## 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.

### Put Call Parity

We verify that the Greeks formulae satisfy the Put Call parity, which in the case of binary Cash Options states that sum of a call and a put cash option is equivalent to a position paying one unit of cash at maturity. In symbols:

$$\text{Digital Cash Call}+\text{Digital Cash Put} = {\text {1 Unit of Cash}}$$ $$\text{Present Value of Call}+\text{Present Value of Put} = \text {Present Value of 1 Unit of Cash}$$ $$\mbox{Digital Cash or Nothing Call Price} + \mbox{Digital Cash or Nothing Put Price} = e^{-r_d \tau}$$

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

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

The equalities are easily verified:

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

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

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

$$\rho (\mbox{Digital Cash or Nothing Call Price})+ \rho (\mbox{Digital Cash or Nothing Put Price})$$ $$= \left. e^{- r_{d} \tau} \left( \frac{\phi \sqrt{\tau}}{\sigma} n{\left ( d_{2} \right )} - \tau N{\left( \phi d_{2} \right )} \right) \right|_{\phi=1}+ \left. e^{- r_{d} \tau} \left( \frac{\phi \sqrt{\tau}}{\sigma} n{\left ( d_{2} \right )} - \tau N{\left( \phi d_{2} \right )} \right) \right|_{\phi=-1}$$ $$= e^{- r_{d} \tau} \left( \frac{\sqrt{\tau}}{\sigma} n{\left ( d_{2} \right )} - \tau N{\left( d_{2} \right )} \right) + e^{- r_{d} \tau} \left( -\frac{ \sqrt{\tau}}{\sigma} n{\left ( d_{2} \right )} - \tau N{\left( - d_{2} \right )} \right)$$ $$= e^{- r_{d} \tau} \tau \left(-N\left( d_{2} \right ) - N\left( - d_{2} \right ) \right)$$ $$= e^{- r_{d} \tau} \tau \left(-N\left( d_{2} \right ) - 1 + N\left(d_{2} \right ) \right)$$ $$= -e^{- r_{d} \tau} \tau$$

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