The Annals of Applied Probability

Simple arbitrage

Christian Bender

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We characterize absence of arbitrage with simple trading strategies in a discounted market with a constant bond and several risky assets. We show that if there is a simple arbitrage, then there is a 0-admissible one or an obvious one, that is, a simple arbitrage which promises a minimal riskless gain of $\varepsilon$, if the investor trades at all. For continuous stock models, we provide an equivalent condition for absence of 0-admissible simple arbitrage in terms of a property of the fine structure of the paths, which we call “two-way crossing.” This property can be verified for many models by the law of the iterated logarithm. As an application we show that the mixed fractional Black–Scholes model, with Hurst parameter bigger than a half, is free of simple arbitrage on a compact time horizon. More generally, we discuss the absence of simple arbitrage for stochastic volatility models and local volatility models which are perturbed by an independent 1$/$2-Hölder continuous process.

Article information

Ann. Appl. Probab., Volume 22, Number 5 (2012), 2067-2085.

First available in Project Euclid: 12 October 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 91G10: Portfolio theory
Secondary: 60G44: Martingales with continuous parameter 60G22: Fractional processes, including fractional Brownian motion

Arbitrage simple strategies fractional Brownian motion law of the iterated logarithm conditional full support


Bender, Christian. Simple arbitrage. Ann. Appl. Probab. 22 (2012), no. 5, 2067--2085. doi:10.1214/11-AAP830.

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