Contributor

Dr. Johannes Tausch, Dr. Arthur Selender

Subject Area

Economics

Abstract

This dissertation presents a computational framework for high-frequency options trading that combines Cross-Data-Type 1-D Convolutional Neural Networks (CDT-1D CNN) with Simpson-Sobolev regularization for directional prediction, and finite element methods (FEM) for realistic option pricing during backtesting. The core innovation lies in developing a mathematically rigorous regularization approach that maintains the adaptability of modern deep learning while enabling accurate evaluation through stochastic volatility models. The primary contribution is the Simpson-Sobolev regularization scheme, which extends traditional Sobolev regularization by incorporating Simpson’s rule for numerical integration. This approach achieves higher-order accuracy in approximating the Sobolev norms that control function smoothness. Simpson’s rule attains O(h^5) local truncation error (O(h^4) global error) for integrating smooth functions, compared to the O(h^3) local error (O(h^2) global error) of trapezoidal methods. In our implementation, this O(h^4) accuracy applies to the H^0 (function value) component, while derivative components achieve O(h^2) accuracy via central differencing schemes. We establish theoretical convergence guarantees for the regularized neural network, demonstrating that the solution converges to the optimal at a rate of O(k^{−1/2}), where k denotes the iteration count. The framework incorporates an adaptive boundary selection mechanism that automatically identifies optimal integration points based on the data distribution. The CDT-1D CNN component learns directly from raw price data, reducing the need for manual feature engineering by approximately 85% compared to traditional approaches— an estimate based on comparing the 5 raw OHLCV features used in CDT-1D against the typical 30–50 technical indicators employed in traditional quantitative trading systems. This efficiency gain is crucial for high-frequency trading applications where computational speed is paramount. Our dual-mode architecture supports both minimal feature extraction and comprehensive technical indicator analysis, with adaptive configuration management. Trading signals generated by the CNN are evaluated using a Heston stochastic volatility model solved via finite elements, providing realistic option pricing that accounts for volatility skew and the leverage effect. This sequential architecture—where the CNN predicts market direction and the FEM solver prices options for execution—enables rigorous backtesting while maintaining computational efficiency during training. Empirical validation on S&P 500 equity data spanning seven distinct market regimes (1998– 2025) demonstrates directional prediction accuracy approaching 50% on a three-class task (validation range 48.8–52.0%, mean 50.5%, versus 33.3% random baseline)—with confidenceconditional accuracy substantially higher in the threshold regime where trades are actually executed—and a median Sharpe ratio of 2.07 across all market periods, with profitable trading in all seven regimes. A small in-sample-to-out-of-sample generalization gap of −1.2 ± 1.7 percentage points across all seven periods (Table 6.32)—with validation accuracy meeting or exceeding training in five of the seven regimes—indicates that the Simpson- Sobolev regularization is effective at preventing overfitting despite the model’s capacity. The implementation framework comprises 22 specialized modules—including parallel execution components for multi-GPU distributed training—supporting both research experimentation and production deployment, with inference latency under 2 milliseconds suitable for real-time trading.

Degree Date

Summer 2026

Document Type

Dissertation

Degree Name

Ph.D.

Department

Mathematics

Advisor

Dr. Johannes Tausch

Acknowledgements

Dr. Johannes Tausch, Dr. Mikhail Zaslavskiy, Dr. Wei Cai, Dr. Arthur Selender, and my entire immediate family.

Number of Pages

470

Format

.pdf

Creative Commons License

Creative Commons Attribution-Noncommercial 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License

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