Utility Maximization in a Binomial Model with transaction costs: a Duality Approach Based on the Shadow Price Process

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We consider the problem of optimizing the expected logarithmic utility of the value of a portfolio in a binomial model with proportional transaction costs with a long time horizon. By duality methods, we can find expressions for the boundaries of the no-trade-region and the asymptotic optimal growth rate, which can be made explicit for small transaction costs (in the sense of an asymptotic expansion). Here we find that, contrary to the classical results in continuous time, see Janeu010dek and Shreve (2004), Finance and Stochastics8, 181u2013206, the size of the no-trade-region as well as the asymptotic growth rate depend analytically on the level u03bb of transaction costs, implying a linear first-order effect of perturbations of (small) transaction costs, in contrast to effects of orders u03bb1/3 and u03bb2/3, respectively, as in continuous time models. Following the recent study by Gerhold et al. (2013), Finance and Stochastics17, 325u2013354, we obtain the asymptotic expansion by an almost explicit construction of the shadow price process.

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Theoretical or Mathematical/ duality (mathematics) investment optimisation pricing/ utility maximization binomial model duality approach shadow price process expected logarithmic utility optimization portfolio proportional transaction costs long time horizon no-trade-region boundary asymptotic optimal growth rate level lambda linear first-order effect continuous time models asymptotic expansion/ C1290D Systems theory applications in economics and business C1180 Optimisation techniques