Neural networks can detect model-free static arbitrage strategies

TL;DR

The paper addresses the detection of model-free static arbitrage in high-dimensional financial markets and proposes neural-network detectors that can output actionable arbitrage strategies in real time after offline training. It establishes a theoretical bridge between arbitrage detection and convex semi-infinite programs (CSIP), proving that a single neural network can approximate solutions to CSIP and, consequently, identify and exploit arbitrage when it exists. The work provides key results (including cor_ftpa and thm:epsilon-arbitrage) and a practical Algorithm 3 for offline training that approximates LSIP-based bounds, with numerical demonstrations on S&P 500 data and historical option prices showing strong detection accuracy and profitable backtests. The approach offers near-instantaneous arbitrate detection and strategy execution in fast-moving markets, with robust performance across hyperparameters and market conditions.

Abstract

In this paper we demonstrate both theoretically as well as numerically that neural networks can detect model-free static arbitrage opportunities whenever the market admits some. Due to the use of neural networks, our method can be applied to financial markets with a high number of traded securities and ensures almost immediate execution of the corresponding trading strategies. To demonstrate its tractability, effectiveness, and robustness we provide examples using real financial data. From a technical point of view, we prove that a single neural network can approximately solve a class of convex semi-infinite programs, which is the key result in order to derive our theoretical results that neural networks can detect model-free static arbitrage strategies whenever the financial market admits such opportunities.

Figures (4)

• Figure 1: The loss function as well as the training and test set accuracy in dependenceof the number of trained epochs.
• Figure 2: Left: The histogram shows the distribution of the net profit $\mathcal{I}_{S_{i,j}}(K_i,a_i,h_i)-f(\pi_i,a_i,h_i)$ for $i = 1,\dots,5000$, $j =1,\dots, 200$ of the strategy trained as described in Section \ref{['sec_exa_1']}. Right: The histogram shows the distribution of the net profit conditional on wrong identification of arbitrage.
• Figure 3: The histogram shows the distribution of the net profit $\mathcal{I}_{S_{i,j}}(K_i,a_i,h_i)-f(\pi_i,a_i,h_i)$ for $i = 1,\dots,5000$, $j =1,\dots, 200$ of the strategy trained as described in Section \ref{['sec_exa_1']} but on a reduced and balanced training set where 50 % of the sampled constitute arbitrage situations.
• Figure 4: In the setting of Section \ref{['sec_exa_2']}, the histogram depicts the net profits $\mathcal{I}_{S_T}(K_i,a_i,h_i)-f(\pi_i,a_i,h_i)$ for $i = 1,\dots,33$, where here $S_T\in \mathbb{R}^5$ refers to the observed realization of the $5$ underlying securities at maturity $T=$$24$ March $2023$.

Theorems & Definitions (27)

• Definition 2.1: Model-free static arbitrage
• Proposition 2.2: Universal approximation theorem
• Remark 2.4
• Theorem 2.5: Neural networks can detect static arbitrage
• Theorem 2.6: A single neural network can detect static arbitrage of magnitude $\varepsilon$
• Proposition 2.7: Approximating $V$ with neural networks
• Remark 3.1
• Remark 4.4: On the assumptions
• Theorem 4.5: Single neural network provides corresponding feasible $\varepsilon$-optimizer for class of (CSIP)
• proof : Proof of Proposition \ref{['prop_arbitrage']}