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Jump Diffusion-Informed Neural Networks with Transfer Learning for Accurate American Option Pricing under Data Scarcity

Qiguo Sun, Hanyue Huang, XiBei Yang, Yuwei Zhang

TL;DR

This study presents a comprehensive framework for American option pricing consisting of six interrelated modules, which combine nonlinear optimization algorithms, analytical and numerical models, and neural networks to improve pricing performance.

Abstract

Option pricing models, essential in financial mathematics and risk management, have been extensively studied and recently advanced by AI methodologies. However, American option pricing remains challenging due to the complexity of determining optimal exercise times and modeling non-linear payoffs resulting from stochastic paths. Moreover, the prevalent use of the Black-Scholes formula in hybrid models fails to accurately capture the discontinuity in the price process, limiting model performance, especially under scarce data conditions. To address these issues, this study presents a comprehensive framework for American option pricing consisting of six interrelated modules, which combine nonlinear optimization algorithms, analytical and numerical models, and neural networks to improve pricing performance. Additionally, to handle the scarce data challenge, this framework integrates the transfer learning through numerical data augmentation and a physically constrained, jump diffusion process-informed neural network to capture the leptokurtosis of the log return distribution. To increase training efficiency, a warm-up period using Bayesian optimization is designed to provide optimal data loss and physical loss coefficients. Experimental results of six case studies demonstrate the accuracy, convergence, physical effectiveness, and generalization of the framework. Moreover, the proposed model shows superior performance in pricing deep out-of-the-money options.

Jump Diffusion-Informed Neural Networks with Transfer Learning for Accurate American Option Pricing under Data Scarcity

TL;DR

This study presents a comprehensive framework for American option pricing consisting of six interrelated modules, which combine nonlinear optimization algorithms, analytical and numerical models, and neural networks to improve pricing performance.

Abstract

Option pricing models, essential in financial mathematics and risk management, have been extensively studied and recently advanced by AI methodologies. However, American option pricing remains challenging due to the complexity of determining optimal exercise times and modeling non-linear payoffs resulting from stochastic paths. Moreover, the prevalent use of the Black-Scholes formula in hybrid models fails to accurately capture the discontinuity in the price process, limiting model performance, especially under scarce data conditions. To address these issues, this study presents a comprehensive framework for American option pricing consisting of six interrelated modules, which combine nonlinear optimization algorithms, analytical and numerical models, and neural networks to improve pricing performance. Additionally, to handle the scarce data challenge, this framework integrates the transfer learning through numerical data augmentation and a physically constrained, jump diffusion process-informed neural network to capture the leptokurtosis of the log return distribution. To increase training efficiency, a warm-up period using Bayesian optimization is designed to provide optimal data loss and physical loss coefficients. Experimental results of six case studies demonstrate the accuracy, convergence, physical effectiveness, and generalization of the framework. Moreover, the proposed model shows superior performance in pricing deep out-of-the-money options.
Paper Structure (18 sections, 39 equations, 10 figures, 1 table, 2 algorithms)

This paper contains 18 sections, 39 equations, 10 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Option Pricing Under Parametric Models
  4. Option Pricing Based on Machine Learning
  5. Methodology
  6. System Framework
  7. I.Parameter Estimation Module
  8. II.American Option Pricing Module
  9. III.Analytical Merton Model
  10. IV.Loss coefficients Optimization Module
  11. V. Numerical PIDE Solver Module
  12. VI. Training Module
  13. Experiments and Analysis
  14. Dataset and Baselines
  15. Framework Performance
  16. ...and 3 more sections

Figures (10)

  • Figure 1: (a) Close price of S&P 500 index with large jumps ( $|log(dS)|> 0.03$) highlighted in red; and (b) distribution of log returns and fitted Normal in black.
  • Figure 2: Framework of PINN-Merton model
  • Figure 3: Eight stochastic price paths generated by the Merton jump diffusion model with optimized parameters
  • Figure 4: Comparison of the Log return distributions of real SPY close price and the eight stochastic paths generated by the Merton jump diffusion model with optimized parameters
  • Figure 5: Contour plot of validation losses under different physics loss $\alpha$ and data loss $\beta$ for (a) PINN-Merton for call options pricing; (b) PINN-Merton for put options pricing; (c) PINN-BS for call options pricing; (d) PINN-BS for put options pricing. The number of trials for Bayesian optimization and number of epochs in each trial are set to 50 and 50, respectively.
  • ...and 5 more figures