# 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?

• You can treat the rate as 1+r and do away with negative rate woes...
– will
May 6 '17 at 0:11
• 1 is a big displacement! Most places add a number between .01 and .03 and do the displaced lognormal vol model or they simply use a normal vol model. Normal vols make sense except at these extremely low levels - when the central banks target a rate > 1%, the changes to policy that they make are pretty symmetric like a normal model would suggest - i.e. +/-25bps. When the policy rate gets to 25bps, then they start reducing rates by 10-20bps - even 5 bps, whereas an increase in rates would be larger and the asymmetry shows itself. Swaptions traders think in terms of normal bps vol/day. May 6 '17 at 14:30
• I mean to convert 2% to 102% and then not add the 1 later when using it.
– will
May 6 '17 at 14:37
• right - and what I m saying is that you might convert 2% to 5% if we displace by 0.03. A displacement by 100% is bigger than what practitioners use as far as I know. The SuperDerivatives data we get displaces by about 1-3% depending on the currency - when they displace. After displacement do lognormal returns as usual - then convert back as you say - so one could have a swaption with a strike as -1%. It is such a hack of a model (because the displacement is totally made up), but good enough for most purposes. May 6 '17 at 16:41
• Yes I know, shifted log normal. Im not taking about that.
– will
May 6 '17 at 17:05