This paper studies the "overpriced puts puzzle" — the finding that historical prices of the S&P 500 put options have been too high and incompatible with the canonical asset-pricing models. To investigate whether put returns could be rationalized by another, possibly non-standard equilibrium model, we implement the model-free methodology of Bondarenko [2003a, Statistical Arbitrage and Securities Prices, Review of Financial Studies 16, 875–919]. The methodology requires no parametric assumptions on investors' preferences. Furthermore, it can be applied even when the sample is affected by certain selection biases (such as the Peso problem) and when investors' beliefs are incorrect. The main finding of the paper is that no model within a studied class of models can possibly explain the put anomaly.