Hahn hybrid functions for solving distributed order fractional Black–Scholes European option pricing problem arising in financial market
Parisa Rahimkhani
Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran
Search for more papers by this authorCorresponding Author
Yadollah Ordokhani
Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran
Correspondence
Yadollah Ordokhani, Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran.
Email: [email protected]
Communicated by: M. Rasmussen
Search for more papers by this authorSedigheh Sabermahani
Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran
Search for more papers by this authorParisa Rahimkhani
Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran
Search for more papers by this authorCorresponding Author
Yadollah Ordokhani
Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran
Correspondence
Yadollah Ordokhani, Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran.
Email: [email protected]
Communicated by: M. Rasmussen
Search for more papers by this authorSedigheh Sabermahani
Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran
Search for more papers by this authorAbstract
The main purpose of this work is to present a new numerical method based on Hahn hybrid functions (HHFs) for solving of Black–Scholes option pricing distributed order time-fractional partial differential equation. To this end, HHFs are introduced and their fractional integral operator with some properties of HHFs is calculated. In the next, with the help of fractional integral operator of HHFs, Gauss–Legendre quadrature formula and collocation method, distributed order time-fractional Black–Scholes model is reduced to a system of algebraic equations. Furthermore, convergence analysis of the mentioned scheme is discussed. Finally, some test problems have been included to confirm the validity and efficiency of the mentioned numerical scheme. Moreover, Black–Scholes equations are studied through a bibliometric viewpoint.
CONFLICT OF INTEREST
This work does not have any conflict of interest.
REFERENCES
- 1Rahimkhani P, Ordokhani Y, Babolian E. Fractional-order Bernoulli wavelets and their applications. Appl Math Model. 2016; 40(17–18): 8087-8107.
- 2Sabermahani S, Ordokhani Y, Yousefi SA. Numerical approach based on fractional-order Lagrange polynomials for solving a class of fractional differential equations. Comput Appl Math. 2018; 37: 3846-3868.
- 3Ngo HT, Vo TN, Razzaghi M. An effective method for solving nonlinear fractional differential equations. Eng Comput. 2020: 1-12.
- 4Nikan O, Golbabai A, Tenreiro Machado JA, Nikazad T. Numerical approximation of the time fractional cable model arising in neuronal dynamics. Eng Comput. 2022; 38: 155-173.
- 5Nikana O, Avazzadeh Z, Tenreiro Machado JA. Numerical approach for modeling fractional heat conduction in porous medium with the generalized Cattaneo model. Appl Math Model. 2021; 100: 107-124.
- 6Nikana O, Avazzadeh Z. Numerical simulation of fractional evolution model arising in viscoelastic mechanics. Appl Numer Math. 2021; 169: 303-320.
- 7Black F, Scholes M. The pricing of options and corporate liabilities. J Polit Econ. 1973; 81(3): 637-654.
- 8Amster P, Averbuj CG, Mariani MC. Solutions to a stationary nonlinear Black-Scholes type equation. J Math Anal Appl. 2002; 276(1): 231-38.
- 9Fabiao F, Grossinho MR, Simoes OA. Positive solutions of a Dirichlet problem for a stationary nonlinear Black–Scholes equation. Nonlinear Anal Theor. 2009; 71(10): 4624-4631.
- 10Kumar S, Kumar D, Singh J. Numerical computation of fractional Black-Scholes equation arising in financial market. Egypt J Basic Appl Sci. 2014; 1(3-4): 177-183.
10.1016/j.ejbas.2014.10.003 Google Scholar
- 11Song L, Weiguo W. Solution of the fractional Black-Scholes option pricing model by finite difference method. Abstr Appl Anal. 2013; 2013: 194286.
10.1155/2013/194286 Google Scholar
- 12Golbabai A, Nikan O, Nikazad T. Numerical analysis of time fractional Black-Scholes European option pricing model arising in financial market. Comput Appl Math. 2019; 38(4): 173.
- 13Saratha SR, Sai Sundara Krishnan G, Bagyalakshmi M, Lim CP. Solving Black-Scholes equations using fractional generalized homotopy analysis method. Comp Appl Math. 2020; 39: 1-35. doi:10.1007/s40314-020-01306-4
- 14Nikan O, Avazzadeh Z, Tenreiro Machado JA. Localized kernel-based meshless method for pricing financial options underlying fractal transmission system. Math Methods Appl Sci. 2021. doi:10.1002/mma.7968
- 15Golbabai A, Nikan O. A computational method based on the moving least-squares approach for pricing double barrier options in a time-fractional Black-Scholes model. Comput Econ. 2020; 55: 119-141.
- 16So M, Kim J, Choi S, Park HW. Factors affecting citation networks in science and technology: focused on non-quality factors. Qual Quant. 2015; 49(4): 1513-1530.
10.1007/s11135-014-0110-z Google Scholar
- 17Sabermahani S, Ordokhani Y. Fibonacci wavelets and Galerkin method to investigate fractional optimal control problems with bibliometric analysis. J Vibrat Control. 2021; 27(15–16): 1778-1792.
- 18Ghanbari Baghestan A, Khaniki H, Kalantari A, et al. A crisis in open access: should communication scholarly outputs take 77 years to become open access? SAGE Open. 2019; 9: 1-8.
- 19Chen C, Dubin R, Kim MC. Orphan drugs and rare diseases: a scientometric review (2000–2014). Expert Op Orphan Drugs. 2014; 2(7): 709-724.
- 20Deuflhard P, Wulkow MM. Computational treatment of polyreaction kinetics by orthogonal polynomials of a discrete variable. IMPACT Comput Sci Eng. 1989; 1(3): 269-301.
10.1016/0899-8248(89)90013-X Google Scholar
- 21Baik J, Kriecherbauer T, McLaughlin K, Miller P. Discrete Orthogonal Polynomials: Asymptotics and Applications, Annals of Mathematics Studies. Princeton University Press; 2007.
- 22Karlin S, McGregor J. Linear growth birth-and-death processes. J Math Mech (Now Indiana Univ Math J). 1958; 7: 643-662.
10.1512/iumj.1958.7.57037 Google Scholar
- 23Stritzke P, King M, Vaknine R, Goldsmith S. Deconvolution using orthogonal polynomials in nuclear medicine: a method for forming quantitative functional images from kinetic studies. IEEE Trans Med Imaging. 1990; 9(1): 11-23.
- 24Mandyam G, Ahmed N. The discrete Laguerre transform: derivation and applications. IEEE Trans Signal Process. 1996; 44(12): 2925-2931.
- 25Salehi F, Saeedi H, Moghadam Moghadam M. Discrete Hahn polynomials for numerical solution of two-dimensional variable-order fractional Rayleigh-Stokes problem. Comp Appl Math. 2018; 37: 5274-5292.
- 26Hariharan G, Padma S, Pirabaharan P. An efficient wavelet based approximation method to time fractional Black-Scholes European option pricing problem arising in financial market. Appl Math Sci. 2013; 7(69): 3445-3456.
- 27Heydari MH, Avazzadeh Z. Numerical study of non-singular variable-order time fractional coupled Burgers' equations by using the Hahn polynomials. Eng Computers. 2022; 38: 101-110.
- 28Rahimkhani P, Ordokhani Y. Approximate solution of nonlinear fractional integro-differential equations using fractional alternative Legendre functions. J Comput Appl Math. 2020; 365: 112365.
- 29Golbabai A, Nikan O, Nikazad T. Numerical analysis of time fractional Black-Scholes European option pricing model arising in financial market. Comput Appl Math. 2019; 38(4): 173.
- 30Rogers LCG. Arbitrage with fractional Brownian motion. Math Financ. 1997; 7(1): 95-105.
- 31Wang J, Liang JR, Lv LJ, Qiu W, Ren F. Continuous time Black-Scholes equation with transaction costs in subdiffusive fractional Brownian motion regime. Phys A. 2012; 391: 750-763.
- 32Jumarie G. Derivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time. Application to Merton's optima portfolio. Comput Math Appl. 2010; 59: 1142-1164.
- 33Wyss W. The fractional Black-Scholes equation. Fract Calc Appl Anal. 2017; 3: 51-62.
- 34Liang JR, Wang J, Zhang WJ, Qiu WY, Ren FY. The solution to a bifractional Black-Scholes-Merton differential equation. Int Pure Appl Math. 2010; 58(1): 99-112.
- 35Cartea A. Derivatives pricing with marked point processes using tick-by-tick data. Quant Financ. 2013; 13(1): 111-123.
- 36Leonenko NN, Meerschaert MM, Sikorskii A. Fractional Pearson diffusions. J Math Anal Appl. 2013; 403(2): 532-546.
- 37Koleva MN, Vulkov LG. Numerical solution of time-fractional Black-Scholes equation. Comput Appl Math. 2017; 36: 1699-1715.
- 38Stoer J, Bulirsch R. Introduction to Numerical Analysis, 2nd ed. Springer; 2002.
- 39Rahimkhani P, Ordokhani Y. A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions. Numerical Method Partial Differ Equ. 2019; 35(1): 34-59.
- 40Morgado ML, Rebelo M, Ferras LL, Ford NJ. Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method. Appl Numer Math. 2017; 114: 108-123.
- 41Kumar Y, Singh S, Srivastava N, Singh A, Singh VK. Wavelet approximation scheme for distributed order fractional differential equations. Comput Math Appl. 2020; 80: 1985-2017.
- 42Morgado L, Rebelo M. Black-Scholes equation with distributed order in time. Progress in Industrial Mathematics at ECMI 2018. Springer; 2019: 313-319.
10.1007/978-3-030-27550-1_39 Google Scholar
