 Solution

For a terminal price distinct from $K$, differentiating the put's financial payoff gives ${\partial }_K(K-S_T{)}^{+}=1_{\{S_T<K\}}$. 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:Equivalently, risk-neutral pricing discounts the probability $\mathbb{Q}(S_T<K\mid S_t=S)=\Phi (-d_{-})$.