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

Vega

We derive the formua for the Vega of a European Swaption. Differentiating the price formula with respect to \(\sigma\), we get

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

We know from Black's section that:

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

Hence:

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