## 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.

### Theta

And in the final section, we derive formula for Theta, which is the first derivative of the option price with respect to t (Note that in our representation so far $$\tau=(T-t)$$ so the dependence on t comes through $$\tau$$).

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

Now, by definition:

$$\frac{\partial}{\partial \tau} \left( d_{1} \right)=\frac{\partial}{\partial \tau} \left( \frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) + \left(r_{d} - r_{f} + \frac{\sigma^{2}}{2}\right) \tau}\right) \right)$$ $$=\frac{\partial}{\partial \tau} \left( \frac{1}{\sigma \sqrt{\tau}}\ln\left (\frac{S}{K} \right )+ \frac{1}{\sigma \sqrt{\tau}} \left(r_{d} - r_{f} + \frac{\sigma^{2}}{2}\right) \tau \right)$$ $$=\frac{\partial}{\partial \tau} \left( \frac{1}{\sigma \sqrt{\tau}}\ln\left (\frac{S}{K} \right )+ \frac{1}{\sigma } \left(r_{d} - r_{f} + \frac{\sigma^{2}}{2}\right) \sqrt{\tau} \right)$$ $$= - \frac{1}{2\sigma \sqrt[3]{\tau}}\ln\left (\frac{S}{K} \right )+ \frac{1}{2\sigma \sqrt{\tau} } \left(r_{d} - r_{f} + \frac{\sigma^{2}}{2}\right)$$ $$=-\frac{1}{2\tau}\frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) - \left(r_{d} - r_{f} + \frac{\sigma^{2}}{2}\right) \tau}\right)$$

Thus

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

Now

$$\frac {\partial Price} {\partial t}=\frac {\partial Price} {\partial \tau}\frac {\partial \tau} {\partial t}=\frac {\partial Price} {\partial \tau}\frac {\partial \left(T-t \right)} {\partial t}=-\frac {\partial Price} {\partial \tau}$$

Hence

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