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:

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

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

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

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

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