## European Swaption's Price and Greeks

We derive the formulae for the Greeks/derivatives of the Black equation for a European Receiver and Payer Swaptions

### 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:

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