could anyone help me understand the definition of "co-terminal" swaptions? What are they? Can you provide an example to illustrate? And why such instruments are important in model calibration?

Thanks vm in advance.


1 Answer 1


This question has partially already been answered here.

Let's do a simple example to illustrate the idea though. Take a 5y Bermudan callable S/A USD bond. How would you reconstruct the multi-call feature using vanilla swaptions? Well, the issuer can call every 6 months so they're long a 6m4.5y swaption on the first call date. For the second call date they're long a 1y4y, on the 3rd a 1.5y3.5y, on the fourth a 2y3y,...., and on the penultimate call date a 4.5y6m swaption. These instruments are obviously important because they are the main building blocks of a volatility surface.

What does that mean for the calibration process? Some of these swaptions are actually traded e.g. 2y3y but for others you will need to find the closest match. What you end up with is a diagonal, or co-terminal, set of calibration instruments. You can see that the tenor of each swaption is decreasing as $T_{bond} - T_j$ for each expiry $T_j$ and fixed bond maturity $T_{bond}$. Often these are chosen to be ATMF (you can also include skew and calibrate diagonally to moneyness). The strike calibration depends a bit on the instrument/bond and its structure (e.g. amortizing, floating, fixed coupon).

In more visual terms, the calibration procedure looks like this:

Diagonal Swaptions

This was obtained from this paper.

  • $\begingroup$ Thanks very much for your help. $\endgroup$
    – pqsn
    Jun 16, 2021 at 14:23

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