Portfolio Management with Transaction Costs

It is known that the optimal trading strategy for a certain portfolio problem featuring fixed transaction costs is obtained from the solution of a free boundary problem. The latter can only be solved with numerical methods, and computations become formidable when the number of available securities is larger than three or four. This paper shows how a transformation of the free boundary problem together with an asymptotic analysis (performed about the solution when the transaction cost is zero) leads to solutions which are shown to be good approximations for cases which can be solved by numerical methods. These approximately optimal trading strategies are easy to compute, even when there are many risky securities, as is illustrated for the case of the 30 Dow Jones Industrials.