Why Rates Bermudan Swaption Value Is Convex in Strike

Summary

The note addresses whether a rates Bermudan callable or swaption remains convex as its strike changes, even though possible exercise dates correspond to swaps with different remaining maturities. It represents the exercise payoff at a given date as the positive part of an affine function of strike, with coefficients that depend on the date and market outcome.

A positive-part affine payoff is convex in strike. Applying the American-option convexity argument across the Bermudan exercise structure supports the conclusion that the callable or swaption value is convex in strike. The explanation gives a structural payoff argument rather than numerical evidence. It assumes the stated payoff form and relies on the option convexity result; it does not discuss model calibration, rate dynamics, or practical hedging implications.

Key ideas

• At each exercise date, the payoff is modeled as the positive part of an affine function of strike.
• That payoff form is convex with respect to strike.
• The convexity argument extends to the Bermudan rates callable or swaption under the stated assumptions.
• Different underlying swap maturities at different exercise dates do not alter the payoff-based argument.
• The note offers a theoretical argument and no numerical or empirical validation.

Tags

# Convexity of a rates Bermudan w.r.t strike


# Convexity of a rates Bermudan w.r.t strike












Recently there was a nice question asked on convexity of American put w.r.t strike: Convexity of an American put option

Does the same hold for a Bermudan option in rates, where they underlyings are different swap rates? Say you have a Bermudan callable every one year from now for the next 5 years, and the underlyings are fixed-float swaps which have a tenor+maturity=10y (i.e, if I exercise at 1y, I enter into a 9 y swap, if I exercise at 2 y, I enter into an 8y swap..).

## Answer by Hans (score 1, accepted)

https://quant.stackexchange.com/a/55064

The Bermudan (American) callable/swaption is convex with respect to the strike.

The payoff function of the Bermudan (American) callable/swaption is of the form, with implicit dependence on sample $\omega$, $$g(t,K)=\big(a(t)-b(t)K\big)_+$$ where $t$ is the time the swap (interest) rate is set and $K$ is the strike. $g(t,K)$ is obviously convex with respect to $K$. Apply the answer to Convexity of an American Option and you obtain the desired result.

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