Many times, we want to calculate VaR using some parametric approach (delta-normal approximation for instance) when historical simulation or monte carlo are simply to slow. This is fine as long as only the deltas are needed and the instruments are reasonably linear.
But when the approach is extended to use both deltas and gammas, it is no longer certain that the approach is computationally efficient compared to the simulation-based methods since the calculation of a complete matrix of cross-gammas between the risk factors (nbr of RFs >10000) becomes a very heavy task.
Are there any "justifications"/old-wives-tales/adhoc methods on how to limit the nbr of gammas to calculate that has been used used with good result in the industry? One obvious simplification is of course to skip all cross-gammas and only calc the diagonal of the matrix but that feels dangerous.