R16.P1.T2. Hull - expected value of u(n+t-1)^2

[uness_o7]

In Hull - Risk Management and Financial Institutions, it is stated, in page 222 (10.10 using GARCH(1,1) to forecase future volatility), that: "the expected value of u(n+t−1)^2 is σ(n+t−1)^2".
Is this something obvious? Can anybody explain why this should be the case?
Thanks!

 

[ami44]

In Hull - Risk Management and Financial Institutions, it is stated, in page 222 (10.10 using GARCH(1,1) to forecase future volatility), that: "the expected value of u(n+t−1)^2 is σ(n+t−1)^2".
Is this something obvious? Can anybody explain why this should be the case?
Thanks!

You need the assumption, that the drift term of u can be neglected. If u(t) is a random variable than is it's variance defined as
σ(t)^2 = E( u(t)^2 ) - E( u(t) )^2
If you now assume, that the E( u(t) )^2 can be neglected against the E( u(t)^2 ) term than you get to your result.
u(t) is modelling the return for a time period dt:
u(t) = ( s(t) - s(t - dt) ) / s(t - dt)
that means, that E( u(t) )^2 is proportional to dt^2 while E( u(t)^2 ) scales with dt. So if dt is small enough, you can neglect the drift term.