance - long run variance]
E[variance in 4 days] = 0.0009 + [(0.8)^4][0.00036 - 0.0009] = 0.00068
E[volatility in 4 days] = SQRT() = about 2.61%