Option pricing model under the G-expectation framework

TL;DR

This paper introduces a nonlinear G-Black-Scholes model for option pricing under model uncertainty using G-expectation, along with efficient finite difference schemes validated by numerical experiments.

Contribution

It develops a unified risk-neutral valuation framework under G-expectation and proposes a logarithmic transformation to improve computational efficiency.

Findings

The schemes are consistent, stable, monotone, and converge to the viscosity solution.

Numerical results show high accuracy and improved efficiency with the logarithmic transformation.

The G-Black-Scholes model generalizes classical models to account for uncertainty.

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.