Optimal portfolios of a small investor in a limit order market: a shadow price approach

We study Merton’s portfolio optimization problem in a limit order market. An investor trading in a limit order market has the choice between market orders that allow immediate transactions and limit orders that trade at more favorable prices but are executed only when another market participant places a corresponding market order. Assuming Poisson arrivals of market orders from other traders we use a shadow price approach, similar to Kallsen and Muhle-Karbe (Ann Appl Probab, forthcoming) for models with proportional transaction costs, to show that the optimal strategy consists of using market orders to keep the proportion of wealth invested in the risky asset within certain boundaries, similar to the result for proportional transaction costs, while within these boundaries limit orders are used to profit from the bid–ask spread. Although the given best-bid and best-ask price processes are geometric Brownian motions the resulting shadow price process possesses jumps.