{c}

In the <Black-Scholes model>, a <digital call option> and <digital put option> with remaining maturity $\tau$ have values
$$
D_{\rm call}(t,S)=e^{-r\tau}\Phi(d_-),
\qquad
D_{\rm put}(t,S)=e^{-r\tau}\Phi(-d_-),
$$
where $d_-$ is defined in the <Black-Scholes formula>. For $t<T$, the digital-call <delta hedge> is
$$
\partial_SD_{\rm call}(t,S)
=\frac{e^{-r\tau}\phi(d_-)}{S\sigma\sqrt\tau}.
$$

Black-Scholes digital option formula

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

{title2=$\mathbf1_{\{S_T<K\}}$}

A digital put option with strike $K$ pays one unit if $S_T<K$ and zero otherwise. With these complementary boundary conventions, one digital call plus one digital put equals a zero-coupon bond paying one unit at maturity.

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