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Dynamic portfolio selection with nonlinear transaction costs

The dynamic portfolio selection problem with bankruptcy and nonlinear transaction costs is studied. The portfolio consists of a risk-free asset, and a risky asset whose price dynamics is governed by geometric Brownian motion. The investor pays transaction costs as a (piecewise linear) function of the traded volume of the risky asset. The objective is to find the stochastic controls (amounts invested in the risky and risk-free assets) that maximize the expected value of the discounted utility of terminal wealth. The problem is formulated as a non-singular stochastic optimal control problem in the sense that the necessary condition for optimality leads to explicit relations between the controls and the value function. The formulation follows along the lines of Merton (Merton 1969 Rev. Econ. Stat. 51, 247–257; Merton 1971 J. Econ. Theory 3, 373–413) and Bensoussan & Julien (Bensoussan & Julien 2000 Math. Finance 10, 89–108) in the sense that the controls are the amounts of the risky asset bought and sold, and they are bounded. It differs from the works of Davis & Norman (Davis & Norman 1990 Math. Oper. Res. 15, 676–713), who use, in the presence of proportional transaction costs, a singular-control formulation in which the controls are rates of buying and selling of the risky asset, and they are unbounded. Numerical results are presented for buy/no transaction and sell/no transaction interfaces, which characterize the optimal policies of a constant relative risk aversion investor. The no transaction region, in the presence of nonlinear transaction costs, is not a cone. The Merton line, on which no transaction takes place in the limiting case of zero transaction costs, need not lie inside the no transaction region for all values of wealth.