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

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