Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Padé Approximation

• Authors: N. Özdemir , M. Yavuz
• Languages of publication: EN

Abstracts

EN

In this study, a new application of multivariate Padé approximation method has been used for solving European vanilla call option pricing problem. Padé polynomials have occurred for the fractional Black-Scholes equation, according to the relations of "smaller than", or "greater than", between stock price and exercise price of the option. Using these polynomials, we have applied the multivariate Padé approximation method to our fractional equation and we have calculated numerical solutions of fractional Black-Scholes equation for both of two situations. The obtained results show that the multivariate Padé approximation is a very quick and accurate method for fractional Black-Scholes equation. The fractional derivative is understood in the Caputo sense.

Keywords

EN

02.60.-x; 02.30.Mv; 02.30.Jr

Discipline

• 02.60.-x: Numerical approximation and analysis
• 02.30.Mv: Approximations and expansions
• 02.30.Jr: Partial differential equations

• Publisher: Institute of Physics, Polish Academy of Sciences
• Journal: Acta Physica Polonica A
• Year: 2017
• Volume: 132
• Issue: 3
• Pages: 1050-1053

Dates

published

2017-09

Contributors

author

N. Özdemir

• Faculty of Sciences and Arts, Department of Mathematics, Balıkesir University, Balıkesir, Turkey

author

M. Yavuz

• Faculty of Science, Department of Mathematics-Computer Sciences, Necmettin Erbakan University, Konya, Turkey

References

• [1] L. Debnath, D.D. Bhatta, Fractional Calculus Appl. Anal. 7, 21 (2004)
• [2] A. Carpinteri, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer, 1997, p. 291
• [3] B. Gürbüz, M. Sezer, Acta Phys. Pol. A 130, 194 (2016), doi: 10.12693/APhysPolA.130.194
• [4] F. Evirgen, N. Özdemir, J. Comput. Nonlinear Dynam. 6, 021003 (2011), doi: 10.1115/1.4002393
• [5] B.B. İskender, N. Özdemir, A.D. Karaoglan, Discontinuity and Complexity in Nonlinear Physical Systems, Springer International Publishing, 2014
• [6] V. Turut, N. Güzel, Europ. J. Pure Appl. Math. 6, 147 (2013)
• [7] A.A. Elbeleze, A. Kılıçman, B.M. Taib, Math. Prob. Engin. 2013, 524852 (2013)
• [8] S. Kumar, A. Yildirim, Y. Khan, H. Jafari, K. Sayevand, L. Wei, J. Fractional Calculus Appl. 2, 1 (2012)
• [9] M.A.M. Ghandehari, M. Ranjbar, Computational Methods Different. Equat. 2, 1 (2014)
• [10] S.-H. Park, J.-H. Kim, Appl. Math. Lett. 24, 1740 (2011)
• [11] A.A. Elbeleze, A. Kılıçman, B.M. Taib, Mathematical Problems Engin. 2013, 543848 (2013), doi: 10.1155/2013/543848
• [12] W. Chen, S. Wang, Management. 11, 241 (2015)
• [13] A. Cartea, D. del-Castillo-Negrete, Phys. A 374, 749 (2007), doi: 10.1016/j.physa.2006.08.071
• [14] N. Özdemir, D. Avcı, B.B. İskender, Int. J. Optimization Control: Theories Applicat. (IJOCTA) 1, 17 (2011), doi: 10.11121/ijocta.01.2011.0028
• [15] Z. Akhmetova, S. Zhuzbaev, S. Boranbayev, Acta Phys. Pol. A 130, 352 (2016), doi: 10.12693/APhysPolA.130.352
• [16] A.T. Özturan, Acta Phys. Pol. A 128, B-93 (2015), doi: 10.12693/APhysPolA.128.B-93
• [17] F. Black, M. Scholes, J. Political Economy 81, 637 (1973)
• [18] I. Podlubny, Fractional differential equations, Academic Press, New York 1999
• [19] G. Mittag-Leffler, Rend. R. Acc. Lincei 13, 3 (1904)
• [20] A. Cuyt, L. Wuytack, Nonlinear methods in numerical analysis, Elsevier 1987
• [21] M. Yavuz, N. Özdemir, Y.Y. Okur, in: Int. Conf. Fractional Differentiation and its Applications (ICFDA), vol. 2, Novi Sad, Serbia 2016, p. 778
• [22] R. Company, E. Navarro, J.R. Pintos, E. Ponsoda, Computers Mathematics Appl. 56, 813 (2008)

Identifiers

DOI

10.12693/APhysPolA.132.1050