We present a market microstructure model of stock splits in the presence of minimum tick size rules. The key feature of the model is that discretionary trading is endogenously determined. There exists a tradeoff between adverse selection costs on the one hand and discreteness related costs and opportunity costs of monitoring the market on the other hand. Under certain parameter values, there exists an optimal price. We document an inverse relation between the coefficient of variation of intraday trading volume and the stock price level. This empirical evidence and other existing evidence are consistent with the model.