## 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

### European Swaption Price Formula

We summarise the formulae for the Price and Greeks/derivatives of Receiver and Payer Swaptions under the Black's assumptions below: Note the formulae assume a Notional of 1, so multiply all the results by the Notional amount if it is different.

$$Swaption Price = A \; Black$$ $$Delta (\Delta) = e^{ -r \tau}A\phi \frac{K}{S} N{\left( \phi d_{2} \right)} + \frac{1}{S} \frac{ M }{ {\left( 1+ \frac{S}{f} \right)}^{f M + 1}} Black$$ $$Gamma (\Gamma) = A \frac {e^{-r\tau}} {S \sigma \sqrt{\tau}} n{\left (d_{1} \right)} - \frac{ M \left( M+ \frac{1}{f} \right)}{ {\left( 1+ \frac{S}{f} \right)}^{f M + 2}} \frac{Black}{S} - e^{-r\tau} \phi K N{\left( \phi d_{2}\right )} \frac{2}{S^2} \left( A - \frac{ M }{ {\left( 1+ \frac{S}{f} \right)}^{f M + 1}} \right)$$ $$Vega = A e^{ -r \tau} S \sqrt{\tau} n{\left (d_{1} \right)}$$ $$Theta (\Theta) = -A \frac{e^{-r\tau}S \sigma}{2 \sqrt{\tau}} n \left( d_{1}\right) + r A\; Black$$

Where,

$$A =\frac{1}{S} {\left( 1-\frac{1}{{(1+S/f)}^{fM}} \right)}$$ $$Black = e^{-r\tau} \left[ \phi S N{\left( \phi d_{1} \right )}- \phi K N{\left( \phi d_{1} - \phi \sigma \sqrt{\tau} \right )} \right]$$ $$\phi= \left\{ \begin{array}{rl} 1 & \mbox{if Call}\\ -1 & \mbox {if Put}\\ \end{array} \right.$$ $$d_1 =\frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) + \frac{\sigma^{2}}{2} \tau}\right)$$ $$d_{2}=d_{1} - \sigma \sqrt{\tau}$$ $$N\left( y \right)= \int_{-\infty}^{y}{\frac{1}{\sqrt{2\pi}} {e^{-\frac{x^{2}}{2}}}dx}={\text {Cumulative Standard Normal}}$$ $$\frac{d N\left( y \right)}{d y}=\frac{1}{\sqrt{2\pi}} {e^{-\frac{y^{2}}{2}}}=n\left( y \right)={\text {Standard Normal Density}}$$