Numerically pricing American and European options using a time fractional Black–Scholes model in financial decision-making
The time fractional Black–Scholes equation (TFBSE) is designed to evaluate price fluctuations within a correlated fractal transmission system. This model prices American or European put and call options on non-dividend-paying stocks. Reliable and efficient numerical techniques are essential for solv...
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Elsevier
2025-01-01
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| Series: | Alexandria Engineering Journal |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S1110016824012481 |
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| author | Omid Nikan Jalil Rashidinia Hossein Jafari |
| author_facet | Omid Nikan Jalil Rashidinia Hossein Jafari |
| author_sort | Omid Nikan |
| collection | DOAJ |
| description | The time fractional Black–Scholes equation (TFBSE) is designed to evaluate price fluctuations within a correlated fractal transmission system. This model prices American or European put and call options on non-dividend-paying stocks. Reliable and efficient numerical techniques are essential for solving fractional differential models due to the global characteristics of fractional calculus. This paper focuses on the numerical solution for the TFBSE for American and European option pricing models by means of the local meshless radial basis function (RBF) interpolation. This problem is temporally approximated using a finite difference scheme with 2−β order accuracy for 0<β<1, and spatially discretized using the localizing RBF partition of unity method (LRBFPUM). The theoretical discussion confirms the convergence analysis and unconditional stability of the semi time-discretized formulation in the perspective of the H1-norm. A main disadvantage of global RBF-based (GRBF) methods is high computational burden required to solve large linear systems. The LRBFPUM overcomes the ill-conditioning that arises in the GRBF methods. It allows for significant sparsification of the algebraic system, leading to a lower condition number and reduced computational effort, while keeping high accuracy. Numerical examples and applications highlight the accuracy of the LRBFPUM technique and confirm the theoretical prediction. |
| format | Article |
| id | doaj-art-dceb552227eb40e186f8a6650a185827 |
| institution | Kabale University |
| issn | 1110-0168 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Alexandria Engineering Journal |
| spelling | doaj-art-dceb552227eb40e186f8a6650a1858272025-01-29T05:00:07ZengElsevierAlexandria Engineering Journal1110-01682025-01-01112235245Numerically pricing American and European options using a time fractional Black–Scholes model in financial decision-makingOmid Nikan0Jalil Rashidinia1Hossein Jafari2School of Mathematics and Computer Science, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran; Corresponding authors.School of Mathematics and Computer Science, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran; Corresponding authors.Department of Applied Mathematics, University of Mazandaran, Babolsar, Iran; Department of Mathematical Sciences, University of South Africa, UNISA0003, Pretoria, South Africa; Department of Medical Research, China Medical University Hospital, China Medical University, 110122, Taichung , Taiwan; Department of Mathematics and Informatics, Azerbaijan University, AZ1007, Jeyhun Hajibeyli, 71, Baku, AzerbaijanThe time fractional Black–Scholes equation (TFBSE) is designed to evaluate price fluctuations within a correlated fractal transmission system. This model prices American or European put and call options on non-dividend-paying stocks. Reliable and efficient numerical techniques are essential for solving fractional differential models due to the global characteristics of fractional calculus. This paper focuses on the numerical solution for the TFBSE for American and European option pricing models by means of the local meshless radial basis function (RBF) interpolation. This problem is temporally approximated using a finite difference scheme with 2−β order accuracy for 0<β<1, and spatially discretized using the localizing RBF partition of unity method (LRBFPUM). The theoretical discussion confirms the convergence analysis and unconditional stability of the semi time-discretized formulation in the perspective of the H1-norm. A main disadvantage of global RBF-based (GRBF) methods is high computational burden required to solve large linear systems. The LRBFPUM overcomes the ill-conditioning that arises in the GRBF methods. It allows for significant sparsification of the algebraic system, leading to a lower condition number and reduced computational effort, while keeping high accuracy. Numerical examples and applications highlight the accuracy of the LRBFPUM technique and confirm the theoretical prediction.http://www.sciencedirect.com/science/article/pii/S1110016824012481Fractional Black–Scholes equationAmerican optionEuropean optionLRBFPUMConvergenceStability |
| spellingShingle | Omid Nikan Jalil Rashidinia Hossein Jafari Numerically pricing American and European options using a time fractional Black–Scholes model in financial decision-making Alexandria Engineering Journal Fractional Black–Scholes equation American option European option LRBFPUM Convergence Stability |
| title | Numerically pricing American and European options using a time fractional Black–Scholes model in financial decision-making |
| title_full | Numerically pricing American and European options using a time fractional Black–Scholes model in financial decision-making |
| title_fullStr | Numerically pricing American and European options using a time fractional Black–Scholes model in financial decision-making |
| title_full_unstemmed | Numerically pricing American and European options using a time fractional Black–Scholes model in financial decision-making |
| title_short | Numerically pricing American and European options using a time fractional Black–Scholes model in financial decision-making |
| title_sort | numerically pricing american and european options using a time fractional black scholes model in financial decision making |
| topic | Fractional Black–Scholes equation American option European option LRBFPUM Convergence Stability |
| url | http://www.sciencedirect.com/science/article/pii/S1110016824012481 |
| work_keys_str_mv | AT omidnikan numericallypricingamericanandeuropeanoptionsusingatimefractionalblackscholesmodelinfinancialdecisionmaking AT jalilrashidinia numericallypricingamericanandeuropeanoptionsusingatimefractionalblackscholesmodelinfinancialdecisionmaking AT hosseinjafari numericallypricingamericanandeuropeanoptionsusingatimefractionalblackscholesmodelinfinancialdecisionmaking |