|M.Sc Thesis||Department of Industrial Engineering and Management|
|Supervisor:||Dr. Zohar-Zalman Gady|
We consider a financial market with two assets: a risk free asset, and a stock that behaves like an Ito process. At any instance the investor may rebalance her portfolio by moving capital from stocks to the risk free asset and vice versa in order to maximize an expected utility at some terminal time T.
For common utility functions (power, log), when there are no transaction costs and also when transaction costs are proportional to the amount traded, the solution is well known and the trading strategy is continuous in time.
But the nature of the optimal policy changes dramatically when the investor has to pay some constant fixed transaction cost upon any intervention. Any trading strategy which is continuous in time, clearly leads to bankruptcy and the optimal policy must become trading at a carefully chosen discrete sequence of instances. With fixed transaction costs the optimization problem becomes much harder to solve.
In the present work we suggest a method to approximate the optimal policy for this problem when transaction costs are small. We also discuss the possible errors of the approximation and analyze some features of the optimal strategies.
In addition to maximizing expected utilities, we also address the problem of maximizing the probability of achieving a pre-determined goal at some terminal time. We present here some simple combinatorial solution for the no-transactions case and
then we apply the approximation method for the case when fixed transactions are present.
We also use simulation in order to compare the performance of our optimal policy versus some other non-trivial policies.