Optimal trading with transaction costs using a PMP gradient method

Sammanfattning: This thesis considers a portfolio optimization problem with linear transaction costs, as interpreted by Ampfield Aktiebolag, and analyses it by using a gradient method based on Pontryagin's maximum principle (PMP). First the problem is outlined and afterwards it turns out that a gradient PMP method is easy to employ and gives reasonable solutions. As with many gradient methods the convergence is very slow, but a good estimate could possibly be found in sub-second time with the right implementation and computer. The strength of the method is the good complexity, linear in the number of time steps and quadratic in the number of dimensions for each iteration. This is compared with quadratic and dynamic programming which have polynomial and exponential complexity respectively. The main weakness, apart from slow convergence, lies in the assumptions that have to be made. All functions, such as the volatility and transaction costs, are considered to only depend on time, not the transactions made. Using the method in this thesis on a more realistic problem would be difficult, why the PMP gradient method is most suited for a preliminary analysis of the problem.