Transaction Costs, Shadow Prices, and Duality in Discrete Time

TL;DR

This paper investigates the existence of shadow prices in discrete-time portfolio optimization with transaction costs, revealing that shadow prices may not always exist even in simple cases, and clarifies their relation to duality theory.

Contribution

It provides an explicit counter-example demonstrating the non-existence of shadow prices in seemingly simple market settings and clarifies the connection between shadow prices and duality theory.

Findings

Shadow prices may fail to exist even in arbitrage-free markets with small transaction costs.

Dual minimizers correspond to a local version of shadow prices.

The paper clarifies the relationship between shadow prices and duality theory.

Abstract

For portfolio choice problems with proportional transaction costs, we discuss whether or not there exists a "shadow price", i.e., a least favorable frictionless market extension leading to the same optimal strategy and utility. By means of an explicit counter-example, we show that shadow prices may fail to exist even in seemingly perfectly benign situations, i.e., for a log-investor trading in an arbitrage-free market with bounded prices and arbitrarily small transaction costs. We also clarify the connection between shadow prices and duality theory. Whereas dual minimizers need not lead to shadow prices in the above "global" sense, we show that they always correspond to a "local" version.