23.6 USING GARCH(1,1) TO FORECAST FUTURE VOLATILITY

The variance rate estimated at the end of day n1 for day n, when GARCH(1,1) is used, is

σn2=(1αβ)VL+αun12+βσn12

so that

σn2VL=α(un12VL)+β(σn12VL)

On day n+t in the future,

σn+t2VL=α(un+t12VL)+β(σn+t12VL)

The expected value of un+t12 is σn+t12. Hence,

E[ σn+t2VL ]=(α+β)E(σn+t12VL)

where E denotes expected value. Using this equation repeatedly yields

E[ σn+t2VL ]=(α+β)t(σn2VL)

or

E[σn+t2]VL+(α+β)t(σn2VL)(12.13)

This equation forecasts the volatility on day n+t using the information available at the end of day n1. In the EWMA model, α+β=1 and equation (23.13) shows that the expected future variance ...

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