Instabilities of explicit finite difference schemes with ghost points on the diffusion equation

ArXiv ID: 2308.04629 “View on arXiv”

Authors: Unknown

Abstract

Ghost, or fictitious points allow to capture boundary conditions that are not located on the finite difference grid discretization. We explore in this paper the impact of ghost points on the stability of the explicit Euler finite difference scheme in the context of the diffusion equation. In particular, we consider the case of a one-touch option under the Black-Scholes model. The observations and results are however valid for a much wider range of financial contracts and models.

Keywords: Ghost Points, Finite Difference Method, Explicit Euler Scheme, Black-Scholes Model, One-Touch Option, Derivatives

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper contains dense mathematical derivations for stability analysis of finite difference schemes, but the empirical validation is limited to a single specific numerical example with theoretical comparisons rather than a comprehensive backtesting framework.

  flowchart TD
    A["Research Goal<br>Impact of Ghost Points<br>on Explicit Scheme Stability"] --> B["Methodology"]
    B --> C["Model Selection<br>Black-Scholes Diffusion Eq"]
    C --> D["Implementation<br>Explicit Euler FD<br>with Ghost Points"]
    D --> E["Computational Process<br>Stability Analysis &<br>One-Touch Option Pricing"]
    E --> F["Key Findings<br>Stability Conditions &<br>Validation on Financial Contracts"]