Existence of Shadow Prices in Finite Probability Spaces

TL;DR

This paper proves that in markets with finite probability spaces and proportional transaction costs, a shadow price process exists that aligns the utility maximization problems of frictionless and transaction-cost markets.

Contribution

It provides an elementary proof for the existence of shadow prices specifically in finite probability space markets, clarifying their theoretical foundation.

Findings

Shadow prices exist in finite probability spaces with transaction costs.

The proof is elementary and accessible.

Aligns utility maximization in frictionless and transaction-cost markets.

Abstract

A shadow price is a process lying within the bid/ask prices of a market with proportional transaction costs, such that maximizing expected utility from consumption in the frictionless market with this price process leads to the same maximal utility as in the original market with transaction costs. For finite probability spaces, this note provides an elementary proof for the existence of such a shadow price.