Shadow prices for continuous processes

TL;DR

This paper studies utility maximization in financial markets with continuous prices and transaction costs, establishing conditions for the existence of a shadow price process that simplifies analysis.

Contribution

It provides sufficient conditions for the existence of shadow prices in markets with continuous processes and transaction costs, highlighting the importance of continuity and no unbounded profit assumptions.

Findings

Shadow prices exist under certain continuity and no arbitrage conditions.

Counter-example shows these conditions are necessary.

Results facilitate easier analysis of utility maximization problems.

Abstract

In a financial market with a continuous price process and proportional transaction costs we investigate the problem of utility maximization of terminal wealth. We give sufficient conditions for the existence of a shadow price process, i.e.~a least favorable frictionless market leading to the same optimal strategy and utility as in the original market under transaction costs. The crucial ingredients are the continuity of the price process and the hypothesis of "no unbounded profit with bounded risk". A counter-example reveals that these hypotheses cannot be relaxed.