The dual optimizer for the growth-optimal portfolio under transaction costs

TL;DR

This paper develops an explicit dual optimizer approach for maximizing long-term growth in a Black-Scholes model with proportional transaction costs, providing analytical formulas for the no-trade region and growth rate.

Contribution

It introduces a method to explicitly compute the shadow price and fractional Taylor expansions for the optimal strategy under transaction costs.

Findings

Explicit formulas for the shadow price and no-trade region.

Analytical expressions for the optimal growth rate.

Fractional Taylor expansions for strategy and growth rate.

Abstract

We consider the maximization of the long-term growth rate in the Black-Scholes model under proportional transaction costs as in Taksar, Klass and Assaf [Math. Oper. Res. 13, 1988]. Similarly as in Kallsen and Muhle-Karbe [Ann. Appl. Probab., 20, 2010] for optimal consumption over an infinite horizon, we tackle this problem by determining a shadow price, which is the solution of the dual problem. It can be calculated explicitly up to determining the root of a deterministic function. This in turn allows to explicitly compute fractional Taylor expansions, both for the no-trade region of the optimal strategy and for the optimal growth rate.