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.

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 asset or nothing:

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

Differentiating both sides with respect to S

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

We know from the Delta section that:

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

Thus

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