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.

翻译：G-期望作为一种次线性期望，为金融市场中的不确定性建模提供了强有力的框架。受模型不确定性下稳健估值的需求驱动，本文在G-期望环境中发展了一种统一的风险中性估值方法，得到了Black-Scholes模型的非线性推广，称为G-Black-Scholes方程。为提升计算效率并降低数值成本，我们引入了资产价格的对数变换，从而推导出一个替代的非线性偏微分方程。基于该变换形式，我们设计了显式和隐式有限差分格式，并严格证明其具有相容性、稳定性、单调性，且收敛于粘性解。数值算例证明，所提格式实现了高精度，而对数变换放宽了显式格式的稳定性约束，并提高了计算效率。