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Option pricing model under the G-expectation framework

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.

Option pricing model under the G-expectation framework

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.
Paper Structure (19 sections, 12 theorems, 45 equations, 2 figures, 8 tables)

This paper contains 19 sections, 12 theorems, 45 equations, 2 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Risk-neutral option pricing under G-expectation
  3. G-Black-Scholes equation
  4. Logarithmic Transformation and Alternative Formulation
  5. Finite difference methods
  6. Discretization
  7. Explicit difference method
  8. Implicit difference method
  9. Numerical analysis for numerical methods
  10. Monotonicity and Convergence of explicit scheme explicit scheme
  11. Numerical analysis for Implicit method
  12. Convergence of inner iteration
  13. Monotonicity and Convergence of implicit scheme implicit scheme
  14. Numerical Examples
  15. Butterfly Spread
  16. ...and 4 more sections

Key Result

Lemma 4.1

(Consistency) The explicit scheme explicit1 is consistent.

Figures (2)

  • Figure 1: The number of iterations within each time step.
  • Figure 2: The number of iterations within each time step.

Theorems & Definitions (18)

  • Definition 2.1: Risk-neutral pricing under G-expectation
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 4.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Theorem 4.1
  • Remark 4.1
  • ...and 8 more