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 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.

中文翻译：

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广义统计套利概念及相关收益策略

Bondarenko（2003）中引入的统计套利概念被广义化为与交易策略相对应的统计套利，该交易策略在代数描述的一类情景中平均产生正收益。此概念包含古典套利作为特例。在一项定价措施的情况下，如Kassberger和Liebmann（2017）所做的那样，将一般静态收益作为广义策略来接受，导致了广义统计的概念-套利。我们证明，即使在标准无套利的情况下，也可能存在通用的收益策略，在特定情况下平均产生正收益。在本文的第一部分中，我们证明了邦达连科（2003）的刻画，在统计上的套利没有等效于存在与路径无关的密度的等效局部local测度，这通常是不正确的。我们确定这种等价性在完整的市场中成立，并且得出了统计套利的一般充分条件。作为主要结果，我们得出了无统计套利与无广义统计套利的等效性。在本文的第二部分中，我们针对不同的选择，构造了几类有利可图的广义策略。代数。特别是，我们考虑了几种形式的嵌入式二项式策略，遵循趋势策略以及分区类型策略。我们研究和比较它们在模拟数据上的行为，并评估其在市场数据上的表现。