Gauge theory of Finance?

TL;DR

The paper critiques the Gauge Theory of Finance by showing its derivations are equivalent to classical results and highlights issues like unjustified assumptions and potential arbitrage opportunities, aiming to improve its formulation.

Contribution

It critically analyzes the Gauge Theory of Finance, identifying redundancies and issues, and suggests improvements for a more coherent framework.

Findings

Derivations of log-normal distribution and Black-Scholes are equivalent to classical methods.

The theory's assumptions, like the exponential weight, lack sufficient justification.

Potential arbitrage opportunities are identified in the model.

Abstract

Some problems with the recent stimulating proposal of a ``Gauge Theory of Finance'' by Ilinski and collaborators are outlined. First, the derivation of the log-normal distribution is shown equivalent both in information and mathematical content to the simpler and well-known derivation, dating back from Bachelier and Samuelson. Similarly, the re-derivation of Black-Scholes equation is shown equivalent to the standard one because the limit of no uncertainty is equivalent to the standard risk-free replication argument. Both re-derivations of the log-normality and Black-Scholes result do not provide a test of the theory because it is degenerate in the limits where these results apply. Third, the choice of the exponential form a la Boltzmann, of the weight of a given market configuration, is a key postulate that requires justification. In addition, the ``Gauge Theory of Finance'' seems to lead to ``virtual'' arbitrage opportunities for pure Markov random walk market when there should be none. These remarks are offered in the hope to improve the formulation of the ``Gauge Theory of Finance'' into a coherent and useful framework.