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February 2018 Volatility and arbitrage
E. Robert Fernholz, Ioannis Karatzas, Johannes Ruf
Ann. Appl. Probab. 28(1): 378-417 (February 2018). DOI: 10.1214/17-AAP1308

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.

Citation

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E. Robert Fernholz. Ioannis Karatzas. Johannes Ruf. "Volatility and arbitrage." Ann. Appl. Probab. 28 (1) 378 - 417, February 2018. https://doi.org/10.1214/17-AAP1308

Information

Received: 1 August 2016; Revised: 1 February 2017; Published: February 2018
First available in Project Euclid: 3 March 2018

zbMATH: 06873687
MathSciNet: MR3770880
Digital Object Identifier: 10.1214/17-AAP1308

Subjects:
Primary: 60G44, 60H05, 60H30, 91G10

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.28 • No. 1 • February 2018
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