Novel Approaches for Getting the Solution of the Fractional Black–Scholes Equation Described by Mittag-Leffler Fractional Derivative
The value of an option plays an important role in finance. In this paper, we use the Black–Scholes equation, which is described by the nonsingular fractional-order derivative, to determine the value of an option. We propose both a numerical scheme and an analytical solution. Recent studies in fracti...
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2020-01-01
|
| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2020/8047347 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832566140667166720 |
|---|---|
| author | Ndolane Sene Babacar Sène Seydou Nourou Ndiaye Awa Traoré |
| author_facet | Ndolane Sene Babacar Sène Seydou Nourou Ndiaye Awa Traoré |
| author_sort | Ndolane Sene |
| collection | DOAJ |
| description | The value of an option plays an important role in finance. In this paper, we use the Black–Scholes equation, which is described by the nonsingular fractional-order derivative, to determine the value of an option. We propose both a numerical scheme and an analytical solution. Recent studies in fractional calculus have included new fractional derivatives with exponential kernels and Mittag-Leffler kernels. These derivatives have been found to be applicable in many real-world problems. As fractional derivatives without nonsingular kernels, we use a Caputo–Fabrizio fractional derivative and a Mittag-Leffler fractional derivative. Furthermore, we use the Adams–Bashforth numerical scheme and fractional integration to obtain the numerical scheme and the analytical solution, and we provide graphical representations to illustrate these methods. The graphical representations prove that the Adams–Bashforth approach is helpful in getting the approximate solution for the fractional Black–Scholes equation. Finally, we investigate the volatility of the proposed model and discuss the use of the model in finance. We mainly notice in our results that the fractional-order derivative plays a regulator role in the diffusion process of the Black–Scholes equation. |
| format | Article |
| id | doaj-art-79884ecff34e453ba3fbec573b0d997f |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-79884ecff34e453ba3fbec573b0d997f2025-02-03T01:05:04ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2020-01-01202010.1155/2020/80473478047347Novel Approaches for Getting the Solution of the Fractional Black–Scholes Equation Described by Mittag-Leffler Fractional DerivativeNdolane Sene0Babacar Sène1Seydou Nourou Ndiaye2Awa Traoré3Laboratoire Lmdan, Département de Mathématiques de la Décision, Université Cheikh Anta Diop de Dakar, Faculté des Sciences Economiques et Gestion, BP 5683 Dakar Fann, SenegalCentre de Recherches Economiques Appliquées, Directeur du Laboratoire de Finances pour le Developpement (LAFIDEV), Faculté des Sciences Economiques et Gestion, Université Cheikh Anta Diop de Dakar, BP 5683 Dakar Fann, SenegalCentre de Recherches Economiques Appliquées, Laboratoire de Finances pour le Development (LAFIDEV), Faculté des Sciences Economiques et Gestion, Université Cheikh Anta Diop de Dakar, BP 5683 Dakar Fann, SenegalCentre de Recherches Economiques Appliqúees, Facultédes Sciences Economiques et Gestion, Université Cheikh Anta Diop de Dakar, BP 5683 Dakar Fann, SenegalThe value of an option plays an important role in finance. In this paper, we use the Black–Scholes equation, which is described by the nonsingular fractional-order derivative, to determine the value of an option. We propose both a numerical scheme and an analytical solution. Recent studies in fractional calculus have included new fractional derivatives with exponential kernels and Mittag-Leffler kernels. These derivatives have been found to be applicable in many real-world problems. As fractional derivatives without nonsingular kernels, we use a Caputo–Fabrizio fractional derivative and a Mittag-Leffler fractional derivative. Furthermore, we use the Adams–Bashforth numerical scheme and fractional integration to obtain the numerical scheme and the analytical solution, and we provide graphical representations to illustrate these methods. The graphical representations prove that the Adams–Bashforth approach is helpful in getting the approximate solution for the fractional Black–Scholes equation. Finally, we investigate the volatility of the proposed model and discuss the use of the model in finance. We mainly notice in our results that the fractional-order derivative plays a regulator role in the diffusion process of the Black–Scholes equation.http://dx.doi.org/10.1155/2020/8047347 |
| spellingShingle | Ndolane Sene Babacar Sène Seydou Nourou Ndiaye Awa Traoré Novel Approaches for Getting the Solution of the Fractional Black–Scholes Equation Described by Mittag-Leffler Fractional Derivative Discrete Dynamics in Nature and Society |
| title | Novel Approaches for Getting the Solution of the Fractional Black–Scholes Equation Described by Mittag-Leffler Fractional Derivative |
| title_full | Novel Approaches for Getting the Solution of the Fractional Black–Scholes Equation Described by Mittag-Leffler Fractional Derivative |
| title_fullStr | Novel Approaches for Getting the Solution of the Fractional Black–Scholes Equation Described by Mittag-Leffler Fractional Derivative |
| title_full_unstemmed | Novel Approaches for Getting the Solution of the Fractional Black–Scholes Equation Described by Mittag-Leffler Fractional Derivative |
| title_short | Novel Approaches for Getting the Solution of the Fractional Black–Scholes Equation Described by Mittag-Leffler Fractional Derivative |
| title_sort | novel approaches for getting the solution of the fractional black scholes equation described by mittag leffler fractional derivative |
| url | http://dx.doi.org/10.1155/2020/8047347 |
| work_keys_str_mv | AT ndolanesene novelapproachesforgettingthesolutionofthefractionalblackscholesequationdescribedbymittaglefflerfractionalderivative AT babacarsene novelapproachesforgettingthesolutionofthefractionalblackscholesequationdescribedbymittaglefflerfractionalderivative AT seydounouroundiaye novelapproachesforgettingthesolutionofthefractionalblackscholesequationdescribedbymittaglefflerfractionalderivative AT awatraore novelapproachesforgettingthesolutionofthefractionalblackscholesequationdescribedbymittaglefflerfractionalderivative |