Abstract: This paper shows how to infer information about any random variable from trading volume, assuming that the random variable and the traders' demands are symmetrically (and then normally) distributed around zero. The volume-based conditional expectation of such a random variable is zero, while the covariance between its absolute value and volume is positive if the variable is jointly normally distributed with the traders' demands. In that case, numerical examples indicate that the volume-based conditional probability of extreme asset value realizations (positive or negative) increases with volume. These results, developed in a market-clearing framework, apply also to market-making frameworks. Finally, the paper develops a simple model where transaction costs can generate a positive covariance between price and trading volume.

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