## 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.

### Rho

We now derive the formula for the first derivative of the Asset or Nothing digital option price formula with respect to the discount rate. Differentiating both sides of the Black-Scholes Call option formula wrt $$r_{d}$$, we get:

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

Now, by definition:

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

Thus

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