Formulation and analysis of an implicit non-standard finite difference scheme for the Black-Scholes option pricing model

Deepak Singh; Vikas Gupta; Mohammad Sajid; Deepak Singh; Vikas Gupta; Mohammad Sajid

doi:10.3934/math.2026457

AIMS Mathematics

2026, Volume 11, Issue 4: 11099-11115. doi: 10.3934/math.2026457

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Formulation and analysis of an implicit non-standard finite difference scheme for the Black-Scholes option pricing model

1.
Centre for Mathematial & Financial Modeling, Department of Mathematics, The LNM Institute of Information Technology, Jaipur-302031, India
2.
Department of Mechanical Engineering, College of Engineering, Qassim University, Buraydah, Saudi Arabia

Received: 10 February 2026 Revised: 03 April 2026 Accepted: 14 April 2026 Published: 21 April 2026
MSC : 65M06, 65M12

This paper introduces a novel numerical scheme for estimating option prices under the Black-Scholes (B-S) model. The proposed method utilizes a non-standard finite difference (NSFD) approach that incorporates the powerful techniques of methods of sub-equation and exact finite difference (EFD). The proposed technique exhibits several positive characteristics: It preserves positivity by design, works with large step sizes, ensures dynamic consistency, and enhances stability. Notably, its implicit scheme and construction ensures that the fundamental properties of the solution are accurately captured. Finally, some numerical simulations are provided to demonstrate the effectiveness of the proposed implicit NSFD scheme.
- exact finite difference scheme,
- sub-equation,
- non-standard finite difference scheme,
- stability,
- simulation,
- numerical results,
- convergence analysis
Citation: Deepak Singh, Vikas Gupta, Mohammad Sajid. Formulation and analysis of an implicit non-standard finite difference scheme for the Black-Scholes option pricing model[J]. AIMS Mathematics, 2026, 11(4): 11099-11115. doi: 10.3934/math.2026457

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Abstract

This paper introduces a novel numerical scheme for estimating option prices under the Black-Scholes (B-S) model. The proposed method utilizes a non-standard finite difference (NSFD) approach that incorporates the powerful techniques of methods of sub-equation and exact finite difference (EFD). The proposed technique exhibits several positive characteristics: It preserves positivity by design, works with large step sizes, ensures dynamic consistency, and enhances stability. Notably, its implicit scheme and construction ensures that the fundamental properties of the solution are accurately captured. Finally, some numerical simulations are provided to demonstrate the effectiveness of the proposed implicit NSFD scheme.

References

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