The Approximate Analytic Solution of the Time-Fractional Black-Scholes Equation with a European Option Based on the Katugampola Fractional Derivative

Ampun, Sivaporn; Sawangtong, Panumart

doi:10.3390/math9030214

Cited by 9 publications

(2 citation statements)

References 23 publications

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“…While considering the K-C derivative as time FD, based on the characteristics of the integral kernel in the definition of the K-C derivative, Fall et al [124] replaced Laplace transform with ρ− Laplace transform [125] and combined HPM to solve the Equation (49) with the usual initial condition. Furthermore, this technique has also been employed by Ampun and Sawangtong [126] to address the same equation subject to a varying initial condition, i.e., the Katugampola integral initial condition. Moreover, Thanompolkrang et al [127] extended the aforementioned method to a scenario involving two assets.…”

Section: The Hpmmentioning

confidence: 99%

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 Hpmmentioning

confidence: 99%

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

Zhang,

Liu

et al. 2024

Fractal Fract

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“…In the past decades, several techniques have been proposed to get the approximate analytic solution of N/ MEMS problems such as the homotopy perturbation method (HPM), higher-order HPM [8], Taylor series [9], energy balance technique [10], spreading residual harmonic balance method [11], higher-order Hamiltonian method [12], Adomian decomposition method (ADM) [13], Li-He modified HPM [14], modified ADM [15], variational approach [16], Galerkin decomposition method [17], and so on. It is also noted that, besides these methods, there are various analytical techniques for getting the approximate solution to the nonlinear equations, for example, the He-Laplace method [18], global residual harmonic balance method [19], integral transform-based methods [20][21][22], max-min approach [23], frequency-amplitude formulation method [24], Hamiltonian approach [25], and others [26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

An Efficient Analytical Approach for the Periodicity of Nano/Microelectromechanical Systems’ Oscillators

Anjum

Rahman

et al. 2022

Mathematical Problems in Engineering

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Periodic behavior analysis of nano/microelectromechanical systems (N/MEMS) is an essential field owing to their many promising applications in microinstruments. The interesting and unique properties of these systems, particularly, small size, batch fabrication, low power consumption, and high reliability, have fascinated researchers and industries to implement these structures for the production of different microdevices. The dynamic oscillatory behavior of N/MEMS is very intricate due to the various types of nonlinearities present in these structures. The foremost objective of this study is to explore the periodicity of oscillatory problems from N/MEMS. The variational iteration method (VIM), which has been considered as an effective approach for nonlinear oscillators, is coupled with the Laplace transform to obtain the approximate analytic solution of these nonlinear vibratory systems with fewer computations. This coupling of VIM and Laplace transform not only helps in the identification of the Lagrange multiplier without getting into the details of the cryptic theory of variations, but also finds the frequency-amplitude relationship and the analytic approximate solution of N/MEMS. A generalized vibratory equation for N/MEMS is followed by three examples as special cases of this generalized equation are given to elucidate the effectivity of the coupling. The solution obtained from the Laplace-based VIM not only exhibits good agreement with observations numerically but also higher accuracy yields when compared to other established techniques in the open literature.

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Analytical and Numerical Solution for the Time Fractional Black-Scholes Model Under Jump-Diffusion

2023

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In this work, we study the numerical solution for time fractional Black-Scholes model under jump-diffusion involving a Caputo differential operator. For simplicity of the analysis, the model problem is converted into a time fractional partial integro-differential equation with a Fredholm integral operator. The L1 discretization is introduced on a graded mesh to approximate the temporal derivative. A second order central difference scheme is used to replace the spatial derivatives and the composite trapezoidal approximation is employed to discretize the integral part. The stability results for the proposed numerical scheme are derived with a sharp error estimation. A rigorous analysis proves that the optimal rate of convergence is obtained for a suitable choice of the grading parameter. Further, we introduce the Adomian decomposition method to find out an analytical approximate solution of the given model and the results are compared with the numerical solutions. The main advantage of the fully discretized numerical method is that it not only resolves the initial singularity occurred due to the presence of the fractional operator, but it also gives a higher rate of convergence compared to the uniform mesh. On the other hand, the Adomian decomposition method gives the analytical solution as well as a numerical approximation of the solution which does not involve any mesh discretization. Furthermore, the method does not require a large amount of computer memory and is free of rounding errors. Some experiments are performed for both methods and it is shown that the results agree well with the theoretical findings. In addition, the proposed schemes are investigated on numerous European option pricing jump-diffusion models such as Merton’s jump-diffusion and Kou’s jump-diffusion for both European call and put options.

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The Approximate Analytic Solution of the Time-Fractional Black-Scholes Equation with a European Option Based on the Katugampola Fractional Derivative

Cited by 9 publications

References 23 publications

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

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

An Efficient Analytical Approach for the Periodicity of Nano/Microelectromechanical Systems’ Oscillators

Analytical and Numerical Solution for the Time Fractional Black-Scholes Model Under Jump-Diffusion

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