Dynamic portfolio choice with market impact costs

Illiquidity and market impact refer to the situation where it may be costly or difficult to trade a desired quantity of assets over a desire period of time. In this paper, we formulate a simple model of dynamic portfolio choice that incorporates liquidity effects. The resulting problem is a stochastic linear quadratic control problem where liquidity costs are modeled as a quadratic penalty on the trading rate. Though easily computable via Riccati equations, we also derive a multiple time scale asymptotic expansion of the value function and optimal trading rate in the regime of vanishing market impact costs. This expansion reveals an interesting but intuitive relationship between the optimal trading rate for the “illiquid” problem and the classical Merton model for dynamic portfolio selection in perfectly liquid markets. It also gives rise to the notion of a “liquidity time scale” which shows how trading horizon and market impact costs affect the optimal trading rate.