Cash or Nothing options Greeks under Black Scholes

We derive the formulae for the Greeks, derivatives with respect to inputs, of a digital or a binary option that pays one unit of cash or nothing.

Vega

We now differentiate the Cash or Nothing digital option price formula with respect to $$\sigma$$, the volatility of the underlying, in order to derive a formula for the Vega:

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

Now, by definition:

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

Then

$$\frac {\partial Price} {\partial \sigma}= \phi e^{- r_{d} \tau} n{\left (\phi d_{2} \right )} \frac{\partial}{\partial \sigma} d_{2}$$ $$=-\phi e^{- r_{d} \tau} n{\left (\phi d_{2} \right )}\frac{d_{1}}{\sigma}$$ $$=- \frac{\phi e^{- r_{d} \tau}}{\sigma} d_{1}n{\left (d_{2} \right )}$$