How to limit the nbr of cross-gamma calculations in a delta-gamma VaR calculation?

Question

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.

Answer 1

Using the delta-gamma approximation is still significantly faster even if you incorporate all of the nonzero cross-gammas. This speedup comes from the fact that you're using just the delta/gamma/cross-gamma parameters to calculate your VaR instead of 10,000+ Monte Carlo simulations. The calculations of each cross-gamma to the overall VaR, furthermore, can be parallelized across CPU/GPU threads and cores.