Call-price decay and moment threshold (${\sup }_K{K}^{\delta }C(K)<\infty ⟹\mathbb{E}{S}^{1+\epsilon }<\infty (\epsilon <\delta )$)

For $C(K)=\mathbb{E}(S-K{)}_{+}$ and $\mathbb{E}S<\infty$, 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 $0<\epsilon <\delta$. The endpoint can fail, as shown by a Pareto distribution with survival exponent $1+\delta$.