Is there a way to use normal volatility in the Black–Scholes–Merton model to value interest rate caps? [duplicate]

Question

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?

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).