# Exercise Probabilities Vanilla Cap/Floor

When looking at the discounted pay-off formulas of a vanilla caplet and a vanilla floorlet

$\frac{\Delta\tau}{1+r_k\Delta\tau}\max(r_k-r_{cap},0)$

$\frac{\Delta\tau}{1+r_k\Delta\tau}\max(r_{floor}-r_k,0)$

$r_{cap/floor} =$ cap/floor rate between $t_k$ and $t_{k+1}$

$r_{k} =$ realised interest rate between $t_k$ and $t_{k+1}$

$\tau =$ reset period

then my intuition tells me that $N(d_2)$ could possibly be the Black risk-neutral exercise probability of the caplet.

• Is this assumption correct?
• If it is, what would be the correct risk-neutral probability of the floorlet? My guess is that it wouldn't be $N(-d_2)$, but I'm not sure.

Seems like you are asking for the risk neutral probability that a digital option on the respective rates will be ITM at expiration. Also, since you say you are assuming Black's model, you are implicitly assuming that rates can never go below zero - which we know to be wrong of course. As such, your intuition that the probability of the caplet exercising is indeed $N(d_2)$. Also, the probability of the floorlet exercising is $N(-d_2)$ just like any other digital put. Unless I am missing something?