Gauge Theory of Finance?

The recent stimulating proposal of a "Gauge Theory of Finance" by Ilinsky et al. is connected here with traditional approaches. First, the derivation of the log-normal distribution is shown to be 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 not uniquely specified 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 a 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.