Efficient numerical method for pricing multi-asset options with the time-fractional Black-Scholes model: focus on American and digital options

Imtiaz Ahmad; Muhammad Nawaz Khan; Rashid Jan; Normy Norfiza Abdul Razak; Imtiaz Ahmad; Muhammad Nawaz Khan; Rashid Jan; Normy Norfiza Abdul Razak

doi:10.3934/mmc.2025011

Mathematical Modelling and Control

2025, Volume 5, Issue 2: 147-163. doi: 10.3934/mmc.2025011

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Research article

Efficient numerical method for pricing multi-asset options with the time-fractional Black-Scholes model: focus on American and digital options

1.
Institute of Informatics and Computing in Energy (ⅡCE), Universiti Tenaga Nasional, Kajang, Selangor, Malaysia
2.
Department of Mathematics, Saveetha School of Engineering (SIMATS), Thandalam 600124, Chennai, Tamil Nadu, India
3.
Mathematics in Applied Sciences and Engineering Research Group, Scientific Research Center, Al-Ayen University, Nasiriyah 64001, Iraq
4.
Department of Mathematics, Khazar University AZ1096, Baku, Azerbaijan
5.
Institute of Energy Infrastructure (IEI), Department of Civil Engineering, College of Engineering, Universiti Tenaga Nasional (UNITEN), Putrajaya Campus, Jalan IKRAM-UNITEN, 43000 Kajang, Selangor, Malaysia

Received: 18 September 2024 Revised: 23 November 2024 Accepted: 09 December 2024 Published: 21 May 2025

This study presents a numerical solution for the two-asset time-fractional Black-Scholes model, which governs American and digital options, using a local meshless collocation method based on Gaussian radial basis functions. The proposed meshless approach effectively discretized the spatial derivatives of the model, while the Caputo derivative was employed to represent the time-fractional component, capturing the memory effects and non-local properties characteristic of fractional-order models. Numerical assessments were conducted to evaluate the method's performance across these option models. The study discusses the handling of interest rates, highlighting the method's capability to manage the complexities inherent in multi-asset options. The efficacy and accuracy of the proposed meshless approach were evaluated using the $ L_{\infty} $ error norms. In the absence of exact solutions for these option models, the double mesh technique was utilized to validate the accuracy and efficiency of the proposed method, ensuring the robustness and reliability of the numerical results.
- meshless method,
- American and digital options,
- Caputo derivative,
- gaussian radial function,
- numerical analysis
Citation: Imtiaz Ahmad, Muhammad Nawaz Khan, Rashid Jan, Normy Norfiza Abdul Razak. Efficient numerical method for pricing multi-asset options with the time-fractional Black-Scholes model: focus on American and digital options[J]. Mathematical Modelling and Control, 2025, 5(2): 147-163. doi: 10.3934/mmc.2025011

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Abstract

This study presents a numerical solution for the two-asset time-fractional Black-Scholes model, which governs American and digital options, using a local meshless collocation method based on Gaussian radial basis functions. The proposed meshless approach effectively discretized the spatial derivatives of the model, while the Caputo derivative was employed to represent the time-fractional component, capturing the memory effects and non-local properties characteristic of fractional-order models. Numerical assessments were conducted to evaluate the method's performance across these option models. The study discusses the handling of interest rates, highlighting the method's capability to manage the complexities inherent in multi-asset options. The efficacy and accuracy of the proposed meshless approach were evaluated using the $ L_{\infty} $ error norms. In the absence of exact solutions for these option models, the double mesh technique was utilized to validate the accuracy and efficiency of the proposed method, ensuring the robustness and reliability of the numerical results.

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