Volatility and Arbitrage

TL;DR

This paper investigates conditions under which arbitrage opportunities exist in equity markets based on market volatility measures, proving that persistent outperformance over arbitrary horizons is generally impossible without extra assumptions.

Contribution

It resolves a long-standing open question by showing that strong arbitrage over all time horizons is not generally feasible, and identifies conditions enabling outperformance with specific strategies.

Findings

Strong arbitrage is not possible over arbitrary horizons in general.

Additional conditions can enable outperformance over any time horizon.

Constructed strategies can achieve outperformance under these conditions.

Abstract

The capitalization-weighted total relative variation $\sum_{i=1}^d \int_0^\cdot \mu_i (t) \mathrm{d} \langle \log \mu_i \rangle (t)$ in an equity market consisting of a fixed number $d$ of assets with capitalization weights $\mu_i (\cdot)$ is an observable and nondecreasing function of time. If this observable of the market is not just nondecreasing, but actually grows at a rate which is bounded away from zero, then strong arbitrage can be constructed relative to the market over sufficiently long time horizons. It has been an open issue for more than ten years, whether such strong outperformance of the market is possible also over arbitrary time horizons under the stated condition. We show that this is not possible in general, thus settling this long-open question. We also show that, under appropriate additional conditions, outperformance over any time horizon indeed becomes possible, and exhibit investment strategies that effect it.