## The Annals of Applied Probability

### Volatility and arbitrage

#### Abstract

The capitalization-weighted cumulative 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 a nondecreasing function of time. If this observable of the market is not just nondecreasing but actually grows at a rate 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.

#### Article information

Source
Ann. Appl. Probab., Volume 28, Number 1 (2018), 378-417.

Dates
Revised: February 2017
First available in Project Euclid: 3 March 2018

https://projecteuclid.org/euclid.aoap/1520046091

Digital Object Identifier
doi:10.1214/17-AAP1308

Mathematical Reviews number (MathSciNet)
MR3770880

Zentralblatt MATH identifier
06873687

#### Citation

Fernholz, E. Robert; Karatzas, Ioannis; Ruf, Johannes. Volatility and arbitrage. Ann. Appl. Probab. 28 (2018), no. 1, 378--417. doi:10.1214/17-AAP1308. https://projecteuclid.org/euclid.aoap/1520046091

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