Nonuniform Finite Difference Scheme for the Three-Dimensional Time-Fractional Black–Scholes Equation

Kim, Sangkwon; Lee, Chaeyoung; Lee, Won Jin; Kwak, Soobin; Jeong, Darae; Kim, Junseok

doi:10.1155/2021/9984473

Cited by 2 publications

(3 citation statements)

References 31 publications

(33 reference statements)

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“…Consequently, they developed an adaptive moving mesh method to handle potential singularities. Later, Kim et al [119] extended the discrete technique presented in [117] to the three-dimensional (3D) version of (48), and solved the resulting fully discretized equation using an operator splitting method.…”

Section: The Finite Difference Methodsmentioning

confidence: 99%

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Review of the Fractional Black-Scholes Equations and Their Solution Techniques

Zhang,

Liu

et al. 2024

Fractal Fract

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The pioneering work in finance by Black, Scholes and Merton during the 1970s led to the emergence of the Black-Scholes (B-S) equation, which offers a concise and transparent formula for determining the theoretical price of an option. The establishment of the B-S equation, however, relies on a set of rigorous assumptions that give rise to several limitations. The non-local property of the fractional derivative (FD) and the identification of fractal characteristics in financial markets have paved the way for the introduction and rapid development of fractional calculus in finance. In comparison to the classical B-S equation, the fractional B-S equations (FBSEs) offer a more flexible representation of market behavior by incorporating long-range dependence, heavy-tailed and leptokurtic distributions, as well as multifractality. This enables better modeling of extreme events and complex market phenomena, The fractional B-S equations can more accurately depict the price fluctuations in actual financial markets, thereby providing a more reliable basis for derivative pricing and risk management. This paper aims to offer a comprehensive review of various FBSEs for pricing European options, including associated solution techniques. It contributes to a deeper understanding of financial model development and its practical implications, thereby assisting researchers in making informed decisions about the most suitable approach for their needs.

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Section: The Finite Difference Methodsmentioning

confidence: 99%

“…Several researchers extended Equation ( 49) to a more general case, such as incorporating interest rate and volatility as functional variables [50,117,118], and considering a broader range of assets [119].…”

Section: Simple Time Fractional B-s Equationsmentioning

confidence: 99%

Review of the Fractional Black-Scholes Equations and Their Solution Techniques

Zhang,

Liu

et al. 2024

Fractal Fract

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“…For example, Jeong et al [14] proposed a finite difference method for solving the BS equation without boundary conditions. Kim et al [17] used a nonuniform finite difference method for threedimensional (3D) time-fractional BS equations. He and Zhang [11] proposed the Fractional Black-Scholes Model (FBSM) of option pricing in a fractal transmission system.…”

Section: Introductionmentioning

confidence: 99%

Approximate Solution for Barrier Option Pricing Using Adaptive Differential Evolution With Learning Parameter

Febrianti

Sidarto

Sumarti

2022

mendel

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Black-Scholes (BS) equations, which are in the form of stochastic partial differential equations, are fundamental equations in mathematical finance, especially in option pricing. Even though there exists an analytical solution to the standard form, the equations are not straightforward to be solved numerically. The effective and efficient numerical method will be useful to solve advanced and non-standard forms of BS equations in the future. In this paper, we propose a method to solve BS equations using an approach of optimization problems, where a metaheuristic optimization algorithm is utilized to find the best-approximated solutions of the equations. Here we use the Adaptive Differential Evolution with Learning Parameter (ADELP) algorithm. The BS equations being solved are meant to find values of European option pricing that is equipped with Barrier option pricing. The result of our approximation method fits well to the analytical approximation solutions.

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Nonuniform Finite Difference Scheme for the Three-Dimensional Time-Fractional Black–Scholes Equation

Cited by 2 publications

References 31 publications

Review of the Fractional Black-Scholes Equations and Their Solution Techniques

Review of the Fractional Black-Scholes Equations and Their Solution Techniques

Approximate Solution for Barrier Option Pricing Using Adaptive Differential Evolution With Learning Parameter

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