A Deep Learning-Driven Operator Splitting Scheme for Free Boundary Problems: An Application to American Option Pricing in Incomplete Markets

Paper: arXiv 2610.36921

Authors: Dong Yan, Haocheng Zhu, Yao Li

Abstract

This paper proposes a deep learning-driven operator splitting scheme for high-dimensional free-boundary problems, with American option pricing as a representative application. In a market with proportional transaction costs, a stochastic liquidity risk with an empirically validated nonlinear drift based on S&P 500 data, and two-factor stochastic volatility, the pricing problem takes the form of a five-dimensional highly nonlinear variational inequality. The proposed scheme decomposes this variational inequality at each time layer into an elliptic continuation equation approximated by a physics-informed neural network and a closed-form projection step enforcing the early-exercise constraint. A conditional $\ell^\infty$ stability bound is derived under stated resolvent and Lipschitz assumptions. The scheme is validated against finite-difference and least-squares Monte Carlo benchmarks, and both the option prices and the learned exercise boundary show close agreement with the reference solutions. Sensitivity analyses with respect to the key model parameters are also reported.

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 presents a highly complex mathematical framework involving variational inequalities, operator splitting, and PINNs for a five-dimensional problem. It demonstrates strong empirical rigor through validation against benchmarks and an empirically validated nonlinear drift based on S&P 500 data. The deep learning-driven operator splitting for free boundary problems in this complex market setting is a novel contribution.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve High-Dimensional Free Boundary Problems, e.g., American Option Pricing] --> B{Key Methodology: Deep Learning-Driven Operator Splitting Scheme};
    B --> C[Method Steps: Decompose Variational Inequality into Elliptic Continuation (PINN) & Projection Step];
    C --> D[Inputs: 5D Variational Inequality (Proportional Transaction Costs, Stochastic Liquidity Risk, Two-Factor Stochastic Volatility)];
    D --> E[Computational Processes: PINN for Elliptic Eq., Closed-Form Projection, Conditional L-infinity Stability Analysis];
    E --> F[Outcomes: Validated Option Prices & Learned Exercise Boundary (Close agreement with benchmarks), Sensitivity Analysis];