Optimal portfolio selection under vanishing fixed transaction costs

TL;DR

This paper investigates the asymptotic behavior of a long-term portfolio optimization model with fixed and proportional transaction costs as fixed costs vanish, establishing convergence to a purely proportional cost model and characterizing optimal strategies.

Contribution

It introduces a limit model with proportional costs and proves the convergence of optimal strategies, boundaries, and growth rates as fixed costs tend to zero.

Findings

Optimal strategy involves maintaining risky fraction within a specific interval

Convergence of optimal boundaries and growth rates is rigorously established

Limit model with purely proportional costs accurately describes the asymptotic behavior

Abstract

In this paper, asymptotic results in a long-term growth rate portfolio optimization model under both fixed and proportional transaction costs are obtained. More precisely, the convergence of the model when the fixed costs tend to zero is investigated. A suitable limit model with purely proportional costs is introduced and an optimal strategy is shown to consist of keeping the risky fraction process in a unique interval $[A,B]\subseteq\,]0,1[$ with minimal effort. Furthermore, the convergence of optimal boundaries, asymptotic growth rates, and optimal risky fraction processes is rigorously proved. The results are based on an in-depth analysis of the convergence of the solutions to the corresponding HJB-equations.