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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..).

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  • $\begingroup$ Will take a look in a bit. Thanks for writing. $\endgroup$
    – Arshdeep
    Commented Jun 22, 2020 at 17:55

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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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  • $\begingroup$ Awesome. Thanks. $\endgroup$
    – Arshdeep
    Commented Jun 22, 2020 at 18:15

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