## Black's Price and Greeks

The pricing and Greeks formulae of the European Swaptions, Caps, and Floor use the Black's formula, so we list and derive the Black's formulae here to simplify the presentation.

### Summary of Black Price and Greeks Formula

We first list the Black price formula and its Greeks:

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

Where,

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