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Black-76 Inputs and Gaussian Pricing for Interest Rate Futures Options

Article Quant Q&A · Author: PingPing
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Summary

The document asks how to select the forward and strike inputs for Black-76 when interest rate futures are quoted as a price derived from an interest rate. The included response does not resolve that input convention, but reports using a Gaussian, or Bachelier, model to handle negative rates, which Black’s lognormal framework cannot represent directly.

The author says most interest rate futures options they encountered were short dated and that, for their use, the pricing differences between models were within their tolerance. This is a practitioner’s brief account rather than a derivation or a comparison backed by market data. It offers a possible modeling choice for negative-rate conditions, while leaving contract-specific quoting, conversion of strikes, calibration, and the suitability of the approximation unaddressed.

Key ideas

  • The source raises the question of whether Black-76 inputs should use quoted futures prices or corresponding interest rates.
  • The response describes using the Bachelier model when negative rates must be represented.
  • The reported tolerance for model differences is specific to the author’s short-dated use case.
  • The document does not provide a derivation or settle the conversion of futures prices and strikes into model inputs.

Tags

OptionsFuturesFixed incomeDerivatives pricing
Full text
# Black 76 for Options on Interest Rate Futures


# Black 76 for Options on Interest Rate Futures












This is my first time using Black76 to value options on IR futures and I have a question on $F$ and $K$.

I understand the price for an IR future is usually quoted as $100 - r$. Do I use this price as $F$ in Black76 or do I have to use $r$ and also convert the strike $K$ to an interest rate?

If I have to use interest rates for $F$ and $K$, how should I treat negative rates i.e. strike prices greater than 100?

## Answer by PingPing (score 1)

https://quant.stackexchange.com/a/9615

I've ended up implementing a different model for -ve rates. I've used Bachelier's (Guassian) model that allows negative values. Most of the IR futures options are short-dated so model differences are within my tolerance.

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.