Using Delta-Gamma VaR with Cross-Gamma Terms at Scale

Summary

The document considers the computational cost of delta-gamma value-at-risk when a portfolio has many risk factors. A full cross-gamma matrix can be large, making the calculation appear comparable in difficulty to simulation methods. The question asks whether practitioners commonly limit cross-gamma calculations, noting diagonal-only gamma as a possible but potentially risky simplification.

The accepted response argues that delta-gamma methods can remain faster than Monte Carlo because VaR is calculated from sensitivities rather than thousands of simulated scenarios. It recommends retaining all nonzero cross-gamma terms where feasible and points out that their contributions can be computed in parallel across processors. The answer offers a computational rationale, not a validation study or criteria for deciding which terms are nonzero. Its efficiency claim therefore depends on the model, hardware, and implementation.

Key ideas

• Delta-gamma VaR summarizes portfolio changes using sensitivities rather than repeated full repricing scenarios.
• Cross-gamma terms capture interactions between pairs of risk factors.
• The response argues that calculating nonzero cross-gammas can still be faster than large Monte Carlo runs.
• Cross-gamma calculations can be distributed across CPU or GPU processing resources.
• The document does not provide a rule for approximating or validating a reduced cross-gamma set.

Tags

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


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












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 by phlsmk (score 5, accepted)

https://quant.stackexchange.com/a/414

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.