Paper: arXiv 2412.12213

Abstract

In the realm of option pricing, existing models are typically classified into principle-driven methods, such as solving partial differential equations (PDEs) that pricing function satisfies, and data-driven approaches, such as machine learning (ML) techniques that parameterize the pricing function directly. While principle-driven models offer a rigorous theoretical framework, they often rely on unrealistic assumptions, such as asset processes adhering to fixed stochastic differential equations (SDEs). Moreover, they can become computationally intensive, particularly in high-dimensional settings when analytical solutions are not available and thus numerical solutions are needed. In contrast, data-driven models excel in capturing market data trends, but they often lack alignment with core financial principles, raising concerns about interpretability and predictive accuracy, especially when dealing with limited or biased datasets. This work proposes a hybrid approach to address these limitations by integrating the strengths of both principled and data-driven methodologies. Our framework combines the theoretical rigor and interpretability of PDE-based models with the adaptability of machine learning techniques, yielding a more versatile methodology for pricing a broad spectrum of options. We validate our approach across different volatility modeling approaches-both with constant volatility (Black-Scholes) and stochastic volatility (Heston), demonstrating that our proposed framework, Finance-Informed Neural Network (FINN), not only enhances predictive accuracy but also maintains adherence to core financial principles. FINN presents a promising tool for practitioners, offering robust performance across a variety of market conditions.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper heavily employs advanced mathematical concepts including stochastic differential equations, partial differential equations, and Itô’s lemma, but provides only theoretical validation across volatility models without presenting backtests, specific datasets, or implementation details for live trading readiness.

Research Flowchart

  flowchart TD
  A["Research Goal: Hybrid Model for Option Pricing<br>Integrate Principle-Driven & Data-Driven Approaches"] --> B["Methodology: Finance-Informed Neural Network<br>FINN Architecture"]
  B --> C["Data & Inputs: Option Pricing Data<br>Black-Scholes & Heston Models"]
  C --> D["Computational Process: Train FINN with PDE Constraints<br>Simulate Market Conditions"]
  D --> E["Outcome 1: Enhanced Predictive Accuracy<br>Superior to Pure ML Models"]
  D --> F["Outcome 2: Financial Principle Adherence<br>Maintains Theoretical Rigor"]
  D --> G["Outcome 3: Robust Performance<br>Effective across Market Conditions"]

Cited References (66)

  1. Real-time segmentation of on-line handwritten arabic script (2014)
  2. Fast classification of handwritten on-line Arabic characters (2014) — link
  3. Estimate and Replace: A Novel Approach to Integrating Deep Neural Networks with Existing Applications (2018) — link
  4. The Pricing of Options and Corporate Liabilities (1973) — link
  5. A Nonparametric Approach to Pricing and Hedging Derivative Securities via Learning Networks (1994) — link
  6. Machine learning in finance: the case of deep learning for option pricing (2017)
  7. Research on European Stock Index Option Pricing Based on Deep Learning Algorithms–Evidence from 50ETF Option Market (2018)
  8. A study of option price prediction based on LSTM neural network (2021)
and 58 more
  1. Physics-informed machine learning (2021)
  2. The Kolmogorov forward fractional partial differential equation for the CGMY-process with applications in option pricing (2018)
  3. Quantum-Inspired Tensor Neural Networks for Partial Differential Equations (2022)
  4. Quantum-Inspired Tensor Neural Networks for Option Pricing (2022)
  5. Application of Tensor Neural Networks to Pricing Bermudan Swaptions — → 2304.09750
  6. Stochastic Calculus for Finance II: Continuous Time Models (2004)
  7. Optimal stopping and free-boundary problems (2006) — link
  8. Options, Futures, and Other Derivatives (2018)
  9. A reinforcement learning approach for pricing derivatives (2010)
  10. Reinforcement Learning Applied to Option Pricing (2014)
  11. Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations (2017)
  12. Solving stochastic differential equations and Kolmogorov equations by means of deep learning (2018) — link
  13. Solving high-dimensional partial differential equations using deep learning (2018)
  14. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations (2019)
  15. The Arbitrage Principle in Financial Economics (1987)
  16. Review of \textit Numerical Solution of Stochastic Differential Equations by Kloeden, P.~E. and Platen, E. (1994)
  17. A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options (1993)
  18. A Theory of the Term Structure of Interest Rates (1985)
  19. The little Heston trap (2007)
  20. QLBS: Q-Learner in the Black-Scholes(-Merton) Worlds (2019) — link
  21. A neural network versus Black-Scholes: A comparison of pricing and hedging performances (2003)
  22. Option pricing with modular neural networks (2009)
  23. Prediction of pricing and hedging errors for equity linked warrants with Gaussian process models (2008)
  24. Predicting a distribution of implied volatilities for option pricing (2011)
  25. Forecasting nonnegative option price distributions using Bayesian kernel methods (2012)
  26. Improving the pricing of options: A neural network approach (1998)
  27. Pricing and hedging derivative securities with neural networks and a homogeneity hint (2000)
  28. Option price forecasting using neural networks (2000)
  29. Pricing Options and Computing Implied Volatilities using Neural Networks (2019)
  30. Parametric models and non-parametric machine learning models for predicting option prices: Empirical comparison study over KOSPI 200 Index options (2014)
  31. Option pricing using Machine Learning (2021)
  32. Deep reinforcement learning for option pricing and hedging under dynamic expectile risk measures (2023)
  33. Principles of Mathematical Analysis (1976)
  34. Stochastic Differential Equations and Applications (1975)
  35. A general version of the fundamental theorem of asset pricing (1994)
  36. Martingales and stochastic integrals in the theory of continuous trading (1981)
  37. Methods of Mathematical Finance (1998)
  38. Deep Learning (2016)
  39. Option pricing in the Heston model with physics inspired neural networks (2024) — link
  40. Neural network learning of Black-Scholes equation for option pricing — → 2405.05780
  41. A Deep Learning Based Numerical PDE Method for Option Pricing (2023) — link
  42. The application of improved physics-informed neural network (IPINN) method in finance (2022) — link
  43. Meshless methods for American option pricing through Physics-Informed Neural Networks (2023) — link
  44. Approximation Capabilities of Multilayer Feedforward Networks (1991) — link
  45. Approximation Theory of the MLP Model in Neural Networks (1999)
  46. Error Bounds for Approximations with Deep ReLU Networks (2017) — link
  47. Kolmogorov–Arnold Finance-Informed Neural Network in Option Pricing (2024) — link
  48. KANOP: A Data-Efficient Option Pricing Model using Kolmogorov–Arnold Networks (2024) — link
  49. Physics Informed Neural Network for Option Pricing (2023) — link
  50. A general version of the fundamental theorem of asset pricing (1994)
  51. Stochastic Differential Equations and Applications (1975)
  52. Martingales and stochastic integrals in the theory of continuous trading (1981)
  53. Sobolev Spaces (2003)
  54. Functional Analysis, Sobolev Spaces and Partial Differential Equations (2011)
  55. Linear and Quasi-linear Equations of Parabolic Type (1968)
  56. Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks (1990)
  57. Approximation theory of the MLP model in neural networks (1999)
  58. An uncertainty-aware physics-informed neural network solution for the Black-Scholes equation: a novel framework for option pricing (2025) — link

2 of 66 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.