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In this paper, we derive some asymptotic theory for the extremogram andcross-extremogram of a bivariate GARCH(1,1) process. We show that the tails ofthe components of a bivariate GARCH(1,1) process may exhibit power law behaviorbut, depending on the choice of the parameters, the tail indices of thecomponents may differ. We apply the theory to 5-minute return data of stockprices and foreign exchange rates. We judge the fit of a bivariate GARCH(1,1)model by considering the sample extremogram and cross-extremogram of theresiduals. The results are in agreement with the iid hypothesis of thetwo-dimensional innovations sequence. The cross-extremograms at lag zero have avalue significantly distinct from zero. This fact points at some strongextremal dependence of the components of the innovations.

The extremogram and the cross-extremogram for a bivariate GARCH(1,1) process