= Put-price strike derivative
{title2=$\partial_KP(K)=D\,\mathbb Q(S_T<K)$}
For a <European put option> with deterministic <discount factor> $D$ and an integrable terminal <stock> price, $P(K)=D\mathbb E_{\mathbb Q}[(K-S_T)^+]$. If $\mathbb Q(S_T=K)=0$, bounded difference quotients and the <dominated convergence theorem> give the displayed derivative, the price of a <digital put option>. At an atom, the left derivative is $D\mathbb Q(S_T<K)$ and the right derivative is $D\mathbb Q(S_T\leq K)$, so the ordinary derivative need not exist.
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