Abstract
We derive a diffusion model for trading volume, assuming that a representative investor with constant relative risk aversion trades in a market with finite depth and constant investment opportunities.
Two regimes arise. If borrowing – either leverage or short-selling – is not optimal without liquidity costs, the optimal trading strategy is the solution of an ordinary differential equation, which admits explicit asymptotics for small liquidity costs, and leads to a dynamic model for trading volume. The optimal portfolio is to keep all wealth in the risky asset if the frictionless solution entails leverage, or in the safe asset if frictionless short-selling is optimal.
As liquidity costs vanish, we recover the usual Merton model. For small costs, the liquidity premium equals the one arising with a suitable level of transaction costs, times a universal constant.