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The two binary terminal payoffs are complementary:in every state, including $S_T=X$. Long one of each is therefore exactly a unit zero-coupon bond. The law of one price gives digital put-call parity:With deterministic interest, the two separate values are $B(t,T)Q(S_T\ge X\mid {\mathcal{F}}_t)$ and $B(t,T)Q(S_T<X\mid {\mathcal{F}}_t)$. There is no need to assume a continuous terminal price distribution: the strict inequality for the put and weak inequality for the call allocate any atom at the strike exactly once.

The digital call option and digital put option indicators partition the possible terminal stock prices:Thus the digital put-call parity isAt time zero, part b gives $D_{\mathrm{call}}(0,S_0)=e^{-rT}\Phi (d_{-})$, so