The Annals of Applied Probability

Volatility and arbitrage

E. Robert Fernholz, Ioannis Karatzas, and Johannes Ruf

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G44: Martingales with continuous parameter 60H05: Stochastic integrals 60H30: Applications of stochastic analysis (to PDE, etc.) 91G10: Portfolio theory

Trading strategies functional generation relative arbitrage short-term arbitrage support of diffusions diffusions on manifolds nondegeneracy


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

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