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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?

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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.

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  • $\begingroup$ 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.) $\endgroup$ – Frido Rolloos Jan 26 at 18:48
  • $\begingroup$ 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. $\endgroup$ – Frido Rolloos Jan 26 at 19:00
  • $\begingroup$ 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. $\endgroup$ – user35980 Jan 26 at 19:01
  • $\begingroup$ 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. $\endgroup$ – Frido Rolloos Jan 26 at 19:10
  • $\begingroup$ 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 $\gamma$. $\endgroup$ – Cettt Jan 27 at 7:48

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