Exchange-mandated discrete pricing restrictions create a wedge between the underlying equilibrium price and the observed price. This wedge permits a competitive market maker to realize economic profits that could help recoup fixed costs. The optimal tick size that maximizes the expected profits of the market maker can equal to $1/8 for reasonable parameter values. The optimal tick size is decreasing in the degree of adverse selection. Discreteness per se can cause time-varying bid-ask spreads, asymmetric commissions, and market breakdowns. Discreteness, which imposes additional transaction costs, reduces the value of private information. Liquidity traders can benefit under certain conditions.

[Exchange-mandated discrete pricing restrictions create a wedge between the underlying equilibrium price and the observed price. This wedge permits a competitive market maker to realize economic profits that could help recoup fixed costs. The optimal tick size that maximizes the expected profits of the market maker can be equal to $1/8 for reasonable parameter values. The optimal tick size is decreasing in the degree of adverse selection. Discreteness per se can cause time-varying bid-ask spreads, asymmetric commissions, and market breakdowns. Discreteness, which imposes additional transaction costs, reduces the value of private information. Liquidity traders can benefit under certain conditions.]

Given the evidence that the level of liquidity affects asset returns, a reasonable hypothesis is that the second moment of liquidity should be positively related to asset returns, provided agents care about the risk associated with fluctuations in liquidity. Motivated by this observation, we analyze the relation between expected equity returns and the level as well as the volatility of trading activity, a proxy for liquidity. We document a result contrary to our initial hypothesis, namely, a negative and surprisingly strong cross-sectional relationship between stock returns and the variability of dollar trading volume and share turnover, after controlling for size, book-to-market ratio, momentum, and the level of dollar volume or share turnover. This effect survives a number of robustness checks, and is statistically and economically significant. Our analysis demonstrates the importance of trading activity-related variables in the cross-section of expected stock returns.