Gauge Invariance, Geometry and Arbitrage

TL;DR

This paper introduces a gauge-invariant measure of arbitrage in market models based on Itô processes, revealing its geometric interpretation and providing methods to detect arbitrage, especially in high-frequency trading data.

Contribution

It presents the most general arbitrage measure invariant under market transformations and extends the Martingale pricing theorem to include arbitrage scenarios.

Findings

Market is efficient at daily or longer horizons.

Evidence of arbitrage in high-frequency intraday data.

Arbitrage decay time is approximately one minute.

Abstract

In this work, we identify the most general measure of arbitrage for any market model governed by It\^o processes. We show that our arbitrage measure is invariant under changes of num\'{e}raire and equivalent probability. Moreover, such measure has a geometrical interpretation as a gauge connection. The connection has zero curvature if and only if there is no arbitrage. We prove an extension of the Martingale pricing theorem in the case of arbitrage. In our case, the present value of any traded asset is given by the expectation of future cash-flows discounted by a line integral of the gauge connection. We develop simple strategies to measure arbitrage using both simulated and real market data. We find that, within our limited data sample, the market is efficient at time horizons of one day or longer. However, we provide strong evidence for non-zero arbitrage in high frequency intraday data. Such events seem to have a decay time of the order of one minute.