Balancing Small Transaction Costs with Loss of Optimal Allocation in Single and Multiple Stock Portfolios

We discuss optimal trading strategies in the presence of small proportional transaction costs for general utility functions. We present a new interpretation of scalings found by Soner, Shreve, and others. To leading order in the small transaction cost parameter, the free boundary problem for the expected utility’s value function is shown to be dual, in the sense of Lagrange multipliers for optimal design problems, to a free boundary problem minimizing a cost function. This cost function is the sum of a boundary integral corresponding to the rate of trading and an interior integral corresponding to opportunity loss that results from suboptimal portfolio allocation. Using the dual problem’s formulation, we show that the quasi-steady state probability density of the optimal portfolio is uniform for a single stock, but generally blows up even in the simple case of two uncorrelated stocks.