Pricing Interest Rate Caps with Normal Volatility

Summary

The document describes using a normal, or Gaussian, model to price caplets and floorlets when rates can be negative. Under the stated setup, the forward rate evolves as a Gaussian martingale with constant absolute volatility, so its value at expiry has a normal distribution. The option payoff is then valued with the Bachelier formula, using the forward rate, strike, volatility, and time to expiry.

The answer notes that this approach has been used in markets where negative interest rates became a lasting feature. It also distinguishes the model used to quote volatility: an observed option price can be inverted into either Black implied volatility under lognormal dynamics or Bachelier implied volatility under normal dynamics. The document gives the core pricing relationship but does not discuss calibration, curve construction, or the details of applying the formula to a complete interest-rate cap, so those steps require additional knowledge.

Key ideas

• A Gaussian model allows the underlying forward rate to take negative values.
• With constant normal volatility, the forward rate at expiry is normally distributed around its current value.
• The Bachelier formula prices calls and puts on a Gaussian underlying using the forward, strike, volatility, and expiry.
• The same observed option price can be expressed as either Black implied volatility or Bachelier implied volatility, depending on the assumed dynamics.
• Normal-volatility conventions are relevant in markets where negative rates are possible.

Tags

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


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












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 by siou0107 (score 2, accepted)

https://quant.stackexchange.com/a/74898

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).

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