Generalized statistical arbitrage concepts and related gain strategies

The notion of statistical arbitrage introduced in Bondarenko (2003) is generalized to statistical G-arbitrage corresponding to trading strategies which yield positive gains on average in a class of scenarios described by a σ-algebra G. This notion contains classical arbitrage as a special case. Admitting general static payoffs as generalized strategies, as done in Kassberger and Liebmann (2017) in the case of one pricing measure, leads to the notion of generalized statistical G-arbitrage. We show that even under standard no-arbitrage there may exist generalized gain strategies yielding positive gains on average under the specified scenarios. In the first part of the paper we prove that the characterization in Bondarenko (2003), no statistical arbitrage being equivalent to the existence of an equivalent local martingale measure with a path-independent density, is not correct in general. We establish that this equivalence holds true in complete markets and we derive a general sufficient condition for statistical G-arbitrages. As a main result we derive the equivalence of no statistical G-arbitrage to no generalized statistical G-arbitrage. In the second part of the paper we construct several classes of profitable generalized strategies with respect to various choices of the σ-algebra G. In particular, we consider several forms of embedded binomial strategies and follow-the-trend strategies as well as partition-type strategies. We study and compare their behavior on simulated data and also evaluate their performance on market data.