Simple Arbitrage

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. 1. Introduction. The fundamental theorem of asset pricing characterizes absence of arbitrage in terms of the existence of equivalent martingale measures. More precisely, the version of the fundamental theorem obtained by Delbaen and Schachermayer [7] states that a locally bounded stock model does not admit a free lunch with vanishing risk, if and only if the the model has an equivalent local martingale measure. As absence of arbitrage is generally considered as a minimum requirement for a sensible stock model, nonsemimartingale models have widely been ruled out in financial model-ing. However, absence of arbitrage heavily depends on the class of admissible strategies. In this respect the fundamental theorem of asset prices assumes the largest possible class of admissible strategies, namely all self-financing strategies with wealth processes which are bounded from below. In this paper we discuss absence of arbitrage within the class of simple strategies. The class of simple strategies consists of portfolios which cannot