 Call-price density recovery

In a zero-interest one-period market with a continuous terminal stock law under a pricing measure, $C(K)={\mathbb{E}}_Q(S-K{)}^{+}$ implies $-C^{\prime }(K)=Q(S>K)$ and $C^{\prime \prime }(K)=f_Q(K)$. Thus a full differentiable call curve determines its pricing density. A finite collection of strikes generally does not. With a deterministic nonunit discount factor, divide the call curve by that factor before recovering the probability density.