We study the problem of maximising expected utility of terminal wealthover a nite horizon, with one risky and one riskless asset available, andwith trades in the risky asset subject to proportional transaction costs.In a discrete time setting, using a utility function with hyperbolic riskaversion, we prove that the optimal trading strategy is characterised bya function of time (t), which represents the ratio of wealth held in therisky asset to that held in the riskless asset. There is a time varying notransaction region with boundaries b(t) < s(t), such that the portfo-lio is only rebalanced when (t) is outside this region. The results areconsistent with similar studies of the in nite horizon problem with in-termediate consumption, where the no transaction region has a similar,but time independent, characterisation. We solve the problem numericallyand compute the boundaries of the no transaction region for typical modelparameters. We show how the results can be used to implement optionpricing models with transaction costs based on utility maximisation overa nite horizon
