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Symmetry Analysis and Exact Solutions to the Space-Dependent Coefficient Pdes in Finance

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Symmetry Analysis and Exact Solutions to the Space-Dependent Coefficient Pdes in Finance Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 156965, 10 pages http://dx.doi.org/10.1155/2013/156965 Research Article Symmetry Analysis and Exact Solutions to the Space-Dependent Coefficient PDEs in Finance Hanze Liu1,2 1 School of Mathematical Sciences, Liaocheng University, Liaocheng, Shandong 252059, China 2 Department of Mathematics, Binzhou University, Binzhou, Shandong 256603, China Correspondence should be addressed to Hanze Liu; [email protected] Received 20 June 2013; Accepted 25 August 2013 Academic Editor: Mariano Torrisi Copyright © 2013 Hanze Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The variable-coefficients partial differential equations (vc-PDEs) in finance are investigated by Lie symmetry analysis andthe generalized power series method. All of the geometric vector fields of the equations are obtained; the symmetry reductions and exact solutions to the equations are presented, including the exponentiated solutions and the similarity solutions. Furthermore, the exact analytic solutions are provided by the transformation technique and generalized power series method, which has shown that the combination of Lie symmetry analysis and the generalized power series method is a feasible approach to dealing with exact solutions to the variable-coefficients PDEs. 1. Introduction the generalized power series method for dealing with exact solutions to some nonlinear PDEs based on the symmetry Gazizov and Ibragimov [1] studied the Black-Scholes equa- analysis method. tion of option pricing by Lie equivalence transformations. It is known that the Lie symmetry analysis is a systematic By the optimal system method, some invariant solutions to and powerful method for dealing with symmetries and exact heat and Black-Scholes equations are obtained [2]. In [3– solutions to partial differential equations (see, e.g.,1 [ –6, 8– 5], the fundamental solutions to the bond pricing equations 18] and the references therein). Furthermore, we find that areconsideredbyLiesymmetryanalysisandtheintegral the combination of Lie symmetry analysis and the power transform method. In [6], the invariance properties of the series method is a feasible approach to investigating exact bond pricing equation are studied by the group classification solutions to nonlinear PDEs [8–13]. On the other hand, under method. In [7], the finite element method was adopted the perspective of mathematical physics and Lie symmetry to solve the bond pricing type of PDE system, and the analysis, the space-time dependent coefficients system differs numerical implementation was provided, such as system greatly from its time-dependent counterpart, and it is more that models the TF convertible bonds with credit risk in complicatedthanthelatter.However,mostofthestudies bond pricing theory. However, the similarity reductions and are related to the time-dependent coefficient systems. More- exact solutions to such variable-coefficient equations are not over, the determination of exact solutions to the variable- considered generally in the aforementioned papers. Recently, coefficients PDEs is a complicated problem that challenges we studied some nonlinear PDEs by Lie symmetry analysis researchers greatly. In the present paper, we consider the and the dynamical system method [8–13]; for example, in [8], symmetry reductions and exact solutions to the general we considered Lie group classifications and exact solutions to space-dependent coefficients PDEs in finance as follows: the space-dependent coefficients hanging chain equation and +2 + +]=0, >0, the simplified bond pricing equation. In9 [ ], we investigated (1) the integrable condition and exact solutions to the time- where =(,)denotes the unknown function of the space dependent coefficient Gardner equations by the Painlevetest´ variable and time and the parameters , , , ] ∈ R are and Lie group analysis method. In [10–13], we developed arbitrary constants, ] ≥0and =0̸ . 2 Abstract and Applied Analysis We first note that (1) is the general form of the bond where (, ,, ) (, ,,and ) (, , ) are coefficient func- pricing types of equations [1–7]. In particular, if ] =0,then tions of the vector field to be determined. The symmetry this equation becomes the following Black-Scholes equation groups of (2)and(3) will be generated by the vector field of of option pricing: the form (5), respectively. Applying the second prolongation (2) +2 + + = 0, >0. pr of to (2)and(3), we find that the coefficient functions (2) , ,and must satisfy the following Lie symmetry condition: If ] =1,then(1) is the general bond pricing equation (2) (Δ) =0, given by pr Δ=0 (6) 2 + + +=0, >0. (3) 2 where Δ= + + +for (2)andΔ= 2 Such equations are called bond pricing types of equations, + + +for (3), respectively. Then, the Lie which are of great importance in financial mathematics and symmetry group calculation method leads to the following bond pricing theory [3–7]. For dealing with exact solutions conditions on the coefficient functions , ,and: to the variable-coefficients PDEs, we will introduce the 1 generalized power series method [10–13] in the present paper. = log +,( = ,) +(,) , By a generalized power series solution, we mean a generalized 2 (7) powerseriesisoftheform 1 − 1 = 2+ + +, ∞ log log log 8 4 2 () =() + ∑ , (4) =0 for some functions , ,and.Nowthefunctions, ,and which is a solution to a system with respect to the variable depend only on .Moreover,for(2), we have ,where ( = 0, 1, 2, .) . are constant coefficients to 1 2 1 1 be determined and () is the undetermined function with + + 8 log 2 log 4 respect to the variable .Inparticular,if(),then( ≡0 4)is (8) the regular power series solution. So, the generalized power ( − )2 −4 − − − + =0; series solution is the generalization of the regular power series 4 2 solutionanditnaturallyincludesthelatterasitsspecialcase. If we obtained a generalized power series solution (4)toa for (3), we have system and the convergence of this power series is shown, 1 1 then the exact generalized power series solution is obtained. 2+ This solution sometimes is called the exact analytic solution 8 log 2 log [10–13, 19]. 1 1 The main purpose of this paper is to develop the + +(+)+ 2 log 4 (9) combination of Lie symmetry analysis and the generalized power series method for dealing with symmetries and exact ( − )2 − solutions to the variable-coefficients PDEs in finance. The − − + =0. 4 2 remainder of this paper is organized as follows. In Section 2, we perform Lie symmetry analysis on the bond pricing types These equations fix the functions , , , ,and.Solvingthe of (2)and(3) and give all of the geometric vector fields of the equations, we obtain the vector field of2 ( ) as follows: equations in terms of the arbitrary parameters. In Section 3, we consider the symmetry reductions of the equations and 1 =,2 =,3 =, provide the exponentiated solutions and the similarity solu- tions to the equations. In Section 4, we investigate the exact 4 =2 +[log +(−)], analytic solutions to the variable-coefficient equations by the generalized power series method. In Section 5,wedealwith 5 =2(log ) +4 the vector fields and exact solutions to the bond pricing type 2 of (1) for the general case ] =0,1̸ . Finally, the conclusions +[(−)log +((−) −4)], and some remarks are given in Section 6. 2 (10) 6 =4(log ) +4 2. Lie Symmetry Analysis for and 2 (2) (3) +[2(−)log +log −2 In this section, we will present a complete list of all possible +((−)2 −4)2] , Liesymmetryalgebrasforthebondpricingtypesofequations of the forms (2)and(3). =, Recall that the geometric vector fields of such equations are as follows: where the parameters =0̸ , , are arbitrary constants and =(, , ) +(, , ) +(, , ) , (5) the function =(,)satisfies (2). Abstract and Applied Analysis 3 For (3), we have the vector field as follows: principles, but they are important for us to investigate the exact solutions to PDEs (see, e.g., [3–5, 10]). 1 =,2 =, =, (11) where the function =(,)satisfies (3). 3. Symmetry Reductions and Exact Solutions { ,...,,} Clearly, for (2), a basis of the Lie algebra is 1 6 . to the Bond Pricing Types of Equations For (3), a basis for the Lie algebra is {1,2,}.Thus,thenew symmetries cannot be derived from the Lie brackets for the In the preceding section, we obtained the symmetries and two equations. symmetry groups of (2)and(3). Now, we deal with the Moreover, we can obtain the one-parameter groups gen- symmetry reductions and exact solutions to the equations. erated by , respectively. In fact, for (2), the one-parameter groups generated by ( = 1,...,6,)aregiveninthe 3.1. The Exponentiated Solutions. Since each (= following: 1,...,6,) is a symmetry group, it implies that if =(,) () (= 1,...,6,) 1 : (, ,) → (, +) , , is a solution to (2), then are all solutions to the following equation as well: 2 : (, ,) → ( , , ), 3 : (, ,) →(,, ) , (1) =(, ) − , (13a) : (, ,) → ( 2,, [( − ) 2 + ]) , 4 exp (2) − =( , ) , (13b) 2 − (3) : (, ,) → ( , , [ (−1) = (,) , (13c) 5 exp 2 log (4) 2 −2 = exp [(−) − ] ( ,) , (13d) ( − )2 −4 + 4 − (5) = [ (1 − −1) exp 2 log ×(2 −1)]), 2 ( − ) −4 (13e) + (1 − −2)] 4 6 : (, ,) 1/ −2 ×( , ) , → ( 1/(1−4), , 1 − 4 (6) 1 − 1 2 = exp {[ log + log − √1 + 4 2 4 √1 − 4 {[ exp 2 log ( − )2 −4 4 1 + ] } + 2 4 1 + 4 4log 1/(1+4)
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