# Delta-Gamma VaR approximation and cross-gamma

Suppose we have a portfolio of say two vanilla options (e.g. on two index futures). One option A with underlying X and a second option B with underlying Y. I'm trying to calculate the delta-gamma value-at-risk approximation but I'm a bit confused with the gamma matrix part and especially the non-diagonal elements a.k.a cross-gamma. How exactly can we calculate this?

There is this great discussion here that suggests to use a finite difference scheme to approximate cross-gamma: Compute cross-gamma

The discussion is on option with several underlyings which is different than my problem here. But it is also mentioned in comments that it could very well be applied at the portfolio level (which is my case here).

I followed this approach and calculated the market value of my portfolio of options with epsilon set as +/- 1% on each underlying. The problem is that only option A is impacted by a shift in price of underlying X (option B does not depend on X) and only the second option B is impacted by a shift in price of underlying Y. Thus I always find a cross-gamma equal to 0 when using the formula (elements offset each other). Am I missing something?

Also, another question I have is when I assume a change in price of underlying X by +/- epsilon, should I assume all other parameters equal to calculate the new option price of A? (especially keeping the same implied volatility)

Thanks

• Your cross gamma come from moves in spot and implied volatilities, $d^2f/dSd\sigma$. Cross does not necessarily imply these two factors being from different underlyings - they can can come from different risk types as well (IV and underlying). Jun 27, 2021 at 20:56
• @Kermittfrog Thank you, makes sense. To be frank I was looking at an example from C. Alexander's book. Example is giving the following indications to compute the var: "The bond options portfolio has a position delta of 10, a position gamma of 2.5, a pv of 1000 and a delta equivalent of 1m. The stock options portfolio has a position delta of 20, a position gamma of 0.5, a pv of 250 and a delta equivalent of 5m. The cross-gamma between bonds and equities is −0.15." I was just curious how this cross-gamma could be calculated/approximated.
– CTXR
Jun 27, 2021 at 21:27
• That would be something like the sensitivity to a parallel interest rate curve shift and a stock market index shift at the same time. Jun 28, 2021 at 4:46