### Summary of Price and Greeks Formulae for an Asset or Nothing option

The pricing formula of an ** Asset or Nothing European Call Option **,which pays one unit of the underlying asset at maturity if the price of the underlying is higher than the strike price and nothing otherwise, is:

$$S e^{- r_{f} \tau} N{\left (d_{1}\right )}$$

and of an ** Asset or Nothing European Put Option**,which pays one unit of the underlying asset at maturity if the price of the underlying is lower than the strike price and nothing otherwise, is:

$$S e^{- r_{f} \tau} N{\left (-d_{1}\right )}$$

We notice the only difference between the two formulae is the + or - sign in front of \( d_{1} \). So if we define a variable

$$ \phi= \left\{ \begin{array}{rl} 1 & \mbox{if Call}\\ -1 & \mbox {if Put}\\ \end{array} \right. $$

We get the **generalised version ** of the formula, helping us to compute the Greeks of Call and Put at the same time.

$$Price= S e^{- r_{f} \tau} N{\left (\phi d_{1}\right )}$$

We list the Greeks/derivatives of the above formula below:

$$ Delta (\Delta) = \frac{\phi e^{- r_{f} \tau}}{\sigma \sqrt{\tau}} n{\left (d_{1} \right )} + e^{- r_{f} \tau} N{ \left (\phi d_{1} \right )} $$ $$ Gamma (\Gamma) = -\frac{ \phi e^{- r_{f} \tau}}{S \sigma^{2} \tau} n{\left (d_{1} \right)} d_{2} $$ $$ Vega = -\frac{\phi S e^{- r_{f} \tau}}{\sigma} d_{2}n{\left (d_{1} \right )} $$ $$ Rho (\rho) =\frac{\phi S e^{- r_{f} \tau} \sqrt{\tau}}{\sigma} n{\left (d_{1} \right )} $$ $$ Theta (\Theta) = Se^{- r_{f}\tau} \left( \phi n{\left ( d_{1} \right )} \frac{1}{2\tau}\frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) - \left(r_{d} - r_{f} + \frac{\sigma^{2}}{2}\right) \tau}\right)+ r_{f} N{\left (\phi d_{1} \right )} \right) $$

Where,

$$ d_1 =\frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) + \left(r_{d} - r_{f} + \frac{\sigma^{2}}{2}\right) \tau}\right)$$ $$ N\left( y \right)= \int_{-\infty}^{y}{\frac{1}{\sqrt{2\pi}} {e^{-\frac{x^{2}}{2}}}dx}={\text {Cumulative Standard Normal}}$$ $$ \frac{d N\left( y \right)}{d y}=\frac{1}{\sqrt{2\pi}} {e^{-\frac{y^{2}}{2}}}=n\left( y \right)={\text {Standard Normal Density}} $$