Hermite Finite Difference Through Kernel Approximations to Efficiently Solve Nonlinear Black-Scholes Model

We develop a high-order compact numerical scheme for solving a nonlinear Black–Scholes equation arising in option pricing under transaction costs. By leveraging a Hermite-enhanced Radial Basis Function-Finite Difference (RBF-HFD) method with three-point stencils, we achieve fourth-order spatial accu...

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Bibliographic Details
Published in:Mathematics
Main Authors: Shuai Wang, Jiameihui Zhu, Tao Liu
Format: Article
Language:English
Published: MDPI AG 2025-08-01
Subjects:
nonlinear Black–Scholes equation
Hermite interpolation
high-order
option pricing
transaction costs
Online Access:https://www.mdpi.com/2227-7390/13/17/2727
Description
Summary:We develop a high-order compact numerical scheme for solving a nonlinear Black–Scholes equation arising in option pricing under transaction costs. By leveraging a Hermite-enhanced Radial Basis Function-Finite Difference (RBF-HFD) method with three-point stencils, we achieve fourth-order spatial accuracy. The fully nonlinear PDE, driven by Gamma-dependent volatility models, is discretized via RBF-HFD in space and integrated using an explicit sixth-order Runge–Kutta scheme. Numerical results confirm the proposed method’s accuracy, stability, and its capability to capture sharp gradient behavior near strike prices.
ISSN:2227-7390

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