= Call-price decay and moment threshold
{title2=$\sup_KK^\delta C(K)<\infty\Longrightarrow\mathbb E S^{1+\varepsilon}<\infty\quad(\varepsilon<\delta)$}
For $C(K)=\mathbb E(S-K)_+$ and $\mathbb ES<\infty$, split the static call integral at a fixed positive strike. The first moment bounds the integrand near zero; polynomial call decay controls the integral at infinity for $0<\varepsilon<\delta$. The endpoint can fail, as shown by a <Pareto distribution> with survival exponent $1+\delta$.
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