Forecasting with ARCH models

In view of the fact that ARCH models make better use of the available information
set than standard time series methodology by allowing excess kurtosis in the distribution
of the data, the resulting model fits the observed data set better. However,
perhaps the strongest argument in favor of ARCH models lies in their ability to
predict future variances.
The way the ARCH model is constructed it can ‘predict’ the next period’s variance
without uncertainty. Since the error term at time t, et , is known we can rewrite
equation (2.5) as
htò1óuòa2
t (2.10)
Thus, the next period’s volatility is found recursively by updating the last observed
error in the variance equation (2.5). For the GARCH type of models it is possible to
deliver a multi-step-ahead forecast. For example, for the GARCH(1, 1) it is only
necessary to update forecasts using:
E(htòsD't )óuò(aòb)E(htòsñ1D't ) (2.11)
where u, a and b are GARCH(1, 1) parameters of equation (2.8) and are estimated
using the data set available, until period t.
Of course, the long-run forecasting procedure must be formed on the basis that
the variance equation has been parameterized. For example, the implied volatility at
time t can enter as an exogenous variable in the variance equation:
E(htò1D't )óuòae2
t òbhtòdp2
t if só1 (2.12a)
E(htòsD't )óuòdp2
t ò(aòb)E(htòsñ1D't) ifsP2 (2.12b)
where the term p2
t is the implied volatility of a traded option on the same underlying
asset, Y.