This paper studies quantile-based moment premiums. The quantile-based approach delivers robust and flexible alternatives to premiums for variance, skewness and kurtosis risk and enhances our understanding of the pricing of risks in derivatives markets. To quantify these premiums, the paper introduces a new class of synthetic derivatives contracts: quantile swaps. Such contracts mimic quantile-based moment measures from robust statistics. An empirical study of index options detects two distinct premiums for dispersion and asymmetry, but no premium for steepness. The premium for dispersion can be explained by traditional moment risk premiums, whereas the asymmetry premium is a novel premium that our approach is able to detect. This finding contrasts markedly with results obtained with traditional moment swaps, and warns us to interpret moment premiums cautiously.