= Solution
{c}
For a terminal price distinct from $K$, differentiating the put's <financial payoff> gives $\partial_K(K-S_T)^+=\mathbf1_{\{S_T<K\}}$. The difference quotients are bounded by one, and the continuous <lognormal distribution> has zero <probability> at the strike. The <dominated convergence theorem> therefore permits differentiation inside its discounted <expectation>. The <put-price strike derivative> is exactly the price of the <digital put option>:
$$
\boxed{P_d(S,t)=\frac{\partial P}{\partial K}=e^{-r\tau}\operatorname N(-d_-).}
$$
Equivalently, <risk-neutral pricing> discounts the <probability> $\mathbb Q(S_T<K\mid S_t=S)=\Phi(-d_-)$.
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