On the Existence of Shadow Prices

TL;DR

This paper investigates the existence of shadow prices in utility maximization with transaction costs, providing a counterexample where they fail to exist and proving conditions under which they do exist.

Contribution

It demonstrates that shadow prices may not always exist and establishes short selling constraints as a sufficient condition for their existence in complex market models.

Findings

Counterexample shows shadow prices can fail to exist.

Short selling constraints guarantee shadow price existence.

Results apply to multi-currency markets with discontinuous spreads.

Abstract

For utility maximization problems under proportional transaction costs, it has been observed that the original market with transaction costs can sometimes be replaced by a frictionless "shadow market" that yields the same optimal strategy and utility. However, the question of whether or not this indeed holds in generality has remained elusive so far. In this paper we present a counterexample which shows that shadow prices may fail to exist. On the other hand, we prove that short selling constraints are a sufficient condition to warrant their existence, even in very general multi-currency market models with possibly discontinuous bid-ask-spreads.