REVIEW PERSAMAAN BLACK-SCHOLES FRAKSIONAL DIMODIFIKASI
Review The Modified Fractional Black-Scholes Equation
DOI:
https://doi.org/10.54199/pjse.v1i2.68Keywords:
Adomian decomposition, Laplace-Adomian decomposition, series expansion, Fractional Calculus, Fractional Black-Scholes equation, homotopy perturbation, solutions, Sumudu transformation.Abstract
This paper will discuss the solution of the fractional Black-Scholes equation which is a general form of the Black-Scholes equation and the novelty of research on the modified Black-Scholes equation. The methods for finding solutions to the fractional Black Scholes equation have been written in many international journals. The solution to the fractional Black Scholes equation in this case is reviewed by the Fractional Calculus approach. With the Fractional Calculus approach, the solution process in finding solutions to the Fractional Black Scholes equation becomes more efficient. Some of the methods used to find solutions to the Fractional Black Scholes Equation include the Sumudu transformation method, the series expansion method, the Homotopy perturbation method, the Adomian decomposition method, and the Laplace Adomian decomposition method.
References
Ahmed & Abdusalam, (2004), On Modified Black-Scholes Equation, Chaos, Solutions and Fractals, Volume 22, Number 8, 583-587.
Aguilar J.P & Korbel J.,(2017), Option Priccing Model Driven by the space time fractional Diffusion, Series Representation and Aplication Volume 20,Number 8, 1-15
Batogna, R.G., (2018), Analysis of Option Pricing Within The Scope Of Fractional Calculus, Afrika Selatan: Departement of Mathematics and Applied Mathematics Faculty of Natural and Agricultural Science at the University of the Free State.
Belgacem & Karaballi, (2005), Sumudu Transform Fundamental Properties Investigation and Aplications, Hindawi Publisjing Corporation Journal of Applied Mathematics and Stocahstic Analysis,Volume 2006, Number 10,Article ID 9108, 1-23.
Elbeleze, A.A., Kihcman, & Taib, B.M., (2013), Homotopy Perturbation Method for Fractional Black-Scholes European Oprion Pricing Equations Using Sumudu Transform, Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013 Number 10, Article ID 524852, 1-7.
Eltayeb, H., & Kihcman, A., (2010), A Note On the Sumudu Transform and Differential Equations, Applied Mathematical Sciences, Volume 4, Number 22, 1089-1098.
Febrianti, W., (2018), Penentuan Harga Opsi Dengan Model Black-Scholes Menggunakan Metode Beda Hingga Forward the Central Space, Journal Of Fundamental Mathematics and Applications, Volume I, Number I, 45-61.
Ghaedahari & Ranjbar, (2014), European Option Pricing of Fractional Version of the Black-Scholes Model Aproach Via Expansion in Series, International Journal of Non Linear Science, Volume 17, Number 2, 105-110.
Gupta & Jain, (1986), Lebesgue Measure And Integration, Canada: Wiley Eastern Limited.
Kaya, F., & Yilmas, Y., (2019), Basic Properties of Sumudu Transformation and Its Application to Some Partial Differential Equations, Sakarya University Journal of Science, Volume 23, Number 4, 509-514.
Khaeruddin & Massalesse, (2008), Penentuan Harga Opsi Eropa Menggunakan Persamaan Black-Scholes, Jurnal Matematika, Statistika dan Komputasi,Volum 4 ,Number 2, 104-116.
Khan, A.W., (2016), European Pricing of Fractional Black-Scholes Model Using Sumudu Transform and its Derivatives (General Letters In Mathematics), General Letters In Mathematics, Volume 1, Number 3, Desember 2016, 74-80.
Khan, A.W., & Ansari, F.A., (2016), European Option Pricing of Fractional Black-Scholes Model Using Sumudu Transform and is Derivative, General Latters in Mathematics, Volume 1, Number 3 December 2016, 74-80.
Khan, Y., and Qingbiao, W.U., (2011), Homotopy Perturbation Transform Method for Nonlinear Equation using He’s Polynomials,Computers and Mathematics with Applications, Volume 61, Number 10, 1963-1967.
Kanth, A.S.V.R., & Aruna, K., (2016), Solution of Time fractional Black-Scholes European option pricing equation arising in financial market, Non Linear Engineering,Volume 5, Number 4, 269-176.
Kreyszeig, E., (1978), Introductory Functional Analysis With Applications, Canada: John Wiley and Sons.
Kumar, A., Yildirin & Khan, (2012), Analitical solution of fractional Black-Scholes European Option Pricing Equation by Using Laplace Transform, Journal of Fractional Calculus and Applications, Volume 2 Number 8, 1-9.
Lasota, A., & Mackey, (1994), Chaos, Fractals, and Noise : Stochastic Aspect Of Dynamic, Applied Mathematical Science, Vol. 97, Berlin: Springer-Verlag.
Mathai, H,J., & Haubold, H.J., (2017), An Introduction to Fractional Calculus, New York: Nova Science Publisher Inc.
Mishra, R., Aggarwal, Chaundari, L., & Kumar, A., (2020), Relationship Between Sumudu and Some Efficient Integral Transform, International Journal of Innovative Technology and Exploring Engineering (IJITEE),Volume 9,Number 3, 153-159.
Mohebbi, M.A., & Ranjbar, M., (2014), European Option Pricing of Fractional Black-Scholes Model with New Lagrange Multipliers, Computational Methods for Diffrenetial Equations,Volume 7,,Number 2, ,hal 171-178.
Naghipour, A., & Manafian, J., (2015), Application of the Laplace Adomian Decomposition and Implicit Methods for Solfing Burger’s equation, Journal Pure Appl Math Volume.6.Number.1, 68-77.
Omer, E.M., (2017) , Double Laplace Transform and Double Sumudu Transform , American Journal of Engineering Research, Volume 6, Issue 5, 312-317.
Ozkan, O., & Kurt, A., (2019), A New Method for Solving Fractional Partial Differential Equation, The Journal of Analysis, Volume 87, Number 12, 2786-2797,
Pinsky, A., (1998), Partial Differetnyial Equations and Boundary Value Problem With Applications, New York: MC Graw-Hill.
Ranjbar & Ghandehari, (2015), Barrier Options Pricing of Fractional Version of the Black-Scholes Model, International Journal Industrial Mathematics, Volume 7, Number 2, Article ID IJIM, 105-110.
Royden, H.L., (1989), Real Analysis Third Edition, New York: Macmillan Publishing.
Sefira, Rusyaman, E., & Sukono, (2019), Methods to Solve Fractional Black-Scholes, Proceedings of the International Conference om Industrial Engineering and Opertions Management Pilsen,Czech Republic, July 23-26, 2019.
Siswanto, Purnomo, D.K., & Kusbidono, (2014), Penentuan Harga Opsi Model Black-Scholes Menggunakan Metode Beda Hingga Dufort-Frankel, Prosiding Seminar Nasional Matematika Universitas Jember, 329-334.
Sumiati, I., Rusyaman, E., & Sukono, (2019), Black-Scholes Equation Solution Using Laplace -Adomian Decomposition Method, IAENG International Journal Of Computer Science, Volume 6, Number 4, IJCS_46_4_2, 1-6.
Taib, Kichman. A., & Elbelese, (2013), Homotopy Perturbation Method for Fractional Black-Scholes European Option Pricing Equations Using Sumudu Transform, Hindawi Publishing Corporation Mathematical Problem in Engineering, Volume 2013, Aricle id 524852, 7 pages, 1-7.
Uddin, M., & Taufiq, M., (2019), Approximation of Time Fractional Black-Scholes Equation Via Radial, Kernels and Transformation, Fractional Difrential Calculus, Volume 9, Number 1 , 75-90.
Yavuz, M., & Ozdemir, N., (2018), A Quantitative Approach to Fractional Option Pricing Problems with Docompotition Series, Konuralp journal of Mathematics, Volume 6, Number 1, 102-109.
Walters, P., (1982), An Introduction to Ergodic Theory, New York: Springer-Verlag.
Downloads
Published
Issue
Section
License
Copyright (c) 2021 Universitas Perwira Purbalingga
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
