A Short Note on Application of Bachelier Option Model to FRAs; with Comment on Black 1976

Abstract

This Short Note arises from questions about applying the Bachelier model to FRAs which arose as interest rates went negative leading to a failure of the Black 1976 model used by convention as the market price model.

The paper reviews the Black ’76 model form highlighting reasons for this failure. Alternate Bachelier model based on log normal and yield standard error measures are shown to have similar limitations. Other extant models derived from the Black-Scholes model form have consistent failings.

Given this, the author defines a Bachelier spread model in terms of yields and the related note price. It is shown these create the same price outcomes for all positive, negative and nil values of the note yield. It can then be shown the Black ’76 model for instance systematically misprices as the Note yields go to zero and fails below zero. Recommendation is the price model be moved from the log normal, yield standard error model form to the Bachelier style spread model formulated consistent with that detailed in this paper.