# Gamma of interest rate derivatives

consider an interest rate derivative whose value $$V$$ depends on $$n$$ interest rates $$r_1, \dots, r_n$$. Hence $$V$$ is a function in $$n$$ variables $$V(r_1, \dots, r_n)$$. My question concerns the gamma $$\gamma$$ of this derivative with respect to parallel shifts.

Does it "always" hold that $$\gamma(r_1, \dots, r_n) = \sum_{i = 1}^n \frac{\partial^2 V}{\partial r_i^2}(r_1, \dots, r_n).$$

To clarify the notation: I define the delta $$\delta$$ of the derivative with respect to parallel shifts as $$\delta(r_1, \dots, r_n) = \lim_{h \rightarrow 0} \frac{V(r_1 + h, \dots, r_n + h) - V(r_1, \dots, r_n)}{h},$$ and $$\gamma$$ is then defined as $$\gamma(r_1, \dots, r_n) = \lim_{h \rightarrow 0} \frac{\delta(r_1 + h, \dots, r_n + h) - \delta(r_1, \dots, r_n)}{h}.$$

From a mathematical standpoint, I think the identity should not hold in general since mixed partial derivatives (i.e. $$\frac{\partial^2 V}{\partial r_i r_j}$$) should be considered as well. I also assume that $$V$$ is sufficiently smooth function such that all second partial derivatives exist and are continuous. However, literature seems to suggest that the identity holds in general. What do you guys think?

It's my understanding that indeed the cross partial derivative terms do have a contribution - so you're correct to say that what u really have to work with is a gamma matrix $$\gamma=[\frac{d^2V}{dr_idr_j}]_{i,j}.$$ In essence, the cross partial terms $$\frac{d^2}{dr_idr_j}$$ allude to the correlation between the various rates $$(r_1,...,r_n)$$. Assuming a high (c. 90%) correlation these terms generally can be ignored. For this reason the gamma as computed by your stated formula is a kind of 'local' gamma as opposed to a 'global' gamma given by the entire matrix. For calculating this 'global' gamma, banks generally use some form of principal components analysis (or some other linear transformation) as it's a less computationally expensive way of capturing the majority of the variance in rate sensitivity without having to compute the whole matrix.

• could you also look at / use total dollar gamma $\sum_{i,j} V_{ij} dr_i dr_j$, where $dr_i dr_j = \rho_{ij} dt$ defines the covariance structure, instead of PCA? (Subscripts of $V$ denote partial derivatives.)
– user34971
Commented Jan 26, 2021 at 18:48
• as the total dollar gamma term I wrote above balances the theta, I think it's the logical multi-asset analog of the familiar single asset BS gamma.
– user34971
Commented Jan 26, 2021 at 19:00
• unless i'm misunderstanding you, looking at the total gamma would be enlightening but ultimately not particularly useful if your objective is to hedge risk - this requires knowing which bucket/instrument i.e. which r_i, is contributing to your risk. PCA is shorthand way to isolate these relevant risk contributors. Commented Jan 26, 2021 at 19:01
• I am not sure hedging was the question. I think the OP was asking how to aggregate/calculate gamma wrt parallel risk, which can be done with PCA or with the covariance matrix (as a dollar amount), or perhaps I misunderstood the OPs question.
– user34971
Commented Jan 26, 2021 at 19:10
• Thank you for the answer. One thing is not clear to me. You talk about high correlation between the r_i. Of course they are modeled as random variables in the background, but for this particular problem the r_i are just inputs to a function. So the way I understand it, the stochastic correlation between r_i and r_j should not have any impact on $\gamma$. Commented Jan 27, 2021 at 7:48

Correct. You're starting with specifying the scenario where all rates move parallel. The correlation implicit there is 100%. There is nothing stochastic left. The gamma to a parallel shift is as cited originally. No matrix is needed.