Generalized statistical arbitrage concepts and related gain strategies

Abstract

The notion of statistical arbitrage introduced in Bondarenko (2003) is generalized to statistical-arbitrage corresponding to trading strategies which yield positive gains on average in a class of scenarios described by a-algebra. 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-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-arbitrages. As a main result we derive the equivalence of no statistical-arbitrage to no generalized statistical-arbitrage. In the second part of the paper we construct several classes of profitable generalized strategies with respect to various choices of the-algebra. In particular, we consider several forms This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

Detecting data-driven robust statistical arbitrage strategies with deep neural networks

“…These are P-robust Garbitrage strategies with the specific choice G := σ(S tn ). These strategies have a particular importance as they can be interpreted as profitable trading strategies on average given any terminal value, compare also [10], [38], [44], and [50]. As σ(S tn ) contains an infinite number of sets, it is a priori unclear how to compute numerically a conditional expectation with respect to σ(S tn ).…”

“…Based on this idea, [38] generalized the notion of statistical arbitrage from [10] by introducing Garbitrage defined through zero-cost payoffs Y (which are not necessarily the payoffs of self-financing trading strategies) fulfilling for G being a σ-algebra G ⊆ σ(S), which allows, in particular, to take into account more flexible choices of trading strategies, possibly adjusted to available information. Building on the definition of G-arbitrage, the results from [10, Proposition 1], [38,Proposition 6], and [50,Theorem 3.3] characterize the existence of G-arbitrage strategies by relating the absence of self-financing strategies fulfilling (1.1) to the existence of G-measurable Radon-Nikodym densities, a result which can be considered as an extension of the fundamental theorem of asset-pricing (compare [2,11,19,34,53] for several versions of the fundamental theorem of asset-pricing in different underlying settings) which connects the absence of arbitrage with the existence of pricing measures. The authors from [50] further propose and validate empirically an embedding-methodology to exploit statistical arbitrage on financial markets.…”

“…Building on the definition of G-arbitrage, the results from [10, Proposition 1], [38,Proposition 6], and [50,Theorem 3.3] characterize the existence of G-arbitrage strategies by relating the absence of self-financing strategies fulfilling (1.1) to the existence of G-measurable Radon-Nikodym densities, a result which can be considered as an extension of the fundamental theorem of asset-pricing (compare [2,11,19,34,53] for several versions of the fundamental theorem of asset-pricing in different underlying settings) which connects the absence of arbitrage with the existence of pricing measures. The authors from [50] further propose and validate empirically an embedding-methodology to exploit statistical arbitrage on financial markets. All these mentioned contributions assume that the underlying securities behave according to a previously fixed underlying probability measure P. To account for ambiguity with respect to the choice of an underlying probability measure [44] recently introduced the notion of P-robust G-arbitrage referring to self-financing trading strategies that allow for G-arbitrage independent which measure P from the considered ambiguity set of measures P is the "correct" underlying probability measure, as the strategy is required to be profitable for all measures from the ambiguity set.…”