Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs

TL;DR

This paper extends the existence of shadow prices in portfolio optimization to models driven by fractional Brownian motion, relaxing previous semimartingale restrictions and applying to a broader class of utility functions.

Contribution

It proves the existence of shadow prices under the weaker TWC condition for fractional Brownian motion models, broadening applicability beyond semimartingale processes.

Findings

Shadow prices exist for exponential fractional Brownian motion.

Applicable to utility functions with reasonable asymptotic elasticity.

Extends previous results to models driven by fractional Brownian motion.

Abstract

We continue the analysis of our previous paper (Czichowsky/Schachermayer/Yang 2014) pertaining to the existence of a shadow price process for portfolio optimisation under proportional transaction costs. There, we established a positive answer for a continuous price process $S=(S_t)_{0\leq t\leq T}$ satisfying the condition $(NUPBR)$ of "no unbounded profit with bounded risk". This condition requires that $S$ is a semimartingale and therefore is too restrictive for applications to models driven by fractional Brownian motion. In the present paper, we derive the same conclusion under the weaker condition $(TWC)$ of "two way crossing", which does not require $S$ to be a semimartingale. Using a recent result of R.~Peyre, this allows us to show the existence of a shadow price for exponential fractional Brownian motion and $all$ utility functions defined on the positive half-line having reasonable asymptotic elasticity. Prime examples of such utilities are logarithmic or power utility.