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.

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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

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