Simulation-based portfolio optimization for large portfolios with transaction costs

We consider a portfolio optimization problem where the investor's objective is to maximize the long-term expected growth rate, in the presence of proportional transaction costs. This problem belongs to the class of stochastic control problems with singular controls, which are usually solved by computing solutions to related partial differential equations called the free-boundary Hamilton-Jacobi-Bellman (HJB) equations. The dimensionality of the HJB equals the number of stocks in the portfolio. The runtime of existing solution methods grow super-exponentially with dimension, making them unsuitable to compute optimal solutions to portfolio optimization problems with even four stocks. In this work we first present a boundary update procedure that converts the free boundary problem into a sequence of fixed boundary problems. Then by combining simulation with the boundary update procedure, we provide a computational scheme whose runtime, as shown by the numerical tests, scales polynomially in dimension. The results are compared and corroborated against existing methods that scale super-exponentially in dimension. The method presented herein enables the first ever computational solution to free-boundary problems in dimensions greater than three.