Gauge invariance, geometry and arbitrage

Abstract

We identify the most general measure of arbitrage for any market model governed by Ito processes, and, on that basis, we develop dynamic arbitrage strategies. It is shown that our arbitrage measure is invariant under changes of numeraire and equivalent probability measure. Moreover, such a 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 cashflows discounted by a line integral of the gauge connection. We develop simple dynamic 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 nonzero arbitrage in high-frequency intraday data. Such events seem to have a decay time of the order of one minute.