An Efficient Numerical Model for the Black–Scholes Equations.
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| Title: | An Efficient Numerical Model for the Black–Scholes Equations. |
|---|---|
| Authors: | Zhou, Yan1 (AUTHOR) 202414161@sdtbu.edu.cn, Zhang, Yunxing2 (AUTHOR), Xu, Yufeng (AUTHOR) xuyufeng@csu.edu.cn |
| Source: | Journal of Applied Mathematics. 2/24/2026, Vol. 2026, p1-12. 12p. |
| Subjects: | Crank-Nicolson method, Discretization methods, Financial engineering, Transport equation, Finite difference method, Mathematical models, Iterative methods (Mathematics) |
| Abstract: | In this paper, a novel numerical model for the Black–Scholes equations is developed. To address some potential issues that may arise when solving this equation using the conventional model, the original Black–Scholes equation is reformulated as a convection–diffusion equation. The Crank–Nicolson scheme is utilized to discretize the diffusion and source terms in time, and the convection term is dealt explicitly using the Adams–Bashforth method. Spatial derivatives for the diffusion term are approximated with the central difference scheme. However, the spatial discretization of the convection term can be chosen adaptively according to the problem setting and the parameter values. By employing various discretization schemes for the convection term, a series of models are constructed for solving the Black–Scholes equations. We demonstrated that the discrete matrix with the new models has a better diagonal dominance property than that of the conventional model, which is beneficial for the solution. Subsequently, the new model is validated with several single‐asset and multiasset benchmark problems, and the numerical results are compared with analytic solutions. It is observed that the choice of the discretization scheme for the convection term has little impact on the computational efficiency but can be significant on the accuracy and robustness. We find that the new model shows clear advantages over conventional methods, particularly in stable market conditions. [ABSTRACT FROM AUTHOR] |
| Copyright of Journal of Applied Mathematics is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Header | DbId: egs DbLabel: Engineering Source An: 191893900 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: An Efficient Numerical Model for the Black–Scholes Equations. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Zhou%2C+Yan%22">Zhou, Yan</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> 202414161@sdtbu.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Zhang%2C+Yunxing%22">Zhang, Yunxing</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Xu%2C+Yufeng%22">Xu, Yufeng</searchLink> (AUTHOR)<i> xuyufeng@csu.edu.cn</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Applied+Mathematics%22">Journal of Applied Mathematics</searchLink>. 2/24/2026, Vol. 2026, p1-12. 12p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Crank-Nicolson+method%22">Crank-Nicolson method</searchLink><br /><searchLink fieldCode="DE" term="%22Discretization+methods%22">Discretization methods</searchLink><br /><searchLink fieldCode="DE" term="%22Financial+engineering%22">Financial engineering</searchLink><br /><searchLink fieldCode="DE" term="%22Transport+equation%22">Transport equation</searchLink><br /><searchLink fieldCode="DE" term="%22Finite+difference+method%22">Finite difference method</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+models%22">Mathematical models</searchLink><br /><searchLink fieldCode="DE" term="%22Iterative+methods+%28Mathematics%29%22">Iterative methods (Mathematics)</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: In this paper, a novel numerical model for the Black–Scholes equations is developed. To address some potential issues that may arise when solving this equation using the conventional model, the original Black–Scholes equation is reformulated as a convection–diffusion equation. The Crank–Nicolson scheme is utilized to discretize the diffusion and source terms in time, and the convection term is dealt explicitly using the Adams–Bashforth method. Spatial derivatives for the diffusion term are approximated with the central difference scheme. However, the spatial discretization of the convection term can be chosen adaptively according to the problem setting and the parameter values. By employing various discretization schemes for the convection term, a series of models are constructed for solving the Black–Scholes equations. We demonstrated that the discrete matrix with the new models has a better diagonal dominance property than that of the conventional model, which is beneficial for the solution. Subsequently, the new model is validated with several single‐asset and multiasset benchmark problems, and the numerical results are compared with analytic solutions. It is observed that the choice of the discretization scheme for the convection term has little impact on the computational efficiency but can be significant on the accuracy and robustness. We find that the new model shows clear advantages over conventional methods, particularly in stable market conditions. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Journal of Applied Mathematics is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1155/jama/9383545 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 12 StartPage: 1 Subjects: – SubjectFull: Crank-Nicolson method Type: general – SubjectFull: Discretization methods Type: general – SubjectFull: Financial engineering Type: general – SubjectFull: Transport equation Type: general – SubjectFull: Finite difference method Type: general – SubjectFull: Mathematical models Type: general – SubjectFull: Iterative methods (Mathematics) Type: general Titles: – TitleFull: An Efficient Numerical Model for the Black–Scholes Equations. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Zhou, Yan – PersonEntity: Name: NameFull: Zhang, Yunxing – PersonEntity: Name: NameFull: Xu, Yufeng IsPartOfRelationships: – BibEntity: Dates: – D: 24 M: 02 Text: 2/24/2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 1110757X Numbering: – Type: volume Value: 2026 Titles: – TitleFull: Journal of Applied Mathematics Type: main |
| ResultId | 1 |
