Paper: arXiv 2510.27132

Authors: Wenxuan Zhang, Yixiao Guo, Benzhuo Lu

Abstract

This paper proposes the Exact Terminal Condition Neural Network (ETCNN), a deep learning framework for accurately pricing American options by solving the Black-Scholes-Merton (BSM) equations. The ETCNN incorporates carefully designed functions that ensure the numerical solution not only exactly satisfies the terminal condition of the BSM equations but also matches the non-smooth and singular behavior of the option price near expiration. This method effectively addresses the challenges posed by the inequality constraints in the BSM equations and can be easily extended to high-dimensional scenarios. Additionally, input normalization is employed to maintain the homogeneity. Multiple experiments are conducted to demonstrate that the proposed method achieves high accuracy and exhibits robustness across various situations, outperforming both traditional numerical methods and other machine learning approaches.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper demonstrates high mathematical complexity through its derivation of the Exact Terminal Condition Neural Network (ETCNN) and adaptation of the Black-Scholes-Merton equations, and shows strong empirical rigor by conducting extensive experiments comparing ETCNN to traditional and other ML methods, demonstrating high accuracy and robustness.

Research Flowchart

  flowchart TD
  A["Research Goal: Accurate American Option Pricing"] --> B{"Methodology: Exact Terminal Condition Neural Network"}
  B --> C["Input Data: BS-Merton Equation Parameters"]
  C --> D["Neural Network Architecture"]
  D --> E["Key Innovation: Terminal Constraint Function"]
  E --> F["Training with Homogeneity-Preserving Normalization"]
  F --> G["Output: Robust & Accurate Option Prices"]
  G --> H["Outcome: Outperforms Finite Differences & ML Baselines"]

Cited References (50)

  1. A Deep BSDE Approach for the Simultaneous Pricing and Delta-Gamma Hedging of Large Portfolios Consisting of High-Dimensional Multi-asset Bermudan Options — → 2502.11706
  2. Physics Informed Neural Network for Option Pricing (2023) — link
  3. A Deep Learning Method for Computing Eigenvalues of the Fractional Schrödinger Operator (2023) — link
  4. Meshless methods for American option pricing through Physics-Informed Neural Networks (2023) — link
  5. The deep parametric PDE method and applications to option pricing (2022) — link
  6. CLOSED FORM OPTIMAL EXERCISE BOUNDARY OF THE AMERICAN PUT OPTION (2019) — link
  7. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations (2019) — link
  8. A unified deep artificial neural network approach to partial differential equations in complex geometries (2017) — link
and 42 more
  1. DGM: A deep learning algorithm for solving partial differential equations (2017) — link
  2. Solving high-dimensional partial differential equations using deep learning (2017) — link
  3. Multi-asset Black–Scholes model as a variable second class constrained dynamical system (2016) — link
  4. Efficient estimation of lower and upper bounds for pricing higher-dimensional American arithmetic average options by approximating their payoff functions (2016) — link
  5. Deep Residual Learning for Image Recognition (2015) — link
  6. An efficient finite element method for pricing American multi-asset put options (2015) — link
  7. Theory of Rational Option Pricing (2015) — link
  8. The Early Exercise Premium Representation for American Options on Multiply Assets (2013) — link
  9. American-Style Derivatives: Valuation and Computation (2009)
  10. Adaptive lattice methods for multi-asset models (2008) — link
  11. Pricing Multi-Asset American Options: A Finite Element Method-of-Lines with Smooth Penalty (2007) — link
  12. Numerical computation of rectangular bivariate and trivariate normal and t probabilities (2004) — link
  13. The valuation of American call options on the minimum of two dividend-paying assets (2003) — link
  14. Monte Carlo valuation of American options (2002) — link
  15. Valuing American Options by Simulation: A Simple Least-Squares Approach (2001) — link
  16. The Valuation of American Options on Multiple Assets (1997) — link
  17. Pricing American-style securities using simulation (1997) — link
  18. Artificial neural networks for solving ordinary and partial differential equations (1997) — link
  19. ALTERNATIVE CHARACTERIZATIONS OF AMERICAN PUT OPTIONS (1992) — link
  20. Path Dependent Options: The Case of Lookback Options (1991) — link
  21. A Log-Transformed Binomial Numerical Analysis Method for Valuing Complex Multi-Option Investments (1991) — link
  22. The Analytic Valuation of American Options (1990) — link
  23. Numerical Evaluation of Multivariate Contingent Claims (1989) — link
  24. A Simple Formula to Compute the Implied Standard Deviation (1988) — link
  25. A Lattice Framework for Option Pricing with Two State Variables (1988) — link
  26. Options on the Maximum or the Minimum of Several Assets (1987) — link
  27. Efficient Analytic Approximation of American Option Values (1987) — link
  28. The Pricing of Futures and Options Contracts on the Value Line Index (1986) — link
  29. Options on the minimum or the maximum of two risky assets : Analysis and applications (1982) — link
  30. Martingales and stochastic integrals in the theory of continuous trading (1981) — link
  31. Two-State Option Pricing (1979) — link
  32. Option pricing: A simplified approach☆ (1979) — link
  33. The Value of an Option to Exchange One Asset for Another (1978) — link
  34. The Valuation of American Put Options (1977) — link
  35. The Pricing of Options and Corporate Liabilities (1973) — link
  36. On optimal stopping and free boundary problems (1973) — link
  37. The Binomial, Multinomial Distributions, and Option Pricing Model (2019) — link
  38. A method for stochastic optimization (2014)
  39. Pricing multi-asset American-style options by memory reduction Monte Carlo methods (2006)
  40. Stochastic calculus for finance II: Continuous-time models (2004)
  41. Generalized nonstandard finite differences and physical applications (1998) — link
  42. Computation of the bivariate normal integral (1978) — link

1 of 50 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.