# 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?

## 1 Answer

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.) – Frido Rolloos Jan 26 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. – Frido Rolloos Jan 26 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. – user35980 Jan 26 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. – Frido Rolloos Jan 26 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 $\gamma$. – Cettt Jan 27 at 7:48