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

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

where

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

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

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

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

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

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