### Rho

We now derive the formula for the first derivative of a Cash or Nothing digital option price formula with respect to the discount rate. Differentiating both sides with respect to \( r_{d} \), we get:

$$\frac {\partial Price} {\partial r_d} =\frac {\partial } {\partial r_d} e^{- r_{d} \tau} N \left (\phi d_{2} \right ) $$ $$= e^{- r_{d} \tau} \frac{\partial}{\partial r_d} N{\left( \phi d_{2} \right )}+N{\left( \phi d_{2} \right )} \frac{\partial}{\partial r_d} e^{- r_{d} \tau}$$ $$=\phi e^{- r_{d} \tau} n{\left (\phi d_{2} \right )} \frac{\partial}{\partial r_d} d_{2} - N{\left( \phi d_{2} \right )} \tau e^{- r_{d} \tau}$$

Now, by definition:

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

Thus

$$\frac {\partial Price} {\partial r_d}= \phi e^{- r_{d} \tau} n{\left (\phi d_{2} \right )} \frac{\partial}{\partial r_d} d_{2} - N{\left( \phi d_{2} \right )} \tau e^{- r_{d} \tau}$$ $$ =\phi e^{- r_{d} \tau} n{\left (\phi d_{2} \right )} \frac{\sqrt{\tau}}{\sigma} - N{\left( \phi d_{2} \right )} \tau e^{- r_{d} \tau}$$ $$ =e^{- r_{d} \tau} \left( \frac{\phi \sqrt{\tau}}{\sigma} n{\left ( d_{2} \right )} - \tau N{\left( \phi d_{2} \right )} \right) $$