Binary options. Consider a price process \(\left(S_{t}ight)_{t \in \mathbb{R}_{+}}\)given by

\[ \frac{d S_{t}}{S_{t}}=r d t+\sigma d B_{t}, \quad S_{0}=1 \]

under the risk-neutral probability measure \(\mathbb{P}^{*}\). The binary (or digital) call option is a contract with maturity \(T\), strike price \(K\), and payoff \[
C_{d}:=\mathbb{1}_{[K, \infty)}\left(S_{T}ight)= \begin{cases}\$ 1 & \text { if } S_{T} \geqslant K \\ 0 & \text { if } S_{T}\]

a) Derive the Black-Scholes PDE satisfied by the pricing function \(C_{d}\left(t, S_{t}ight)\) of the binary call option, together with its terminal condition.

b) Show that the solution \(C_{d}(t, x)\) of the Black-Scholes PDE of Question (a) is given by \[
\begin{aligned}
C_{d}(t, x) & =\mathrm{e}^{-(T-t) r} \Phi\left(\frac{\left(r-\sigma^{2} / 2ight)(T-t)+\log (x / K)}{|\sigma| \sqrt{T-t}}ight) \\
& =\mathrm{e}^{-(T-t) r} \Phi\left(d_{-}(T-t)ight)
\end{aligned}
\]
where \[
d_{-}(T-t):=\frac{\left(r-\sigma^{2} / 2ight)(T-t)+\log \left(S_{t} / Kight)}{|\sigma| \sqrt{T-t}}, \quad 0 \leqslant t