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A Finite Difference Method for a Normalized Time-Fractional Black–Scholes Equation

International Journal of Applied and Computational Mathematics 2025年第3期11卷 1-17页

作者： Nam, Yunjae Wang, Jian Lee, Chaeyoung Choi, Yongho Bang, Minjoon Li, Zhengang Kim, Junseok Program in Actuarial Science and Financial Engineering Korea University Seoul 02841 South Korea School of Mathematics and Statistics Nanjing University of Information Science and Technology Nanjing 210044 China Department of Mathematics Kyonggi University Suwon 16227 South Korea Computer and Information Engineering Daegu University Gyeongsan 38453 South Korea Department of Mathematics Korea University Seoul 02841 South Korea

In this article, we propose a normalized time-fractional Black–Scholes (TFBS) equation. The proposed model uses a normalized time-fractional derivative which has a distinctive feature wherein a weight function possesses the property that its integral with respect to time is always equal to one. This feature ensures a well-balanced integration over time and provides a fair comparison between different fractional orders. An implicit finite difference method is used for the numerical solution of the TFBS equation. We perform several standard numerical tests to examine the effect of the fractional parameter on the option pricing. These experiments are designed to investigate how variations in the fractional parameter influence pricing outcomes and provide valuable insights into the model’s behavior under different conditions. The computational results demonstrate the model’s potential for capturing more intricate market dynamics, which makes the proposed model a promising tool for financial analysis. In the appendix, we provide a computer program that implements the numerical methods discussed in the paper for interested readers. © The Author(s), under exclusive licence to Springer Nature India Private Limited 2025.

关键词： Implicit finite difference method Memory effect Normalized time-fractional Black–Scholes equation

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A review of the numerical methods for solving the binary Allen–Cahn equation

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Physica A: Statistical Mechanics and its Applications 2025年 670卷

作者： Lee, Hyun Geun Li, Yibao Yang, Junxiang Kwak, Soobin Hwang, Youngjin Ham, Seokjun Kim, Hyundong Jyoti Nam, Yunjae Kim, Junseok Department of Mathematics Dongguk University Seoul04620 Korea Republic of School of Mathematics and Statistics Xi'an Jiaotong University Xi'an710049 China School of Computer Science and Engineering Faculty of Innovation Engineering Macau University of Science and Technology China Department of Mathematics Korea University Seoul02841 Korea Republic of Department of Mathematics and Physics Gangneung-Wonju National University Gangneung25457 Korea Republic of Smart infrastructure Gangneung-Wonju National University Gangneung25457 Korea Republic of The Institute of Basic Science Korea University Seoul02841 Korea Republic of Program in Actuarial Science and Financial Engineering Korea University Seoul02841 Korea Republic of

In this review, we present an overview of numerical methods to solve the binary Allen–Cahn (AC) equation, which is extensively used to model phase separation processes in materials science. It describes the time-dependent evolution of interfaces between two phases and accounts for both local reaction kinetics and diffusion effects. This equation plays a critical role in understanding the behavior of interfaces. The AC equation has various applications across fields such as materials science, physics, and biology, where it helps to analyze and predict phenomena such as phase transitions, grain boundary motion, and pattern formation in complex systems. Its importance lies in its ability to model the dynamics of interfaces and help the study of pattern formation and phase transitions in diverse environments. We discuss various computational methodologies developed for this important mathematical model and describe their strengths, limitations, and applications in diverse scientific domains. © 2025 Elsevier B.V.

关键词： Rate constants

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