Arbitrage Problems with Reflected Geometric Brownian Motion

TL;DR

This paper demonstrates that financial models based on reflected geometric Brownian motion are not arbitrage-free, contradicting previous claims, and are unsuitable for option pricing due to violations of no-arbitrage conditions.

Contribution

It clarifies that reflected geometric Brownian motion models violate no-arbitrage conditions and cannot be used for proper derivative pricing, correcting misconceptions in prior literature.

Findings

Reflected geometric Brownian motion models violate no-arbitrage conditions

Such models lack risk-neutral measures and numéraire portfolios

Option pricing formulas based on these models violate no-arbitrage bounds

Abstract

Contrary to the claims made by several authors, a financial market model in which the price of a risky security follows a reflected geometric Brownian motion is not arbitrage-free. In fact, such models violate even the weakest no-arbitrage condition considered in the literature. Consequently, they do not admit num\'eraire portfolios or equivalent risk-neutral probability measures, which makes them totally unsuitable for contingent claim valuation. Unsurprisingly, the published option pricing formulae for such models violate textbook no-arbitrage bounds.