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!