Futures Trading with Transaction Costs

We specify a continuous time model of futures contract market where the price follows an arithmetic Brownian motion. In this market an agent is not allowed to have a short position in the contract or borrow money to make investment, but he or she can invest arbitrary portion of his or her wealth. With each transaction proportional transaction cost is paid and agent's goal is to maximize long-term logarithmic utility of the remaining wealth. State of the system is described by a stochastic differential equation (SDE). Using ItO's formula we show that the information about the portion of invested wealth is sufficient for making optimal decisions. We use Markov decision process with finite state space and discrete time to approximate SDE for the portion of invested wealth. The value function for approximating finite Markov chain is found by policy iteration, also known as Howard's algorithm. The optimal policy is determined by an interval which the portion of invested wealth has to be kept in. At the end we provide numerical results.