Numerical computation of fractional Black–Scholes equation arising in financial market
The aim of present paper is to present a numerical algorithm for time-fractional Black–Scholes equation with boundary condition for a European option problem by using homotopy perturbation method and homotopy analysis method. The fractional derivative is described in the Caputo sense. The methods gi...
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| Published in: | Egyptian Journal of Basic and Applied Sciences Vol. 1; no. 3-4; pp. 177 - 183 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
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Elsevier B.V
01.12.2014
Taylor & Francis |
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| ISSN: | 2314-808X, 2314-808X |
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| Abstract | The aim of present paper is to present a numerical algorithm for time-fractional Black–Scholes equation with boundary condition for a European option problem by using homotopy perturbation method and homotopy analysis method. The fractional derivative is described in the Caputo sense. The methods give an analytic solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. The methods show improvements over existing analytical techniques. Two examples are given and show that the homotopy perturbation method and homotopy analysis method are very effective and convenient overcomes the difficulty of traditional methods. The numerical results show that the approaches are easy to implement and accurate when applied to time-fractional Black–Scholes equation.
•We consider a fractional model of Black–Scholes equation.•The HPM and HAM are applied to obtain the solution of the problem.•The results obtained by HPM and HAM are in excellent agreement.•The numerical results show that the techniques are easy to implement and accurate. |
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| AbstractList | The aim of present paper is to present a numerical algorithm for time-fractional Black–Scholes equation with boundary condition for a European option problem by using homotopy perturbation method and homotopy analysis method. The fractional derivative is described in the Caputo sense. The methods give an analytic solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. The methods show improvements over existing analytical techniques. Two examples are given and show that the homotopy perturbation method and homotopy analysis method are very effective and convenient overcomes the difficulty of traditional methods. The numerical results show that the approaches are easy to implement and accurate when applied to time-fractional Black–Scholes equation.
•We consider a fractional model of Black–Scholes equation.•The HPM and HAM are applied to obtain the solution of the problem.•The results obtained by HPM and HAM are in excellent agreement.•The numerical results show that the techniques are easy to implement and accurate. The aim of present paper is to present a numerical algorithm for time-fractional Black-Scholes equation with boundary condition for a European option problem by using homotopy perturbation method and homotopy analysis method. The fractional derivative is described in the Caputo sense. The methods give an analytic solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. The methods show improvements over existing analytical techniques. Two examples are given and show that the homotopy perturbation method and homotopy analysis method are very effective and convenient overcomes the difficulty of traditional methods. The numerical results show that the approaches are easy to implement and accurate when applied to time-fractional Black-Scholes equation. |
| Author | Singh, Jagdev Kumar, Devendra Kumar, Sunil |
| Author_xml | – sequence: 1 givenname: Sunil surname: Kumar fullname: Kumar, Sunil email: skumar.math@nitjsr.ac.in organization: Department of Mathematics, National Institute of Technology, Jamshedpur, 831014, Jharkhand, India – sequence: 2 givenname: Devendra surname: Kumar fullname: Kumar, Devendra email: devendra.maths@gmail.com organization: Department of Mathematics, JECRC University, Jaipur, 303905, Rajasthan, India – sequence: 3 givenname: Jagdev surname: Singh fullname: Singh, Jagdev email: jagdevsinghrathore@gmail.com organization: Department of Mathematics, Jagan Nath University, Jaipur, 303901, Rajasthan, India |
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| Keywords | European option pricing Black–Scholes equation Homotopy analysis method Homotopy perturbation method 35A20 Fractional derivatives Analytical solution 34A08 |
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| SubjectTerms | 2010 34A08 35A20 Analytical solution Black-Scholes equation European option pricing Fractional derivatives Homotopy analysis method Homotopy perturbation method Mathematics Subject Classification |
| Title | Numerical computation of fractional Black–Scholes equation arising in financial market |
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