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Forward start swaption

July 8th, 2022, 12:02 pm

Hi all,

I'm currently looking at a new type of swaption payoff.  Underlying is a 20year fixed vs float swap on 3m USD LIBOR.  The option is European, and can be exercised 10y after trade date.  However, the strike is not set until 5yrs from the valuation date, and will be set to the ATMF for the swap as of that date.

I am struggling a little to understand how best one might approach valuation of this product at trade date, largely because I can't think of the clean hedging strategy for it.  As a result, I also can't seem to see rational upper and lower bounds.  I can think about as of the trade date working out what the ATMF would be for the swap, and then pricing a 10y20y European swaption, but I imagine I'm missing some important factors.

Any help would be much appreciated.


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Re: Forward start swaption

July 10th, 2022, 9:36 pm

This feels more like an exam or interview question than a real life example, if only because 3M USD LIBOR is scheduled to be gone forever in less than a year. But the question is still interesting. You should first put yourself at the 5Y point (trade date is 0). On that date you have an ATMF 5x10 swaption. You should be able to value that option as a function of the state variables in your model. Details of your interest rate model will matter a lot here. In particular, how sensitive is the ATMF option value to the rate level? In a Gaussian rate model, very little. In a lognormal model, a lot. I don’t think it is a bad idea to recast the problem in a Black-Scholes world, where the secondary effect of discounting is eliminated. In that setting, you will initially have a small delta, zero gamma and theta (they go together), and lots of vega. The swaption story is a bit more complicated, but ultimately just depends on the joint distribution of your state variables at two points in time: 5Y and 10Y. So in a single factor model, it’s a (probably messy) 2-D integral.