Paper: arXiv 2605.06688

Authors: Rohan, Siddanth Shetty, Amit N. Kumar

Abstract

In American options, the early exercise feature allows the option to be exercised at any time prior to expiration. However, this flexibility introduces a challenge: the pricing model must value the option while simultaneously determining an unknown, time-varying exercise boundary. The Heston model is one of the most popular ways to model real market behavior because it allows volatility to change over time. However, unlike European options, there is no closed-form solution for American options under the Heston model, so we have to use numerical methods. In this paper, we propose a novel approach to solving the stochastic Heston partial differential equation for American options, using coupled physics-informed neural networks (PINNs) to predict both the option price and the free boundary, while employing curriculum learning and adaptive resampling to stabilize model training. Our work builds on recent deep learning methods but introduces a more effective training strategy to address the limitations of these approaches. The numerical results demonstrate the effectiveness of the proposed learning framework, providing a robust and efficient alternative to pricing American options, enabling rapid inference and accurate estimation under stochastic volatility.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper focuses on advancing numerical PDE solving techniques using sophisticated Physics-Informed Neural Networks and curriculum learning, which involves high-level mathematical modeling of stochastic volatility. However, it lacks empirical validation against real-world market datasets or a production-ready backtesting framework, remaining confined to numerical simulations.

Research Flowchart

  flowchart TD
    A[<b>Goal:</b> Price American Options under Heston Model] --> B[<b>Framework:</b> Coupled PINNs]
    B --> C[<b>Inputs:</b> Stochastic Heston PDE & Boundary Conditions]
    C --> D[<b>Methodology:</b> Curriculum Learning & Adaptive Resampling]
    D --> E[<b>Process:</b> Simultaneous Prediction of Price & Free Boundary]
    E --> F[<b>Outcomes:</b> Rapid Inference & Robust Pricing Accuracy]

Cited References (45)

  1. Option pricing when underlying stock returns are discontinuous (1976) — link
  2. Jumps in Financial Markets: A New Nonparametric Test and Jump Dynamics (2007) — link
  3. Predicting the Price of Bitcoin Using Machine Learning (2018) — link
  4. Bitcoin price prediction using machine learning: An approach to sample dimension engineering (2020) — link
  5. Bitcoin price prediction using Deep Learning Algorithm (2019) — link
  6. Bitcoin Price Forecasting via Ensemble-based LSTM Deep Learning Networks (2021) — link
  7. XGBoost: A Scalable Tree Boosting System (2016)
  8. Greedy function approximation: A gradient boosting machine. (2001) — link
and 37 more
  1. LightGBM: a highly efficient gradient boosting decision tree (2017)
  2. Metaverse: Cryptocurrency Price Analysis Using Monte Carlo Simulation (2023)
  3. Predicting the direction of stock market prices using tree-based classifiers (2019)
  4. Gold price prediction using ensemble based machine learning techniques (2019)
  5. A CNN–LSTM model for gold price time-series forecasting (2020)
  6. Predicting cryptocurrency price bubbles using social media data and epidemic modelling (2017)
  7. Deep learning approach to determine the impact of socio economic factors on bitcoin price prediction (2019)
  8. Is Bitcoin a hedge, a safe haven or a diversifier for oil price movements? A comparison with gold (2018)
  9. A predictive model for digital currencies prices using geometric brownian motion stochastic differential equation: A case study of the bitcoin (2024)
  10. Lead-lag relationship between bitcoin and ethereum: Evidence from hourly and daily data (2019)
  11. Cryptocurrency Price Prediction Using News and Social Media Sentiment (2017)
  12. Machine Learning For Option Pricing: An Empirical Investigation of Network Architectures (2017)
  13. A Hybrid Deep Learning Approach for Crude Oil Price Prediction (2023)
  14. Long and short-term impacts of regulation in the cryptocurrency market (2021)
  15. Volatility estimation for Bitcoin: A comparison of GARCH models (2017)
  16. Bitcoin price prediction using ARIMA model (2020)
  17. High-frequency jump analysis of the bitcoin market (2020)
  18. An empirical investigation of volatility dynamics in the cryptocurrency market (2019) — link
  19. A novel cryptocurrency price trend forecasting model based on LightGBM (2020) — link
  20. A Jump-Diffusion Model for Option Pricing (2002)
  21. Long-run relationship and mean reversion in Bitcoin prices: A cointegration approach (2023)
  22. Volatility and market efficiency of Bitcoin and other cryptocurrencies (2022)
  23. Bitcoin: Medium of exchange or speculative asset? (2023)
  24. Forecasting Bitcoin closing price series using linear regression and neural networks models (2020)
  25. The Python programming language (1991–) — link
  26. Probabilistic deep learning and transfer learning for robust cryptocurrency price prediction (2024)
  27. Forecasting Bitcoin prices using artificial intelligence: Combination of ML, SARIMA, and Facebook Prophet models (2024) — link
  28. Enhancing Price Prediction in Cryptocurrency Using Transformer Neural Network and Technical Indicators — → 2403.03606
  29. LSTM–GARCH hybrid model for the prediction of volatility in cryptocurrency portfolios (2024)
  30. Comparative analysis of machine learning algorithms for daily cryptocurrency price prediction (2024)
  31. Attention-based CNN–LSTM for high-frequency multiple cryptocurrency trend prediction (2024)
  32. Predicting the cryptocurrency market using social media metrics and search trends during COVID-19 (2024)
  33. Profitability trend prediction in crypto financial markets using Fibonacci technical indicator and hybrid CNN model (2024)
  34. Forecasting cryptocurrencies volatility using statistical and machine learning methods: A comparative study (2024)
  35. The Pricing of Options and Corporate Liabilities (1973) — link
  36. A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options (1993) — link
  37. Brownian Motion and Stochastic Calculus (1991)

1 of 45 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.