I know what greeks are for standard options: just take the derivative with respect to some parameter, like spot, time, rate, etc.

But how does one calculate greeks for swaptions and capfloors? I was only able to find information on the delta, but what about gamma, vanna, theta, rho?

It seems only vega and volga are straightforward to calculate as a usual greek by differentiating wrt the volatility, but the others don't make much sense to me. Any information or a reference to a book/paper where this is treated will be appreciated.

What's especially tricky is that the method will obviously depend on how one will do the risk-management, and that is not obvious to me either.


2 Answers 2


Practically, few things in real life have convenient closed-form calculations.

Instead, you price some exotic, then you bump the various inputs, one or several at a time, up and down, by various small amounts, and re-price. There are seldom any short-cuts. (Autodiff can sometimes be a shortcut.)

This Wikipedia article actually has a good list of commonly used risk mesures: https://en.wikipedia.org/wiki/Greeks_(finance)

During model validation and ongoing model performance monitoring, you figure out which risk measures are important (or can become important under plausible large market moves). Then you put limits on them and calculate them a lot. There's no glamorous math here, just lots of brute force automated calculations.

Edit: thanks KermittFrog for reminding that different risk measures can be used for different purposes. Here's an example which actually involves some math. Suppose you hedge your interest rate risk with ED futures until 10 years and IR swaps after 10 years. You fit your IR curve from the hedging instruments. You bump each instrument and refit the IR curve. You reprice each instrument in your portfolio under each bumped IR curve. The resulting sensitivities tell you what notionals of hedging instruments you need to add to the portfolio in order to flatten the IR risk. But suppose further that you want to see the sensivities to IR swap rates from 1 to 10 years, for market risk limits monitoring. Since you don't use these swap rates to fit your IR curve, you can't just perturb them. But you can calculate how these swap rates change then the ED futures change, and multiply the ED futures sentivities of your portfolio by an inverse Jacobian to get a good estimate of the sensitivities to 1-10 swap rates.

As for the book question, I should mention Prof. Carol Alexandar's 4-volume Market Risk Analysis, which is probably an overkill. There is also a discussion of Greeks of exotic options in Chapters 7-9 of Leonardo Marroni, Irene Perdomo. Pricing and Hedging Financial Derivatives: A Guide for Practitioners.

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    $\begingroup$ On spot IMHO. A minor nuisance is the question what to bump (which inputs / quotes ); which is driven by how the model has been setup in the first place. You may see a difference between the front office and the risk controlling implementation, for example; or different bootstrap instruments etc. in short: you calculate sensitivities with respect to those inputs you deem relevant from a hedging / risk perspective ... $\endgroup$ Commented Nov 21, 2020 at 20:37
  • $\begingroup$ Thanks, Dimitri, wondering how we can calculate the swap rates change by the ED futures change? And how we can get the Jacobian matrix? $\endgroup$
    – Parting
    Commented Sep 3, 2021 at 15:27

If the question is how one defines Greeks for interest rate options, then it is a relatively straightforward extension of the concept from the basic idea for say equity options. They are defined as sensitivities to the inputs that go into pricing an option. Any half-decent interest rate derivatives book (search for interest rate modelling on Amazon, say) should cover it in detail. As inputs to interest rate models are fundamentally multi-dimensional as the whole interest rate curve is an input. So Greeks become multidimensional. It is common to think of delta as a vector (sensitivity to each forward rate in the interest rate curve), Gamma is a matrix, etc etc. Various aggregations are then used to make them more easily understood by humans, eg deltas could be summed up to come up with a "parallel" delta etc.

For European-style interest rate options such as swaptions, where they are priced as an option on a single rate (such as a given swap rate for a swaption), one can talk about 'asset delta', a sensitivity of the option to the change in that specific rate (very similar to Black-Scholes delta). Again these should be seen as aggregations as the 'fundamental' bucketed deltas.

If the question if whether one can calculate various Greeks for interest rate models in closed form, this is even less common than for say equity options due to the inherent multi-dimensionality that I mentioned.


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