GARCH(1,1)

 The GARCH process is a popular stochastic process which has been fairly successful in modeling financial time series [27]. In general the GARCH(p,q) process has p+q+1 parameters which must be fit to the data. GARCH(1,1) is the simplest of this class with 3 parameters.

The GARCH(1,1) based risk model is the first model we present for which optimization is involved - and we make full use of all 2251 data points available to us from each of the 10 log differenced price change series k. For convenience we define $t\in[-1249,-1001]$as representing the build up period, $t\in[-1000,0]$ as representing the in-sample period and $t\in[1,1001]$ as the period over which we construct the 1000 prediction-realization pairs associated with this model.