## Asset or Nothing Options' Greeks under Black Scholes

We derive the formulae for the Greeks/derivatives of a digital or binary option that pays one unit of asset or nothing.

### 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,

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