### Theta

We derive the formua for the Theta of a European Swaption. Differentiating the price formula with respect to t, we get

$$ Swaption = A \; Black $$ $$ \frac{\partial Swaption}{\partial t}= \frac{\partial}{ \partial t} \left( A Black \right)= A \frac{\partial Black}{ \partial t} $$

We know from Black's section that:

$$ \frac{\partial Black }{\partial t} = -\frac{e^{-r\tau}S \sigma}{2 \sqrt{\tau}} n \left( d_{1}\right) + r \; Black $$

Hence:

$$ \frac{\partial Swaption}{\partial t}= -A \frac{e^{-r\tau}S \sigma}{2 \sqrt{\tau}} n \left( d_{1}\right) + r A\; Black $$