# Duration/ Convexity of a Swaption

how does one calculate the duration/convexity of a swaption?

First and second derivatives of price wrt interest rate? At a minimum you could do it numerically.

But people wouldn’t normally call those duration and convexity but delta and gamma. I’ll bet a closed form solution is pretty straightforward.

I’m getting the idea there’s no analytic solution. Here’s an example paper: http://129.3.20.41/eps/fin/papers/0411/0411036.pdf

Well, he’s doing a more difficult problem than I was thinkiing of (also, someone should tell the guy who wrote the paper that it is the Heath-Jarrow-MORTON model not the Heath-Jarrow-MERTON model even though Merton is better known than Morton). I think a simple approach would be to just decompose the swap into a bunch of FRA’s, use Black’s model or whatever it’s called to price the option on eacah of those, and say that they all shift in some parallel way and call that the duration. Not nearly as cool as looking at Markovian no-arbitrage interest rate evolution models, but good enough for gov’t work.

ok… thanks ! is the duration of a swaption linear like the bonds? I mean to say… Wld there be a difference in duration if I use different level of shocks? for eg. 10 bps in one try and then 50 bps !

gamma is generally non-zero, so I think the answers to your questions are: no (and neither is it for bonds), and yes.