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:

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

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

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

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

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