The exact traveling wave solutions of a class of generalized Black-Scholes equation

: In this paper, the traveling wave solutions of a class of generalized Black-Scholes equation are considered. By using the ﬁrst integral method and the G (cid:48) / G -expansion method, several exact traveling wave solutions of the equation are obtained.


Introduction
In financial mathematics and financial engineering, the classical Black-Scholes equation is a practical partial differential equation.In 1973, Black and Scholes derived the famous Black-Scholes Option Pricing Model [1].In [2], Sunday O. Edeki etc successfully calculated the European option valuation using the Projected Differential Transformation Method.The results obtained converge faster to their associated exact solutions.In [3], the author studied the Black-Scholes equation in stochastic volatility model.In [4], the author considered to deal with the Black-Scholes equation in financial mathematics by the volatility of a variable and the abstract boundary conditions.
In this paper, we consider to obtain the traveling wave solutions of a class of generalized Black-Scholes equation by using the first integral method and the G /G-expansion method.In 2002, Feng first proposed the first integral method [5].This method has been widely used to solve the exact solutions of some partial differential equations.In [6], authors applied first integral method and functional variable method to obtain optical solitons from the governing nonlinear Schr ödinger equation with spatio-temporal dispersion.In [7], the first integral method is applied for solving the system of nonlinear partial differential equations which are (2 + 1)-dimensional Broer-Kaup-Kupershmidt system and (3 + 1)-dimensional Burgers equations exactly.In [8], authors applied first integral method to construct travelling wave solutions of modified Zakharov-Kuznetsov equation and ZK-MEW equation.This method can also be applied to other systems of nonlinear differential equations [9][10][11][12].The advantage of the first integral method is that the calculation is concise.A more accurate traveling wave solution can be obtained by the first integral method.In 2008, Mingliang Wang et al introduced the G /Gexpansion method in detail [13].In [14], the G /G-expansion method is applied to address the resonant nonlinear Schr ödinger equation with dual-power law nonlinearity.In [15], the author constructed the traveling wave solutions involving parameters for some nonlinear evolution equations in mathematical physics via the (2 + 1)-dimensional Painlevé integrable Burgers equations, the (2 + 1)-dimensional Nizhnik-Novikov-Vesselov equations, the (2 + 1)-dimensional Boiti-Leon-Pempinelli equations and the (2 + 1)-dimensional dispersive long wave equations by using the G /G-expansion method.In [16], a generalized G /G-expansion method is proposed to seek exact solutions of the mKdV equation with variable coefficients.The G /G-expansion method has been proposed to construct more explicit travelling wave solutions to many nonlinear evolution equations [17][18][19].The performance of this method is effective, simple, convenient and gives many new solutions.
The classical celebrated Black-Scholes option pricing model is as follows: We consider the class of generalized Black-Scholes equation is as follows: where v = v(t, x), A, B, C, D 0 are arbitrary constants, when D = 0, (1) changes to the classical Black-Scholes equation.
Using the wave transformation v(t, x) = v(ξ), ξ = ln x − at, and a is wave velocity, we have the following ordinary differential equation, Letting w = v , (2) is equivalent to the autonomous system, (3)

Traveling wave solutions of (1) by using the first integral method
In this section, we apply the first integral method to obtain the traveling wave solution to (1).We assume that p(v, w) Owing to Division Theorem, there exists a polynomial g(v) + h(v)w in the complex domain C(v, w), such that, Here, we mainly consider (4) in two cases: N = 1 and N = 2.

N=1
At present, From (4), we have ( By observing the coefficients of w i (i = 0, 1) of the two sides of ( 5), obviously, we have Since α i (v)(i = 0, 1) are polynomials, then from the first equation of ( 6), we deduce that α 1 (v) is constant and h(v) = 0.For simplification, taking α 1 (v) = 1.In order to keep balancing the degree of g(v), α 1 (v) and α 0 (v) in the second and the third equations of (6), one can conclude that deg g(v) = 1.Suppose that g(v) = g 0 v + d 0 , where g 0 and d 0 are arbitrary constants.By solving the above equations, one can obtain where d 1 is an integration constant.Substituting g(v), α 1 (v) and α 0 (v) into the third equation of ( 6) and setting all the coefficients of v i (i = 0, 1, 2, 3) to be zeros, then one can get From the last equation of ( 7), we consider to solve (7) in two cases d 0 = 0 or d 0 0.
Case 1: d 0 = 0 By solving (7), we have and So, One can obtain the following one order ordinary differential equation, Integrating (8) once with respect to ξ, we can get the following results.
, where ξ 0 is an integration constant.
The traveling wave solution to (1) can be got as follows: .
where ξ 1 is an arbitrary integration constant.The traveling wave solution to (1) can be got as follows: Case 2: d 0 0 From (7), we have and So, One can obtain the following one order ordinary differential equation, Integrating equation ( 9) once with respect to ξ, we can get the following results, where ξ 2 is an integration constant.
The exact traveling wave solution to (1) can be got as follows: (ln x−at) .

N = 2
At present, From ( 4), we have By observing the coefficients of w i (i = 0, 1, 2, 3) of the two sides of ( 11), obviously, we have Similarly, from the first equation of ( 12), we deduce that α 2 (v) is constant and h(v) = 0. From the second and the third equations of ( 12), we assume that deg By the last equation of ( 12), we have the degree of the polynomial and the degree of the polynomial α 0 (v)g(v) is 3k + 2. By balancing the degree of the last equation of ( 12), we have k + 4 = 3k + 2, obviously, k = 1.Specially, we conclude that deg g(v) = 0, then deg α 1 (v) = 1.The degree on both sides of the last equation of ( 12) is still true.
where d 2 , d 3 are integration constants.From the last one of equation ( 12), we have Solving ( 14), we have Then from ( 15), ( 13) and (10), one can obtain the following equation, Solving the above algebraic equation with respect to the variable w, we have Integrating (16) once with respect to ξ, then we have where ξ 3 is an integration constant.
The exact traveling wave solution to (1) can be got as follows: From (12), we have where d 5 , d 6 are integration constants.
From the last equation of ( 12), we have From the last equation of ( 17), we need to discuss (17) in two cases d 4 = 0 or d 4 0.
Case I: and D < 0, A 2 − 2B + 2a = 0, such that, So, Solving the above algebraic equation with respect to the variable w, we have Integrating (18) once with respect to ξ, we can get the following results. .
where ξ 5 is an integration constant.The traveling wave solution to (1) can be got as follows: