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.

Gamma

We now proceed to derive formula for the Gamma, which is the second partial derivative of the price with respect to S. We have, from the Delta section, the formula for the Delta of a digital option, paying one unit of cash or nothing:

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

Differentiating both sides with respect to S

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

We know from the Delta section that:

$$ \frac{\partial}{\partial S} \left( d_{2} \right)=\frac{1}{S \sigma \sqrt{\tau}}$$

Thus

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