Solution
= Solution
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
$$
\boxed{
D_{\rm put}(0,S_0)
=e^{-rT}\bigl(1-\Phi(d_-)\bigr)
=e^{-rT}\Phi(-d_-).}
$$