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The extremal properties of GARCH processes are of wide interest for market risk management. Only for simple GARCH(1,1) extremes have been widely discussed. Much remains to be found about the dependence structure of extreme values for higher order GARCH. Although recent research has identified the multivariate regular variation property of stationary GARCH(p,q) processes, currently there are no methods for numerically evaluating extreme components, like the average length of an extreme period. Only very simple special cases are well understood, but these are of little practical relevance, as bounded distribution of the error term is assumed. We present a unified toolkit that tackles the above critics and it is usable for Integrated GARCH(p,q) processes, assuming innovations with unbounded support or asymmetry. With our method we are able to generate the forward tail chain of the process to derive all extremal features. The convergence of our numerical algorithm is very fast due to an efficient implementation of a particle filtering simulation technique.