The <digital call option> and <digital put option> indicators partition the possible terminal stock prices:
$$
\mathbf1_{\{S_T\geq K\}}
+\mathbf1_{\{S_T<K\}}=1.
$$
Thus the <digital put-call parity> is
$$
\boxed{
D_{\rm call}(t,S_t)+D_{\rm put}(t,S_t)
=e^{-r(T-t)}.}
$$
At time zero, part b gives $D_{\rm call}(0,S_0)=e^{-rT}\Phi(d_-)$, so
$$
\boxed{
D_{\rm put}(0,S_0)
=e^{-rT}\bigl(1-\Phi(d_-)\bigr)
=e^{-rT}\Phi(-d_-).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 29K / c / Solution

Complementary digital call and put payoffs sum to one, so their time-$t$ values satisfy
$$
D_{\rm call}(t,S)+D_{\rm put}(t,S)=e^{-r(T-t)}.
$$


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