Co-Terminal Swaptions as Calibration Instruments for Callable Bonds

Summary

Co-terminal, or diagonal, swaptions share a common final maturity while their expiries step forward and their underlying swap tenors shorten. The document illustrates the structure with a callable bond that can be redeemed at regular intervals before maturity. Each call date is associated with a swaption beginning at that date and ending at the bond’s maturity, creating a sequence that can approximate the bond’s multiple call opportunities.

These swaptions matter in calibration because they form building blocks for the volatility surface and provide instruments aligned with the callable bond’s exercise dates. Some points on the diagonal may be directly traded; for others, a calibrator must select a close market match. Instruments are often chosen at-the-money-forward, though calibration can also incorporate strike skew. Strike selection depends on the bond’s terms and structure, including features such as amortization, floating rates, or fixed coupons. The explanation is illustrative and does not provide a full pricing or calibration algorithm, nor does it quantify how the choice of proxy instruments affects model fit.

Key ideas

• Co-terminal swaptions have expiries that advance while their underlying swap tenors shorten toward a common maturity.
• A sequence of such swaptions can represent the possible call dates of a callable bond.
• The diagonal offers calibration instruments aligned with the bond’s exercise schedule.
• Some diagonal points may require using the closest available traded instruments.
• Calibration can use at-the-money-forward strikes or include strike skew, depending on the application.
• The bond’s structure influences the appropriate swaption strikes and calibration choices.

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# What is the definition of "co-terminal swaptions"? why they are important in the calibration process?


# What is the definition of "co-terminal swaptions"? why they are important in the calibration process?












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.

## Answer by oronimbus (score 5, accepted)

https://quant.stackexchange.com/a/63793

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:

This was obtained from this paper.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.