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.