### 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).