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

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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)

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.

Suggested Citation

  • Shuai Wang & Jiameihui Zhu & Tao Liu, 2025. "Hermite Finite Difference Through Kernel Approximations to Efficiently Solve Nonlinear Black-Scholes Model," Mathematics, MDPI, vol. 13(17), pages 1-18, August.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:17:p:2727-:d:1731986
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