## Cash or Nothing options Greeks under Black Scholes

We derive the formulae for the Greeks, derivatives with respect to inputs, of a digital or a binary option that pays one unit of cash or nothing.

### Summary of Price and Greeks Formulae for a Cash or Nothing option

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

$$e^{- r_{d} \tau} N{\left (d_{2}\right )}$$

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

$$e^{- r_{d} \tau} N{\left (-d_{2}\right )}$$

We notice the only difference between the two formulae is the + or - sign in front of $$d_{2}$$. 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 both a Call and a Put options at the same time.

$$Price= e^{- r_{d} \tau} N \left (\phi d_{2} \right )$$

We list the Greeks/derivatives of the above formula for the price Cash or Nothing European Options below:

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

Where,

$$d_1 =\frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) + \left(r_{d} - r_{f} + \frac{\sigma^{2}}{2}\right) \tau}\right)$$ $$d_2 =\frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) + \left(r_{d} - r_{f} - \frac{\sigma^{2}}{2}\right) \tau}\right)$$ $$N\left( y \right)= \int_{-\infty}^{y}{\frac{1}{\sqrt{2\pi}} {e^{-\frac{x^{2}}{2}}}dx}={\text {Cumulative Standard Normal}}$$ $$\frac{d N\left( y \right)}{d y}=\frac{1}{\sqrt{2\pi}} {e^{-\frac{y^{2}}{2}}}=n\left( y \right)={\text {Standard Normal Density}}$$