# Black-76 Model for Swaption Price and Greeks

I'm in the early stages of developing a swaption pricing model.

Suppose $t_1$ is the tenor of the swap rate in years, $F$ is the forward rate of the underlying swap, $X$ is the strke rate of the swaption, $r$ is the risk-free rate, $T$ is the swaption expiration (term) in years, $\sigma$ is the volatility of the forward-starting swap rate and $m$ is the compounding per year in swap rate.

As I understand, the Black-76 model for the price of a European payer swaption is

$$P_{PS}= \frac{1-(1+\frac{F}{m})^{-t_1m}}{F}\cdot e^{-rT}[F\Phi(d_1)-X\Phi(d_2)],$$

where

$$d_1=\frac{\ln(\frac{F}{X})+ \frac{\sigma^2T}{2}}{\sigma\sqrt{T}}\quad\text{and}\quad d_2 = d_1-\sigma\sqrt{T}.$$

Equivalently, for a receiver swaption, the price is given by the formula

$$P_{RS}= \frac{1-(1+\frac{F}{m})^{-t_1m}}{F}\cdot e^{-rT}[X\Phi(-d_2)-F\Phi(-d_2)].$$

This is like the original formulae in Black's model except for the additional term $\frac{1-(1+\frac{F}{m})^{-t_1m}}{F}$(source). In additional to validating that these are indeed the correct pricing formulae, I'd like to derive formula for two greeks in particular: theta ($\Theta$) and gamma ($\Gamma$).

Theta

\begin{align} \Theta_{PS} =\frac{\partial P_{PS}}{\partial T} = \Bigg[\frac{1-(1+\frac{F}{m})^{-t_1m}}{F}\Bigg]\cdot\frac{\partial}{\partial T}\{e^{-rT}[F\Phi(d_1)-X\Phi(d_2)]\}= \frac{1-(1+\frac{F}{m})^{-t_1m}}{F}\cdot\Bigg[-\frac{Fe^{-rT}\phi(d_1)\sigma}{2\sqrt{T}}-rFe^{-rT}\Phi(-d_1)+rXe^{-rT}\Phi(-d_2)\Bigg] \end{align}

where the term in the square parentheses in the standard formula for the theta of a put option under Black's model. $\Theta_{RS}$ derived analogously.

Does anyone know a source for the delta and gamma of a swaption under Black model?

Many thanks