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

TL;DR

The paper tackles robust statistical arbitrage under model ambiguity in high-dimensional markets. It casts trading as a conditional super-replication problem under an ambiguity set $\mathcal{P}$ and solves a penalized surrogate $\Gamma_{B,L,k}$ using neural networks, with universal approximation ensuring representability of trading rules. The main theoretical result shows convergence $\lim_{k\to\infty} \Gamma_{B,L,k}(\Phi,\mathcal{G}) = \Gamma_{B,L}(\Phi,\mathcal{G})$, enabling computationally feasible, data-driven construction of $\mathcal{P}$-robust $\mathcal{G}$-arbitrage strategies via deep nets. Empirical demonstrations across up to $d=50$ assets and crisis periods show robust outperformance relative to the market and to classical pairs trading, highlighting the method’s scalability and practical impact. The framework thus provides a scalable, data-driven path to robust arbitrage that remains effective when cointegration or mean-reversion assumptions fail.

Abstract

We present an approach, based on deep neural networks, that allows identifying robust statistical arbitrage strategies in financial markets. Robust statistical arbitrage strategies refer to trading strategies that enable profitable trading under model ambiguity. The presented novel methodology allows to consider a large amount of underlying securities simultaneously and does not depend on the identification of cointegrated pairs of assets, hence it is applicable on high-dimensional financial markets or in markets where classical pairs trading approaches fail. Moreover, we provide a method to build an ambiguity set of admissible probability measures that can be derived from observed market data. Thus, the approach can be considered as being model-free and entirely data-driven. We showcase the applicability of our method by providing empirical investigations with highly profitable trading performances even in 50 dimensions, during financial crises, and when the cointegration relationship between asset pairs stops to persist.

Figures (11)

• Figure 1: Illustration of the observation dates $(s_i)_{i=1,\dots,N}$ and the future dates $(t_i)_{i=0,\dots,n-1}$.
• Figure 2: The figure illustrates how the measure $\widehat{\mathbb{P}}$ and $\widehat{\mathbb{P}}_{\tau}$, as defined in \ref{['eq_definition_dirac_measure']} and \ref{['eq_defn_p_tau']} respectively, are constructed. Left: An exemplary observed path $(Y_{s_i})_{i=1,\dots,7}$ is displayed. Centre: In the case $n=2$ the future paths with positive probability under $\widehat{\mathbb{P}}$ are shown. To each of the paths equal probability is assigned. Right: A possible realization of $\widehat{\mathbb{P}}_{\tau}$ is depicted, the paths from $\widehat{\mathbb{P}}$ are deviated according to a normally distributed realization of $\tau$.
• Figure 3: We compare the Sharpe ratio of a trading strategy that was trained in line with Algorithm \ref{['algo_training_nn']} with the Sharpe ratio of a strategy that was trained according to an LP-approach described in lutkebohmert2021robust. We depict the Sharpe ratio in dependence of the width of the bounds $\overline{K}^1-\underline{K}^1$. The underlying security is the EUROSTOXX 50, while the testing period ranges from September $2013$ until July $2018$.
• Figure 4: The figure shows the evolution of the $S\&P~500$ (bottom) and the EUROSTOXX$50$ (top) in the training period from $1986/12/31$ to $2006/11/30$ (blue) and in the testing period from $2006/11/31$ to $2013/01/25$ (red).
• Figure 5: The figure shows the equity curves of the trained strategies with per share transaction costs during the testing period depicted in Figure \ref{['fig_test_period']}. The y-axis represents the cumulated net profits as the trading window propagates. Since a total of 50 independent experiments are conducted, we plot the equity curves by quantiles, where the rank is based on the net profit of the experiment at the end of the testing period.

Theorems & Definitions (23)

• Remark 2.2
• Definition 2.3: $\mathcal{P}$-robust $\mathcal{G}$-arbitrage
• Lemma 2.4
• Lemma 2.7
• Proposition 2.8
• Remark 2.9
• Lemma 2.11
• Proposition 2.12
• Proposition 2.13
• Theorem 2.14