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

Shuai Wang, Jiameihui Zhu and Tao Liu ()
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Shuai Wang: Foundation Department, Changchun Guanghua University, Changchun 130033, China
Jiameihui Zhu: School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China
Tao Liu: School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China

Mathematics, 2025, vol. 13, issue 17, 1-18

Abstract: 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.

Keywords: nonlinear Black–Scholes equation; Hermite interpolation; high-order; option pricing; transaction costs (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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