We study contextual bilateral trade under full feedback when trader valuations have bounded density but infinite variance. We first extend the self-bounding property of Bachoc et al. (ICML 2025) from bounded to real-valued valuations, showing that the expected regret of any price $π$ satisfies $\mathbb{E}[g(m,V,W) - g(π,V,W)] \le L|m-π|^2$ under bounded density alone. Combining this with truncated-mean estimation, we prove that an epoch-based algorithm achieves regret $\widetilde{O}(T^{1-2β(p-1)/(βp + d(p-1))})$ when the noise has finite $p$-th moment for $p \in (1,2)$ and the market value function is $β$-Hölder, and we establish a matching $Ω(\cdot)$ lower bound via Assouad's method with a smoothed moment-matching construction. Our results characterize the exact minimax rate for this problem, interpolating between the classical nonparametric rate at $p=2$ and the trivial linear rate as $p \to 1^+$.

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