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,

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