Simple arbitrage

Abstract

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 ε, 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. © Institute of Mathematical Statistics, 2012.