I am trying to understand if there is a version of the Black–Scholes–Merton model that can use Normal volatilities instead of Lognormal volatilities while valuing interest rate caps and floors?


1 Answer 1


This has actually been done widely in the industry since negative interest rates became a long-term feature of financial markets (JPY, EUR, CHF).

When your underlying is a Gaussian martingale, following SDE \begin{equation} d F_t = \sigma \, dW_t \end{equation} then $F_T \sim \mathcal{N} \left(F_0, \sigma^2 T\right)$ and the (numeraire-rebased) price of a call/put option $\mathbb{E} \left[\left(F_T - K\right)^+\right]$ is given by the Bachelier formula

\begin{equation} \omega \left(F_0 - K\right) \Phi \left(\omega d\right) + \sigma \sqrt{T} \phi \left(d\right) \end{equation} where $\omega = \pm 1$ is a dummy variable to indicate if you are pricing a call or a put, $\Phi$ and $\phi$ are the standard Gaussian CDF and PDF and $d = \frac{F_0 - K}{\sigma \sqrt{T}}$.

You can technically invert any option price to give you either a Black implied volatility (if you assume a lognormal dynamics) or a Bachelier implied volatility (if you assume Gaussian dynamics as above).

  • $\begingroup$ Thank you so much. But I want to understand if there any extensions/modifications to the Black Scholes model that can be use Normal volatility instead of the Lognormal vols. $\endgroup$
    – Ambat
    Commented Mar 11, 2023 at 16:04
  • $\begingroup$ What do you mean? By definition, the Black model assumes a lognormal dynamics, so the volatility in its formulas is a lognormal volatility. $\endgroup$
    – siou0107
    Commented Mar 11, 2023 at 19:50
  • $\begingroup$ @siou0107, I think the OP wants to use a normal vol quote directly in the black model (after some conversion). Also, if there are negative rates, you cannot use Black (vol) directly. $\endgroup$
    – AKdemy
    Commented Mar 11, 2023 at 20:49
  • $\begingroup$ If there are negative rates, but quoted volatilities are those of a Black model, then they are of a shifted lognormal model, and the dealer usually indicates the shift they use. $\endgroup$
    – siou0107
    Commented Mar 12, 2023 at 10:40

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