Paper: arXiv 2504.20088

Authors: Joao Felipe Gueiros, Hemanth Chandravamsi, Steven H. Frankel

Abstract

This paper explores the use of deep residual networks for pricing European options on Petrobras, one of the world’s largest oil and gas producers, and compares its performance with the Black-Scholes (BS) model. Using eight years of historical data from B3 (Brazilian Stock Exchange) collected via web scraping, a deep learning model was trained using a custom built hybrid loss function that incorporates market data and analytical pricing. The data for training and testing were drawn between the period spanning November 2016 to January 2025, using an 80-20 train-test split. The test set consisted of data from the final three months: November, December, and January 2025. The deep residual network model achieved a 64.3% reduction in the mean absolute error for the 3-19 BRL (Brazilian Real) range when compared to the Black-Scholes model on the test set. Furthermore, unlike the Black-Scholes solution, which tends to decrease its accuracy for longer periods of time, the deep learning model performed accurately for longer expiration periods. These findings highlight the potential of deep learning in financial modeling, with future work focusing on specialized models for different price ranges.

Complexity vs Empirical Score

  • Math Complexity: 6.0/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper features significant mathematical content, including partial differential equations and neural network architectures, while also demonstrating strong empirical rigor through multi-year Brazilian market data collection, hybrid loss functions, and clear performance metrics.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Compare Deep Learning vs<br>Black-Scholes on Petrobras Options"] --> B
  
  subgraph B ["Methodology & Data"]
      direction LR
      B1["Web Scraping<br>B3 Historical Data"] --> B2["Preprocessing<br>Nov 2016 - Jan 2025"]
      B2 --> B3["Split: 80% Train / 20% Test<br>Test: Nov-Jan 2025"]
  end

  B --> C
  
  subgraph C ["Computational Processes"]
      direction LR
      C1["Black-Scholes<br>Standard Analytical Model"] --- C2["Deep Learning<br>ResNet + Hybrid Loss Function"]
  end

  C --> D
  
  subgraph D ["Key Findings"]
      D1["64.3% reduction in MAE<br>for 3-19 BRL price range"]
      D2["DL accuracy maintained<br>for longer expiration periods"]
      D3["BS model accuracy<br>decreases over time"]
  end

Cited References (30)

  1. Théorie de la spéculation (1900)
  2. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations (2019) — link
  3. CS230: Deep Learning - Option Pricing with Deep Learning (2019) — link
  4. Neural Network Learning of Black-Scholes Equation for Option Pricing (2024) — link
  5. Deep Residual Learning for Image Recognition (2016)
  6. Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift (2015)
  7. Optuna: A Next-generation Hyperparameter Optimization Framework (2019)
  8. Adam: A Method for Stochastic Optimization (2015) — link
and 22 more
  1. Option pricing based on the residual neural network (2024)
  2. On the numerical solution of nonlinear Black–Scholes equations (2008)
  3. A Deep Learning Approach for Option Pricing in Emerging Markets (2021) — link
  4. Deep Learning Artificial Neural Network for Pricing Multi-Asset European Options (2023)
  5. Accelerated American Option Pricing with Deep Neural Networks (2023)
  6. Stable Multilevel Deep Neural Networks for Option Pricing and xVAs Using Forward-Backward Stochastic Differential Equations (2022)
  7. Deep Neural Network Framework Based on Backward Stochastic Differential Equations for Pricing and Hedging American Options in High Dimensions (2018)
  8. Option pricing using deep learning approach based on LSTM-GRU neural networks: Case of London stock exchange (2022)
  9. A neural network model for estimating option prices (1993) — link
  10. Proof that properly anticipated prices fluctuate randomly (2016)
  11. The pricing of options and corporate liabilities (1973)
  12. Theory of rational option pricing (1971)
  13. Market and industry factors in stock price behavior (1966)
  14. A non-random walk down Wall Street (2011)
  15. The valuation of American put options (1977)
  16. Deep optimal stopping (2019)
  17. A nonparametric approach to pricing and hedging derivative securities via learning networks (1994)
  18. Pricing and hedging derivative securities with neural networks and a homogeneity hint (2000)
  19. Deep learning volatility: a deep neural network perspective on pricing and calibration in (rough) volatility models (2021)
  20. A neural network model for estimating option prices (1993)
  21. Option pricing model combining ensemble learning methods and network learning structure (2022)
  22. Neural networks for option pricing and hedging: a literature review (2019) — link

0 of 30 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.