Paper: arXiv 2603.22831

Authors: Ziting Pei, Xingye Yue, Xiaotao Zheng

Abstract

G-expectation, as a sublinear expectation, provides a powerful framework for modeling uncertainty in financial markets. Motivated by the need for robust valuation under model uncertainty, this work develops a unified risk-neutral valuation approach within the G-expectation environment, yielding a nonlinear generalization of the Black-Scholes model, termed the G-Black-Scholes equation. To enhance computational efficiency and reduce numerical cost, we introduce a logarithmic transformation of the asset price, which yields an alternative nonlinear PDE. Based on this transformed formulation, we design both explicit and implicit finite difference schemes that are rigorously demonstrated to be consistent, stable, monotone, and convergent to the viscosity solution. Numerical examples confirm that the proposed schemes achieve high accuracy, while the logarithmic transformation relaxes the stability constraints of explicit schemes and improves computational efficiency.

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 presents a strong theoretical framework using sublinear expectations (G-expectation) to generalize Black-Scholes and proves convergence to viscosity solutions via finite difference schemes. However, it lacks empirical rigor, relying primarily on stylized numerical examples rather than real-world market data backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Option Pricing under Model Uncertainty<br><i>(G-Expectation Framework)</i>"] --> B["Mathematical Formulation:<br>Derive Nonlinear G-Black-Scholes PDE"]
    
    B --> C["Transformation & Inputs:<br>Apply Log Price Transformation<br>& Set Market Parameters"]
    
    C --> D["Numerical Scheme Design:<br>Construct Explicit & Implicit<br>Finite Difference Schemes"]
    
    D --> E["Theoretical Verification:<br>Prove Consistency, Stability,<br>Monotonicity & Convergence"]
    
    E --> F["Outcomes & Key Findings:<br>Highly Accurate Valuation &<br>Improved Efficiency/Relaxed Stability"]