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.

Delta

Delta is the first derivative of the option price with respect to the underlying price (e.g., stock price). The generalised formula for the price of a digital option, paying one unit of cash or nothing, under the Black Scholes assumptions, is:

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

Where

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

Differentiating both sides with respect to S, we get

$$\frac{\partial Price}{\partial S}=\frac{\partial}{\partial S}\left( e^{- r_{d} \tau} N\left( \phi d_{2} \right )\right)$$ $$= e^{- r_{d} \tau}\frac{\partial}{\partial S} \left( N\left( \phi d_{1} \right ) \right)$$ $$=\phi e^{- r_{d} \tau} n{\left (\phi d_{2} \right )} \frac{\partial}{\partial S} d_{2}$$

Now, by definition:

$$ \frac{\partial}{\partial S} \left( d_{2} \right)=\frac{\partial}{\partial S} \left( \frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) + \left(r_{d} - r_{f} - \frac{\sigma^{2}}{2}\right) \tau}\right) \right)$$ $$=\frac{\partial}{\partial S} \left( \frac{1}{\sigma \sqrt{\tau}} \ln \left( S \right ) + \frac{1}{\sigma \sqrt{\tau}} \left(-\ln{\left (K \right ) + \left(r_{d} - r_{f} - \frac{\sigma^{2}}{2}\right) \tau}\right) \right)$$ $$=\frac{1}{\sigma \sqrt{\tau}} \frac{\partial}{\partial S} \left( \ln \left( S \right ) \right)$$ $$=\frac{1}{S \sigma \sqrt{\tau}}$$

Then

$$\frac{\partial Price}{\partial S}=\phi e^{- r_{d} \tau} n{\left (\phi d_{2} \right )} \frac{\partial}{\partial S} d_{2}$$ $$=\phi e^{- r_{d} \tau} n{\left (\phi d_{2} \right )} \frac{1}{S \sigma \sqrt{\tau}}$$ $$= \frac{\phi e^{- r_{d} \tau}}{S \sigma \sqrt{\tau}}n{\left (d_{2} \right)}$$

Where we have used \( n(-x)=n(x)\) because the function \( n(x) \) is symmetric around 0 (visualise the bell shaped standard normal distribution curve).