Skip to content
All library documents

Choosing a Rate Model for Pricing a Zero-Strike Floor

Article Quant Q&A · Author: Ladislao Vidal
Read the original

Summary

The document discusses why the standard Black formula cannot be applied directly to a zero-strike interest-rate floor: its log-forward-to-strike term is undefined when the strike is zero. The answers explain that model suitability depends on the forward rate. Under a lognormal Black–Scholes assumption, a positive forward implies zero probability of the rate falling to or below zero, making the zero-strike floor worthless within that model.

When rates can be negative, the lognormal assumption is unsuitable. The suggested alternatives are a displaced lognormal model, which shifts the underlying before assuming lognormal behavior, or a normal Gaussian model. These are brief conceptual recommendations rather than a pricing derivation or comparison of model calibration and performance. The discussion gives no market data or evidence for choosing between the alternatives, so the appropriate model depends on the rate dynamics being represented.

Key ideas

  • The Black formula’s logarithmic term is undefined at a zero strike.
  • With a positive forward, a lognormal model assigns no probability to a rate at or below zero.
  • A displaced lognormal model can accommodate negative rates by shifting the modeled variable.
  • A normal model is another suggested choice when rates may be negative.

Tags

Fixed incomeDerivatives pricingVolatility
Full text
# Pricing 0% interest rate Floor Black Model


# 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?

## Answer by Antoine Conze (score 3, accepted)

https://quant.stackexchange.com/a/36775

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.

## Answer by Randor (score 0)

https://quant.stackexchange.com/a/37270

in interest rates , the logN model is not correct since rates can be negative. so you should use the Normal (Gaussian) model, this is the standard now.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.