ANALYTICAL SOLUTION OF FRACTIONAL BLACK-SCHOLES EUROPEAN OPTION PRICING EQUATION BY USING LAPLACE TRANSFORM

In this paper, Laplace homotopy perturbation method, which is combined form of the Laplace transform and the homotopy perturbation method, is employed to obtain a quick and accurate solution to the fractional Black Scholes equation with boundary condition for a European option pricing problem. The Black-Scholes formula is used as a model for valuing European or American call and put options on a non-dividend paying stock. The proposed scheme finds the solutions without any discretization or restrictive assumptions and is free from round-off errors and therefore, reduces the numerical computations to a great extent. The analytical solution of the fractional Black Scholes equation is calculated in the form of a convergent power series with easily computable components. Two examples are presented.

[1]  Majid Khan,et al.  Application of Laplace decomposition method on semi-infinite domain , 2011, Numerical Algorithms.

[2]  Vildan Gülkaç,et al.  The homotopy perturbation method for the Black–Scholes equation , 2010 .

[3]  Lucas Jódar,et al.  A numerical method for European Option Pricing with transaction costs nonlinear equation , 2009, Math. Comput. Model..

[4]  Enrique A. Navarro,et al.  Numerical solution of linear and nonlinear Black-Scholes option pricing equations , 2008, Comput. Math. Appl..

[5]  Yasir Khan,et al.  On the coupling of the homotopy perturbation method and Laplace transformation , 2011, Math. Comput. Model..

[6]  Zhongdi Cen,et al.  A robust and accurate finite difference method for a generalized Black-Scholes equation , 2011, J. Comput. Appl. Math..

[7]  Ji-Huan He,et al.  A NEW PERTURBATION TECHNIQUE WHICH IS ALSO VALID FOR LARGE PARAMETERS , 2000 .

[8]  Matthias Ehrhardt,et al.  On the numerical solution of nonlinear Black-Scholes equations , 2008, Comput. Math. Appl..

[9]  R. Gorenflo,et al.  AN OPERATIONAL METHOD FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS WITH THE CAPUTO DERIVATIVES , 1999 .

[10]  Asghar Ghorbani,et al.  He's Homotopy Perturbation Method for Calculating Adomian Polynomials , 2007 .

[11]  Osama L. Moustafa,et al.  On the Cauchy problem for some fractional order partial differential equations , 2003 .

[12]  Hossein Jafari,et al.  Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion-wave equations , 2011, Appl. Math. Lett..

[13]  R. K. Gazizov,et al.  Lie Symmetry Analysis of Differential Equations in Finance , 1998 .

[14]  Ji-Huan He.  A coupling method of a homotopy technique and a perturbation technique for non-linear problems , 2000 .

[15]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[16]  Asghar Ghorbani,et al.  Beyond Adomian polynomials: He polynomials , 2009 .

[17]  Ji-Huan He.  SOME ASYMPTOTIC METHODS FOR STRONGLY NONLINEAR EQUATIONS , 2006 .

[18]  Ji-Huan He.  Application of homotopy perturbation method to nonlinear wave equations , 2005 .

[19]  M. Grossinho,et al.  Positive solutions of a Dirichlet problem for a stationary nonlinear Black-Scholes equation , 2009 .

[20]  Yasir Khan,et al.  Homotopy perturbation transform method for nonlinear equations using He's polynomials , 2011, Comput. Math. Appl..

[21]  Ji-Huan He.  Homotopy perturbation technique , 1999 .

[22]  C. Khalique,et al.  Comments on “He's Homotopy Perturbation Method for Calculating Adomian Polynomials” , 2013 .

[23]  Stationary solutions for two nonlinear Black-Scholes type equations , 2003 .

[24]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[25]  P. Amster,et al.  Solutions to a stationary nonlinear Black–Scholes type equation , 2002 .