Do arbitrage-free prices come from utility maximization?

TL;DR

This paper investigates whether arbitrage-free prices align with utility maximization for investors with various initial endowments, revealing limitations and peculiar phenomena in the boundary cases.

Contribution

It demonstrates that arbitrage-free prices consistent with utility maximization do not always exist for all endowments, especially on the boundary, and explores the resulting unusual trading behaviors.

Findings

No universal arbitrage-free utility-maximizing prices for all endowments.

Existence of arbitrage-free prices depends on the investor's initial endowment.

Boundary endowments can lead to non-intuitive trading decisions near arbitrage prices.

Abstract

In this paper we ask whether, given a stock market and an illiquid derivative, there exists arbitrage-free prices at which an utility-maximizing agent would always want to buy the derivative, irrespectively of his own initial endowment of derivatives and cash. We prove that this is false for any given investor if one considers all initial endowments with finite utility, and that it can instead be true if one restricts to the endowments in the interior. We show however how the endowments on the boundary can give rise to very odd phenomena; for example, an investor with such an endowment would choose not to trade in the derivative even at prices arbitrarily close to some arbitrage price.