A study on existence and global asymptotical mittag-leffler stability of fractional black-scholes european option pricing equation

Document Type : Research Paper

Author

Faculty of Mathematical Sciences, University of Malayer, P. O. Box 65718-18164, Malayer, Iran

Abstract

In this paper, the application of asymptotic expansion method on fractional perturbated equations are studied. Furthermore, the proposed scheme is employed to obtain an analytical solution of fractional BlackScholes equation for a European option pricing problem. Finally, the asymptotical Mittag-Leffler stability of this problem will be discussed.

Keywords


[1] P. Amster, C. G. Averbuj and M. C. Mariani, Solutions to a stationary nonlinear Black-Scholes type equation, J. Math. Anal. Appl. 276 (2002), 231-238.
[2] P. Amster, C. G. Averbuj and M.C. Mariani, Stationary solutions for two nonlinear BlackScholes type equations, Appl. Numer. Math. 47 (2003), 275-280.
[3] J. Ankudinova and M. Ehrhardt, On the numerical solution of nonlinear BlackScholes Equations, Comput. Math. Appl. 56 (2008), 799-812.
[4] C.M. Bender and S. A. Orszag, Advanced Mathematics Methods for Scientists and Engi-neers, McGraw-Hill Inc.,
New York ( 1978).
[5] F. Black and M. S. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ. 61 (1973), 637-654.
[6] M. Bohner and Y. Zheng, On analytical solution of the Black-Scholes equation, Appl. Math. Lett. 22 (2009), 309-313.
[7] Z. Cen and A. Le, A robust and accurate finite difference method for a generalized Black-Scholes equation;J. Comput. Appl. Math. 235 (2011), 3728-3733.
[8] R. Company, L. Jodar, J. R. Pintos, A numerical method for European Option Pricing with transaction costs nonlinear equation, Math. Comput. Modell. 50 (2009) 910-920.
[9] R. Company, E. Navarro, J. R. Pintos and E. Ponsoda, Numerical solution of linear and nonlinear Black-Scholes option pricing equations, Comput. Math. Appl. 56 (2008), 813-821.
[10] W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath,Boston (1965).
[11] F. Fabiao, M. R. Grossinho and O.A. Simoes, Positive solutions of a Dirichlet problem for a stationary nonlinear Black Scholes equation, Nonlinear Anal. 71 (2009), 4624-4631.
[12] R. K. Gazizov, R. K. and N. H. Ibragimov, Lie symmetry analysis of differential equations in Finance, Nonlin. Dynam. 17 (1998), 387-407.
[13] V. Gulkac, The homotopy perturbation method for the Black-Scholes equation, J. Stat. Comput. Simul. 80 (2010), 1349-1354.
[14] R. Haberman, Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, third ed., Prentice-Hall, Inc., Englewood Cliffs, NJ (1998).
[15] H. Jafari and S. Seifi, Homotopy analysis method for solving linear and nonlinear frac-tional diffusion-wave equation,
J. Com. Nonl Sci. Numer. Simulat. 14 (2008), 2006-2012.
[16] H. Jafari and S. Seifi, Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, J. Com. Nonl Sci. Numer. Simu. 14 (2009), 1962-1969.
[17] H. Jafari and S. Momani, Solving fractional diffusion and wave equations by modified homotopy perturbation method, Phys. Lett. A 370 (2007), 388-396.
[18] H. Jafari and M. Nazari, Application of Laplace decomposition method for solving linear and nonlinear fractional diffusion- wave equations, J. Appl. Math. Lett. 24 (2011), 1799-1805.
[19] J. Kevorkian and J.D. Cole, Perturbation Methods in Applied Mathematics, Springer-Verlag, New York ( 1981).
[20] A. Kilbas, H.M. Srivastava and H. M., J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier: Amsterdam ( 2006).
[21] A. I. Klimushev and N. Krasovski, Uniform asymptotic stability of systems of differential equations with a small parameter in the derivative terms, Prikl. Mat. Meh. 25 (1961),680-690 (Russian); translated as J. Appl. Math. Meek. 25 (1962), 1011-1025.
[22] S. Kumar, A. Yildirim, Y. Khan, H. Jafari, K. Sayevand and L. Wei, Analytical solution of fractional Black-Scholes European option pricing equation by using Laplace transform, , J. Fractional Calculus and Appli. 2 ( 2012), 1-9.
[23] J. M. Manale, F. M. Mahomed, A simple formula for valuing American and European all and put options in: J. Banasiak (Ed.), Proceeding of the Hanno Rund Workshop on the Differential Equations, University of Natal (2000), 210-220.
[24] K. S. Miller, B. Ross, An introduction to the fractional calculus and Fractional Differential Equations, Johan Willey and Sons, Inc. New York ( 2003).
[25] M. A. Mohebbi and M. Ranjbar, European option pricing of fractional Black-Scholes model with new Lagrange multipliers, Comput. Methods for Differential Equa. 1
(2013),123-128.
[26] A.H. Nayfeh, Problems in Perturbation, John Wiley and Sons, Inc., New York ( 1993).
[27] K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York (1974).
[28] I. Podlubny, Fractional Differential Equations Calculus, Academic Press, New York (1999).
[29] D. Zwillinger, Handbook of Differential Equations, Academic Press, Inc., New York (1992).