Bermudan option exercise probability when rates rise

Question

I am looking for an explanation of what happens to the Bermudan exercise probability (i.e. does probability of early exercise go higher if rates rise or lower) w.r.t rates. This is of course with regard to a rates bermudan. I pay fixed and recieve floating, and I have a right to enter. Early exercise is the exercise event at the first date.

My thought is that, fixing a particular 'scenario' (i.e. realized evolution of the yield curve'), if i consider 2 exercise dates, then at date 1, when I compare the swap with the European option that corresponds to date 2 exercise, when i use higher rates, the swap (delta=:1) will increase in value more than the european (delta <1); and thus, the early exercise boundary can only widen (i.e. i will get additional scenarios where early exercise is optimal). I'm using delta as 'sensitivity to a swap'.

Edit: I've obviously used some approximations, as in the 2nd swap is not the same as the first swap and therefore should have a different sensitivity. But as long as the difference is not too wild (say they only differ by a FRA payment), this should hold.

A book I'm reading tends to disagree, but gives no reason, stating it should be obvious. Any advice? Thanks!

Answer 1

Ok as an example consider a 1yr-10yr 3pct Bermudan payer (the right to pay fixed at 3pct vs libor starting at any annual date from 1yr onwards with a maturity of 11yrs from today). For simplicity assume a flat yield curve.

If rates are 1pct, the probability of exercise on the first date is low (a long way to cross the 3pct strike). If rates are 6pct, the probability of exercise on the first date is high (deep in the money). Thus probability of exercise on the first date increases montonically with the current swap rate, at least in these idealized circumstances.

One could ask for each rate scenario, what is the most likely exercise date? At 1pct rate level , one might find that the 5th exercise date is most likely , and at 1.5pct it is the 4th, etc until at 3pct it is the second, and above 3pct it is the first. Those are all just estimates based on some experience.

Hope that helps.