## 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.

### Put Call Parity

We verify that the Greeks formulae satisfy the Put Call parity, which in the case of Asset or Nothing Options states that sum of a call and a put option is equivalent to a position paying one unit of asset at maturity. In symbols:

$$\text{Digital Asset or Nothing Call}+\text{Digital Asset or Nothing Put} = {\text {1 Unit of Asset}}$$ $$\text{Present Value of Call}+\text{Present Value of Put} = \text {Present Equivalent of 1 Unit of Asset at maturity}$$ $$\mbox{Digital Asset or Nothing Call Price} + \mbox{Digital Asset or Nothing Put Price} = S e^{-r_f \tau}$$

Taking derivatives of both sides with respect to $$S, S^2, \sigma, r_d, \text{and } t$$, we get:

$$\Delta (\mbox{Digital Asset or Nothing Call Price})+ \Delta (\mbox{Digital Asset or Nothing Put Price}) = e^{-r_f \tau}$$ $$\Gamma (\mbox{Digital Asset or Nothing Call Price} + \Gamma(\mbox{Digital Asset or Nothing Put Price}) = 0$$ $$Vega (\mbox{Digital Asset or Nothing Call Price}) + Vega(\mbox{Digital Asset or Nothing Put Price}) = 0$$ $$\rho (\mbox{Digital Asset or Nothing Call Price})+ \rho (\mbox{Digital Asset or Nothing Put Price}) = 0$$ $$\Theta (\mbox{Digital Asset or Nothing Call Price})- \Theta(\mbox{Digital Asset or Nothing Put Price}) = r_f S e^{-r_f \tau}$$

The equalities are easily verified:

$$\Delta (\mbox{Digital Asset or Nothing Call Price})+ \Delta (\mbox{Digital Asset or Nothing Put Price})$$ $$= \left. \left[ \frac{\phi e^{- r_{f} \tau}}{\sigma \sqrt{\tau}} n{\left (d_{1} \right )} + e^{- r_{f} \tau} N{ \left (\phi d_{1} \right )} \right] \right|_{\phi=1}+\left. \left[ \frac{\phi e^{- r_{f} \tau}}{\sigma \sqrt{\tau}} n{\left (d_{1} \right )} + e^{- r_{f} \tau} N{ \left (\phi d_{1} \right )} \right] \right|_{\phi=-1}$$ $$= e^{- r_{f} \tau} N{ \left (d_{1} \right )}+e^{- r_{f} \tau} N{ \left ( - d_{1} \right )}$$ $$= e^{- r_{f} \tau}$$

$$\Gamma (\mbox{Digital Asset or Nothing Call Price}) + \Gamma(\mbox{Digital Asset or Nothing Put Price})$$ $$=\left. -\frac{ \phi e^{- r_{f} \tau}}{S \sigma^{2} \tau} n{\left (d_{1} \right)} d_{2} \right|_{\phi=1}+ \left. -\frac{ \phi e^{- r_{f} \tau}}{S \sigma^{2} \tau} n{\left (d_{1} \right)} d_{2} \right|_{\phi=-1}$$ $$= -\frac{ e^{- r_{f} \tau}}{S \sigma^{2} \tau} n{\left (d_{1} \right)} d_{2} + \frac{ e^{- r_{f} \tau}}{S \sigma^{2} \tau} n{\left (d_{1} \right)} d_{2}$$ $$= 0$$

$$Vega (\mbox{Digital Asset or Nothing Call Price}) + Vega(\mbox{Digital Asset or Nothing Put Price})$$ $$= \left. -\frac{\phi S e^{- r_{f} \tau}}{\sigma} d_{2}n{\left (d_{1} \right )} \right|_{\phi=1}+ \left. -\frac{\phi S e^{- r_{f} \tau}}{\sigma} d_{2}n{\left (d_{1} \right )} \right|_{\phi=-1}$$ $$= -\frac{S e^{- r_{f} \tau}}{\sigma} d_{2}n{\left (d_{1} \right )} + \frac{S e^{- r_{f} \tau}}{\sigma} d_{2}n{\left (d_{1} \right )}$$ $$= 0$$

$$\rho (\mbox{Digital Asset or Nothing Call Price})+ \rho (\mbox{Digital Asset or Nothing Put Price})$$ $$= \left. \frac{\phi S e^{- r_{f} \tau} \sqrt{\tau}}{\sigma} n{\left (d_{1} \right )} \right|_{\phi=1}+ \left. \frac{\phi S e^{- r_{f} \tau} \sqrt{\tau}}{\sigma} n{\left (d_{1} \right )} \right|_{\phi=-1}$$ $$= 0$$

$$\Theta (\mbox{Digital Asset or Nothing Call Price})+ \Theta(\mbox{Digital Asset or Nothing Put Price})$$ $$= \left. Se^{- r_{f}\tau} \left( \phi n{\left ( d_{1} \right )} \frac{1}{2\tau}\frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) - \left(r_{d} - r_{f} + \frac{\sigma^{2}}{2}\right) \tau}\right)+ r_{f} N{\left (\phi d_{1} \right )} \right) \right|_{\phi=1}$$ $$\; + \left. Se^{- r_{f}\tau} \left( \phi n{\left ( d_{1} \right )} \frac{1}{2\tau}\frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) - \left(r_{d} - r_{f} + \frac{\sigma^{2}}{2}\right) \tau}\right)+ r_{f} N{\left (\phi d_{1} \right )} \right) \right|_{\phi=-1}$$ $$= Se^{- r_{f}\tau} \left( n{\left ( d_{1} \right )} \frac{1}{2\tau}\frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) - \left(r_{d} - r_{f} + \frac{\sigma^{2}}{2}\right) \tau}\right)+ r_{f} N{\left ( d_{1} \right )} \right)$$ $$\; + Se^{- r_{f}\tau} \left( - n{\left ( d_{1} \right )} \frac{1}{2\tau}\frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) - \left(r_{d} - r_{f} + \frac{\sigma^{2}}{2}\right) \tau}\right)+ r_{f} N{\left (- d_{1} \right )} \right)$$ $$= r_{f} S e^{- r_{f}\tau} \left( N{\left ( d_{1} \right )}+N{\left (- d_{1} \right )} \right)$$ $$= r_{f}S e^{- r_{f}\tau} \left( N{\left ( d_{1} \right )}+1 -N{\left ( d_{1} \right )} \right)$$ $$= r_{f}S e^{- r_{f}\tau}$$