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

TL;DR

This paper analyzes optimal portfolio utility in a binomial model with transaction costs, deriving explicit formulas for no-trade regions and growth rates, highlighting linear effects of small costs via a duality approach using shadow prices.

Contribution

It introduces a duality method to explicitly characterize the no-trade region and growth rate dependence on transaction costs in a binomial model.

Findings

No-trade-region size depends analytically on transaction costs.

Asymptotic growth rate has a linear first-order dependence on small transaction costs.

Shadow price process construction enables explicit asymptotic expansion.

Abstract

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. Here we find that, contrary to the classical results in continuous time, the size of the no-trade-region as well as the asymptotic growth rate depend analytically on the level of transaction costs, implying a linear first order effect of perturbations of (small) transaction costs. We obtain the asymptotic expansion by an almost explicit construction of the shadow price process.