A Quantitative Approach to Fractional Option Pricing Problems with Decomposition Series

Yıl 2018, Cilt: 6 Sayı: 1, 102 - 109, 15.04.2018

Mehmet Yavuz Necati Özdemir

Öz

This study addresses a novel identification of Adomian Decomposition Method (ADM) to have an accurate and quick solution for the European option pricing problem by using Black-Scholes equation of time-fractional order (FBSE) with the initial condition and generalized Black-Scholes equation of fractional order (GFBSE). The fractional operator is understood in the Caputo mean. First of all, we redefine the Black-Scholes equation as fractional mean which computes the option price for fractional values. Then we have applied the ADM to the FBSE and GFBSE, so we have obtained accurate and quick approximate analytical solutions for these equations. The results related to the solutions have been presented in figures.

Anahtar Kelimeler

Adomian decomposition method, convergence analysis, fractional Black-Scholes model, option pricing

Kaynakça

• [1] Abdulaziz, O., Hashim, I. and Saif, A.: Series solutions of time-fractional PDEs by homotopy analysis method. Differ. Equ. Nonlinear Mech., 2008, Article ID 686512, (2008).
• [2] Adomian, G.: A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135 (2), 501-544, (1988).
• [3] Adomian, G.: A review of the decomposition method and some recent results for nonlinear equations, Math. Comp. Model., 13 (7), 17-43, (1990).
• [4] Ahmad, J., Shakeel, M., Hassan, Q. M. U. and Mohyud-Din, S.: Analytical solution of Black-Scholes model using fractional variational iteration method, Int. J. Mod. Math. Sci, 5 133-142, (2013).
• [5] Ankudinova, J. and Ehrhardt, M.: On the numerical solution of nonlinear Black–Scholes equations, Comp. Math. with Appl., 56 (3), 799-812, (2008).
• [6] Avci, D., Eroglu, B.B.I. and Ozdemir, N.: Conformable heat equation on a radial symmetric plate, Therm. Sci., 21 (2), 819-826, (2017).
• [7] Baleanu, D., Srivastava, H.M. and Yang, X.-J.: Local fractional variational iteration algorithms for the parabolic Fokker-Planck equation defined on Cantor sets, Prog. Fract. Differ. Appl., 1 (1), 1-11, (2015).
• [8] Bildik, N. and Konuralp, A.: The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations, Int. J. Nonlin. Sci. Num. Sim., 7 (1), 65-70, (2006).
• [9] Black, F. and Scholes, M.: The pricing of options and corporate liabilities, J. Pol. Eco., 637-654, (1973).
• [10] Caputo, M.: Linear models of dissipation whose Q is almost frequency independent, Ann. Geophys., 19 (4), 383-393, (1966).
• [11] Carpinteri, A. and Mainardi, F.: Fractional calculus: some basic problems in countinuum and statistical mechanics, Frac. Fract. Cal. Cont. Mech., 291-348, (1997).
• [12] Cen, Z. and Le, A.: A robust and accurate finite difference method for a generalized Black–Scholes equation, J. Comp. Appl. Math., 235 (13), 3728-3733, (2011).
• [13] Chen, W. and Wang, S.: A finite difference method for pricing European and American options under a geometric L´evy process, Management, 11 (1), 241-264, (2015).
• [14] Cherruault, Y. and Adomian, G.: Decomposition methods: a new proof of convergence, Math. Comp. Model., 18 (12), 103-106, (1993).
• [15] Daftardar-Gejji, V. and Jafari, H.: Adomian decomposition: a tool for solving a system of fractional differential equations, J. Math. Anal. Appl., 301 (2), 508-518, (2005).
• [16] Debnath, L. and Bhatta, D.D.: Solutions to few linear fractional inhomogeneous partial differential equations in fluid mechanics, Frac. Cal. Appl. Anal., 7 (1), 21-36, (2004).
• [17] El-Sayed, A. and Gaber, M.: The Adomian decomposition method for solving partial differential equations of fractal order in finite domains, Phys. Lett. A, 359 (3), 175-182, (2006).
• [18] Elbeleze, A.A., Kılıçman, A. and Taib, B.M.: Homotopy perturbation method for fractional Black-Scholes European option pricing equations using Sumudu transform, Math. Prob. Eng., 2013, Article ID 524852, (2013).
• [19] Evirgen, F. and Özdemir, N.: Multistage Adomian decomposition method for solving NLP problems over a nonlinear fractional dynamical system, J. Comp. and Nonl. Dyn., 6 (2), 021003, (2011).
• [20] Evirgen, F. and Özdemir, N.: A fractional order dynamical trajectory approach for optimization problem with HPM, in: Fractional Dynamics and Control (Ed. D. Baleanu, J.A.T. Machado, and A.C.J. Luo), Springer, pp. 145-155, (2012).
• [21] Ghandehari, M. A. M. and Ranjbar, M.: European option pricing of fractional Black-Scholes model with new Lagrange multipliers, Comp. Met. Diff. Equ., 2 (1), 1-10, (2014).
• [22] Gülkac¸, V.: The homotopy perturbation method for the Black–Scholes equation, J. Stat. Comp. Sim., 80 (12), 1349-1354, (2010).
• [23] He, J.-H.: Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol., 15 (2), 86-90, (1999).
• [24] Kumar, S., Kumar, D. and Singh, J.: Numerical computation of fractional Black–Scholes equation arising in financial market, Egyp. J. Basic Appl. Sci., 1 (3), 177-183, (2014).
• [25] Luchko, Y. and Gorenflo, R.: The initial value problem for some fractional differential equations with the Caputo derivatives, Preprint Series A08-98. Fachbereich Mathematik und Informatic, Berlin, Freie Universitat, (1998).
• [26] Özdemir, N. and Yavuz, M.: Numerical solution of fractional Black-Scholes equation by using the multivariate Pade´ approximation, Acta Phys. Pol. A, 132 (3), 1050-1053, (2017).
• [27] Park, S.-H. and Kim, J.-H.: Homotopy analysis method for option pricing under stochastic volatility, Appl. Math. Lett., 24 (10), 1740-1744, (2011).
• [28] Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Math. Sci. Eng., 198 (1999).
• [29] Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation, ArXiv preprint math/0110241, (2001).
• [30] Song, L. and Wang, W.: Solution of the fractional Black-Scholes option pricing model by finite difference method, Abst. Appl. Anal., 2013, Article ID 194286, (2013).
• [31] Turut, V. and Güzel, N.: On solving partial differential equations of fractional order by using the variational iteration method and multivariate Pad´e approximations, Euro. J. Pure Appl. Math., 6 (2), 147-171, (2013).
• [32] Wilmott, P.: The mathematics of financial derivatives: a student introduction, Cambridge University Press, 1995.
• [33] Yang, X.-J., Srivastava, H. and Cattani, C.: Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics, Rom. Rep. Phys., 67 (3), 752-761, (2015).
• [34] Yavuz, M., Ozdemir, N. and Okur, Y. Y.: Generalized differential transform method for fractional partial differential equation from finance, Proceedings, International Conference on Fractional Differentiation and its Applications, Novi Sad, Serbia, pp. 778-785, (2016).
• [35] Yavuz, M.: Novel solution methods for initial boundary value problems of fractional order with conformable differentiation, An Int. J. Opt. Cont.: Theo. Appl. (IJOCTA), 8 (1), 1-7, (2018).
• [36] Yavuz, M. and O¨ zdemir, N.: A different approach to the European option pricing model with new fractional operator, Math. Model. Nat. Phenom., 13 (1), 1-12, (2018).
• [37] Yavuz, M. and O¨ zdemir, N.: European vanilla option pricing model of fractional order without singular kernel, Fractal Fract., 2 (1), 3, (2018). [38] Yavuz M., Özdemir N.: New numerical techniques for solving fractional partial differential equations in conformable sense. In: Ostalczyk P., Sankowski D., Nowakowski J. (eds) Non-Integer Order Calculus and its Applications. RRNR 2017. Lecture Notes in Electrical Engineering, vol 496. Springer, Cham (2019).
• [39] Yavuz, M. and O¨ zdemir, N.: Numerical inverse Laplace homotopy technique for fractional heat equations, Therm. Sci., 22 (1), 185-194, (2018).
• [40] Yavuz, M. and Yaşkıran, B.: Approximate-analytical solutions of cable equation using conformable fractional operator, New Trends Math. Sci, 5 (4), 209-219, (2017).
• [41] Zhang, H., Liu, F., Turner, I. and Yang, Q.: Numerical solution of the time fractional Black–Scholes model governing European options, Comp. Math. with Appl., 71 (9), 1772-1783, (2016).