Utility maximization in a binomial model with proportional transaction costs

We study the classical problem of maximizing the expected utility of the terminal value of a portfolio in a binomial (Cox-Ross-Rubinstein) model. By classical results [Merton 1969] both in discrete and continuous time, the optimal portfolio strategy in a friction-less market is is given by keeping the proportion between the wealth invested in the stock and the total portfolio wealth constant. For markets with proportional transaction costs in continuous time, it is known that the optimal trading strategy consists in keeping the wealth proportion inside an interval, in the sense that no trade takes place while the proportion remains in the interior of the interval. The size of this interval is asymptotically proportional to the bid-ask-spread to the power 1/3. In the discrete-time case, we show that the size of the no-trade-region is now asymptotically proportional to the bid-ask-spread itself.