## 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.

### 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 the asset or nothing, under the Black Scholes assumptions, is given by

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

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

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

Now, by definition:

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

Then

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

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).