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Hybrid Ridgelet Deep Neural Networks for Data-Driven Arbitrage Strategies

Bahadur Yadav, Sanjay Kumar Mohanty

TL;DR

This work tackles robust data-driven arbitrage in high-dimensional time series by proposing HRDNN, which couples the Ridgelet Transform with deep neural networks to approximate robust super-replication prices under an ambiguity set $\\mathcal{P}$. It introduces a penalized functional $\\mathcal{X}_{B,L,k}$ and proves its convergence to the target value, then shows that HRDNN, under Lipschitz and boundedness constraints, can densely approximate admissible trading strategies via ridgelet components. A key result is that, by discretizing the terminal state via finite partitions $\\mathcal{F}_i$, HRDNN can approximate conditional expectations and robust arbitrage prices with provable convergence to $\\mathcal{X}_{B,L}(\\Phi,\\sigma(S_{t_n}))$. Empirically, HRDNN with various activations outperforms traditional buy-and-hold and LP-based methods across up to 50 assets, including during market stress, underscoring its scalability and practical utility for data-driven, high-dimensional arbitrage detection. These findings suggest that Ridgelet-enabled neural architectures can robustly exploit complex market structures while maintaining stability under uncertainty.

Abstract

In this study, we propose a novel model framework that integrates deep neural networks with the Ridgelet Transform. The Ridgelet Transform on Borel measurable functions is used for arbitrage detection on high-dimensional sparse structures. This transform also enhances the expressive power of neural networks, enabling them to capture complex and high-dimensional market structures. Theoretically, we determine profitable trading strategies by optimizing hybrid ridgelet deep neural networks. Further, we emphasize the role of activation functions in ensuring stability and adaptability under uncertainty. We use a high-performance computing cluster for the detection of arbitrage across multiple assets, ensuring scalability, and processing large-scale financial data. Empirical results demonstrate strong profitability across diverse scenarios involving up to 50 assets, with particularly robust performance during periods of market volatility.

Hybrid Ridgelet Deep Neural Networks for Data-Driven Arbitrage Strategies

TL;DR

This work tackles robust data-driven arbitrage in high-dimensional time series by proposing HRDNN, which couples the Ridgelet Transform with deep neural networks to approximate robust super-replication prices under an ambiguity set . It introduces a penalized functional and proves its convergence to the target value, then shows that HRDNN, under Lipschitz and boundedness constraints, can densely approximate admissible trading strategies via ridgelet components. A key result is that, by discretizing the terminal state via finite partitions , HRDNN can approximate conditional expectations and robust arbitrage prices with provable convergence to . Empirically, HRDNN with various activations outperforms traditional buy-and-hold and LP-based methods across up to 50 assets, including during market stress, underscoring its scalability and practical utility for data-driven, high-dimensional arbitrage detection. These findings suggest that Ridgelet-enabled neural architectures can robustly exploit complex market structures while maintaining stability under uncertainty.

Abstract

In this study, we propose a novel model framework that integrates deep neural networks with the Ridgelet Transform. The Ridgelet Transform on Borel measurable functions is used for arbitrage detection on high-dimensional sparse structures. This transform also enhances the expressive power of neural networks, enabling them to capture complex and high-dimensional market structures. Theoretically, we determine profitable trading strategies by optimizing hybrid ridgelet deep neural networks. Further, we emphasize the role of activation functions in ensuring stability and adaptability under uncertainty. We use a high-performance computing cluster for the detection of arbitrage across multiple assets, ensuring scalability, and processing large-scale financial data. Empirical results demonstrate strong profitability across diverse scenarios involving up to 50 assets, with particularly robust performance during periods of market volatility.

Paper Structure

This paper contains 14 sections, 7 theorems, 46 equations, 3 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Mathematical Setting
  3. Proposed Model
  4. $\mathcal{P}$-Robust $\mathcal{G}$-Arbitrage Strategies
  5. An Approximation of $\mathcal{X}_{B,L}$ through Penalization
  6. An Approximation of $\mathcal{X}_{B,L}$ using HRDNN
  7. Computation of $\mathcal{P}$-Robust Statistical Arbitrage Strategies using HRDNN
  8. Empirical Evaluation on Real-World Financial Data
  9. Neural Network Architecture and Training Setup
  10. Performance Matrices for Trading Strategy
  11. Performance Ratios for Trading Strategy
  12. Comparison with the Linear Programming Approach
  13. Trading in a Large Number of Stocks
  14. Conclusion

Key Result

Lemma 3.1

( neufeld2024detecting): Let Assumption assumption1 holds true. Then, for all $B > 0$ and $L > 0$, the space ${W}_{B,L}$ is compact in the uniform topology on $\Omega$.

Figures (3)

  • Figure 1: Comparison of Sharpe ratios between the Linear Programming approach and the proposed method across iterations (1--50).
  • Figure 2: This figure illustrates the price evolution of 10 selected securities during the training period from 2000/01/03 to 2015/11/25 (shown in blue) and the testing period from 2015/11/26 to 2020/12/31 (shown in red).
  • Figure 3: This figure illustrates the price evolution of 50 selected securities during the training period from 2000/01/03 to 2015/11/25 (shown in blue) and the testing period from 2015/11/26 to 2020/12/31 (shown in red).

Theorems & Definitions (7)

  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Lemma 3.4: Density of Ridgelet Strategies
  • Proposition 3.5
  • Proposition 3.6
  • Theorem