# Pricing 0% interest rate Floor Black Model

I'm having some trouble pricing a 0% interest rate Floor following Black's formula. The term d1 contains the expresion Ln(Forward/Strike) if the strike is exactly 0 this expresion yields an indetermination and therefore we can't compute N(d1) in the pricing formula.

I was wondering how to work around this. A peer suggested to compute the regular formula using a 100% probability, but the results are not really meningful.

Aside from the displaced model is there any other adjustment to the black model to price a 0% interest rate floor?

• Usually for Floors, strike is given in absolute terms with reference to the at-the-money (ATM) level. In your case, the strike would therefore be Forward + 0% i.e. your floor is ATM and Ln(Forward/Strike) = 0. – JejeBelfort Nov 6 '17 at 13:21
• Hi, in my case I only have 0% interest as a given Floor. From what you said I understand that we have 100% probability to execute the Floor which will translate in a N(d2) = 1 and then plug the 1 in BS formula. Is that what you mean? – Ladislao Vidal Nov 6 '17 at 14:25
• Forget about this 100% probability, just plug in Black's formula the relevant parameters (i.e. Forward, Strike, Implied volatility and time-to-maturity) and compute the floor price. – JejeBelfort Nov 6 '17 at 14:47
• Still not sure about how to just "plug" the Strike since this is 0% d1 is simply undetermined – Ladislao Vidal Nov 6 '17 at 15:54
• No, the strike is equal to the forward price as you have an absolute strike of 0% – JejeBelfort Nov 6 '17 at 16:25

If the forward is $> 0$ then under the Black & Scholes model the probability of underlying rate being $\leq 0$ is zero, so that the $0\%$ strike floor is worth zero. If the forward is $\leq 0$ (as has been happening since rates went into the negative territory) then the Black & Scholes model is meaningless since it models the underlying as being log normal. This is why practitioners have resorted to the displaced log normal model, where the variable $underlying + displacement$ is assumed to be log normal, with $displacement$ an additional parameter of the model.