Simple arbitrage

TL;DR

This paper characterizes the absence of simple arbitrage in financial markets with risky assets, introducing the 'two-way crossing' property and applying it to models like mixed fractional Black-Scholes and stochastic volatility.

Contribution

It introduces the 'two-way crossing' property as a criterion for no simple arbitrage in continuous models and applies it to various stochastic volatility models.

Findings

Mixed fractional Black-Scholes model with Hurst > 0.5 has no simple arbitrage.

The 'two-way crossing' property characterizes absence of simple arbitrage in continuous models.

Many models satisfy the no simple arbitrage condition via the law of the iterated logarithm.

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 \epsilon, 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-H\"{o}lder continuous process.