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

Abstract

The variable-coefficients partial differential equations (vc-PDEs) in finance are investigated by Lie symmetry analysis and the 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

Gazizov and Ibragimov [1] studied the Black-Scholes equation of option pricing by Lie equivalence transformations. By the optimal system method, some invariant solutions to heat and Black-Scholes equations are obtained [2]. In [3–5], the fundamental solutions to the bond pricing equations are considered by Lie symmetry analysis and the integral transform method. In [6], the invariance properties of the bond pricing equation are studied by the group classification method. In [7], the finite element method was adopted to solve the bond pricing type of PDE system, and the numerical implementation was provided, such as system that models the TF convertible bonds with credit risk in bond pricing theory. However, the similarity reductions and exact solutions to such variable-coefficient equations are not considered generally in the aforementioned papers. Recently, we studied some nonlinear PDEs by Lie symmetry analysis and the dynamical system method [8–13]; for example, in [8], we considered Lie group classifications and exact solutions to the space-dependent coefficients hanging chain equation and the simplified bond pricing equation. In [9], we investigated the integrable condition and exact solutions to the time-dependent coefficient Gardner equations by the Painlevé test and Lie group analysis method. In [10–13], we developed the generalized power series method for dealing with exact solutions to some nonlinear PDEs based on the symmetry analysis method.

It is known that the Lie symmetry analysis is a systematic and powerful method for dealing with symmetries and exact solutions to partial differential equations (see, e.g., [1–6, 8–18] and the references therein). Furthermore, we find that the combination of Lie symmetry analysis and the power series method is a feasible approach to investigating exact solutions to nonlinear PDEs [8–13]. On the other hand, under the perspective of mathematical physics and Lie symmetry analysis, the space-time dependent coefficients system differs greatly from its time-dependent counterpart, and it is more complicated than the latter. However, most of the studies are related to the time-dependent coefficient systems. Moreover, the determination of exact solutions to the variable-coefficients PDEs is a complicated problem that challenges researchers greatly. In the present paper, we consider the symmetry reductions and exact solutions to the general space-dependent coefficients PDEs in finance as follows: 𝑢_𝑡+𝛼𝑥²𝑢_𝑥𝑥+𝛽𝑥𝑢_𝑥+𝛾𝑥^𝜈𝑢=0,𝑥>0, (1) where 𝑢=𝑢(𝑥,𝑡) denotes the unknown function of the space variable 𝑥 and time 𝑡 and the parameters 𝛼,𝛽,𝛾,𝜈∈ℝ are arbitrary constants, 𝜈≥0 and 𝛼≠0.

We first note that (1) is the general form of the bond pricing types of equations [1–7]. In particular, if 𝜈=0, then this equation becomes the following Black-Scholes equation of option pricing: 𝑢_𝑡+𝛼𝑥²𝑢_𝑥𝑥+𝛽𝑥𝑢_𝑥+𝛾𝑢=0,𝑥>0. (2)

If 𝜈=1, then (1) is the general bond pricing equation given by 𝑢_𝑡+𝛼𝑥²𝑢_𝑥𝑥+𝛽𝑥𝑢_𝑥+𝛾𝑥𝑢=0,𝑥>0. (3)

Such equations are called bond pricing types of equations, which are of great importance in financial mathematics and bond pricing theory [3–7]. For dealing with exact solutions to the variable-coefficients PDEs, we will introduce the generalized power series method [10–13] in the present paper. By a generalized power series solution, we mean a generalized power series is of the form 𝑓(𝜉)=𝐴(𝜉)+^∞_𝑛=0𝑐_𝑛𝜉^𝑛, (4) which is a solution to a system with respect to the variable 𝜉, where 𝑐_𝑛 (𝑛=0,1,2,…) are constant coefficients to be determined and 𝐴(𝜉) is the undetermined function with respect to the variable 𝜉. In particular, if 𝐴(𝜉)≡0, then (4) is the regular power series solution. So, the generalized power series solution is the generalization of the regular power series solution and it naturally includes the latter as its special case. If we obtained a generalized power series solution (4) to a system and the convergence of this power series is shown, then the exact generalized power series solution is obtained. This solution sometimes is called the exact analytic solution [10–13, 19].

The main purpose of this paper is to develop the combination of Lie symmetry analysis and the generalized power series method for dealing with symmetries and exact solutions to the variable-coefficients PDEs in finance. The remainder of this paper is organized as follows. In Section 2, we perform Lie symmetry analysis on the bond pricing types of (2) and (3) and give all of the geometric vector fields of the equations in terms of the arbitrary parameters. In Section 3, we consider the symmetry reductions of the equations and provide the exponentiated solutions and the similarity solutions to the equations. In Section 4, we investigate the exact analytic solutions to the variable-coefficient equations by the generalized power series method. In Section 5, we deal with the vector fields and exact solutions to the bond pricing type of (1) for the general case 𝜈≠0,1. Finally, the conclusions and some remarks are given in Section 6.

2. Lie Symmetry Analysis for (2) and (3)

In this section, we will present a complete list of all possible Lie symmetry algebras for the bond pricing types of equations of the forms (2) and (3).

Recall that the geometric vector fields of such equations are as follows: 𝑉=𝜉(𝑥,𝑡,𝑢)𝜕_𝑥+𝜏(𝑥,𝑡,𝑢)𝜕_𝑡+𝜙(𝑥,𝑡,𝑢)𝜕_𝑢, (5) where 𝜉(𝑥,𝑡,𝑢), 𝜏(𝑥,𝑡,𝑢), and 𝜙(𝑥,𝑡,𝑢) are coefficient functions of the vector field to be determined. The symmetry groups of (2) and (3) will be generated by the vector field of the form (5), respectively. Applying the second prolongation pr⁽²⁾𝑉 of 𝑉 to (2) and (3), we find that the coefficient functions 𝜉, 𝜏, and 𝜙 must satisfy the following Lie symmetry condition: pr⁽²⁾𝑉(Δ)_Δ=0=0, (6) where Δ=𝑢_𝑡+𝛼𝑥²𝑢_𝑥𝑥+𝛽𝑥𝑢_𝑥+𝛾𝑢 for (2) and Δ=𝑢_𝑡+𝛼𝑥²𝑢_𝑥𝑥+𝛽𝑥𝑢_𝑥+𝛾𝑥𝑢 for (3), respectively. Then, the Lie symmetry group calculation method leads to the following conditions on the coefficient functions 𝜉, 𝜏, and 𝜙: 1𝜉=2𝑥𝜏_𝑡()(),1log𝑥+𝑥𝜌,𝜙=𝑟𝑥,𝑡𝑢+𝑠𝑥,𝑡𝑟=𝜏8𝛼^𝑡𝑡log²𝑥+𝛼−𝛽𝜏4𝛼^𝑡1log𝑥+𝜌2𝛼^𝑡log𝑥+𝜎, (7) for some functions 𝜏, 𝜌, and 𝜎. Now the functions 𝜏, 𝜌, and 𝜎 depend only on 𝑡. Moreover, for (2), we have 1𝜏8𝛼^𝑡𝑡𝑡log²1𝑥+𝜌2𝛼^𝑡𝑡1log𝑥+4𝜏_𝑡𝑡−𝛼−𝛽²−4𝛼𝛾𝜏4𝛼^𝑡−𝛼−𝛽𝜌2𝛼^𝑡+𝜎_𝑡=0; (8) for (3), we have 1𝜏8𝛼^𝑡𝑡𝑡log²1𝑥+2𝛾𝜏_𝑡+1𝑥log𝑥𝜌2𝛼^𝑡𝑡log𝑥+𝛾𝜌+𝜏_𝑡1𝑥+4𝜏_𝑡𝑡−𝛼−𝛽²𝜏4𝛼^𝑡−𝛼−𝛽𝜌2𝛼^𝑡+𝜎_𝑡=0. (9) These equations fix the functions 𝜉, 𝜏, 𝜌, 𝜎, and 𝜙. Solving the equations, we obtain the vector field of (2) as follows: 𝑉₁=𝜕_𝑡,𝑉₂=𝑥𝜕_𝑥,𝑉₃=𝑢𝜕_𝑢,𝑉₄=2𝛼𝑥𝑡𝜕_𝑥+log𝑥+𝛼−𝛽𝑡𝑢𝜕_𝑢,𝑉₅=2𝛼𝑥𝑡log𝑥𝜕_𝑥+4𝛼𝑡𝜕_𝑡+𝛼−𝛽log𝑥+𝛼−𝛽²−4𝛼𝛾𝑡𝑢𝜕_𝑢,𝑉₆=4𝛼𝑥𝑡log𝑥𝜕_𝑥+4𝛼𝑡²𝜕_𝑡+2𝛼−𝛽𝑡log𝑥+log²+𝑥−2𝛼𝑡𝛼−𝛽²−4𝛼𝛾𝑡²𝑢𝜕_𝑢,𝑉_𝑠=𝑠𝜕_𝑢, (10) where the parameters 𝛼≠0, 𝛽, 𝛾 are arbitrary constants and the function 𝑠=𝑠(𝑥,𝑡) satisfies (2).

For (3), we have the vector field as follows: 𝑉₁=𝜕_𝑡,𝑉₂=𝑢𝜕_𝑢,𝑉_𝑠=𝑠𝜕_𝑢, (11) where the function 𝑠=𝑠(𝑥,𝑡) satisfies (3).

Clearly, for (2), a basis of the Lie algebra is {𝑉₁,…,𝑉₆,𝑉_𝑠}. For (3), a basis for the Lie algebra is {𝑉₁,𝑉₂,𝑉_𝑠}. Thus, the new symmetries cannot be derived from the Lie brackets for the two equations.

Moreover, we can obtain the one-parameter groups generated by 𝑉_𝑖, respectively. In fact, for (2), the one-parameter groups 𝐺_𝑖 generated by 𝑉_𝑖 (𝑖=1,…,6,𝑠) are given in the following: 𝐺₁∶()⟶(),𝐺𝑥,𝑡,𝑢𝑥,𝑡+𝜖,𝑢₂∶(𝑥,𝑡,𝑢)⟶𝑒^𝜖,𝐺𝑥,𝑡,𝑢₃∶(𝑥,𝑡,𝑢)⟶𝑥,𝑡,𝑒^𝜖𝑢,𝐺₄∶(𝑥,𝑡,𝑢)⟶𝑥𝑒^{2𝛼𝜖𝑡},𝑡,𝑥^𝜖𝑢exp𝛼−𝛽𝜖𝑡+𝛼𝜖²𝑡,𝐺₅∶(𝑥,𝑡,𝑢)⟶𝑥^𝛿,𝛿²𝑡,𝑢exp𝛼−𝛽()+2𝛼𝛿−1log𝑥𝛼−𝛽²−4𝛼𝛾×𝛿4𝛼₂𝑡,𝐺−1₆∶()⟶𝑥𝑥,𝑡,𝑢_{1/(1−4𝛼𝜖𝑡)},𝑡,𝑢√1−4𝛼𝜖𝑡1−4𝛼𝜖𝑡exp𝛼−𝛽+12𝛼log𝑥4𝛼𝑡log²𝑥+𝛼−𝛽²−4𝛼𝛾𝑡×4𝛼4𝛼𝜖𝑡,𝐺1−4𝛼𝜖𝑡_𝑠∶(𝑥,𝑡,𝑢)⟶(𝑥,𝑡,𝑢+𝜖𝑠), (12) where 𝛿=𝑒^2𝛼𝜖, 𝜖≪1, and the function 𝑠=𝑠(𝑥,𝑡) is an arbitrary solution to (2). For (3), the one-parameter groups are 𝐺_𝑖 (𝑖=1,3,𝑠) as above, while 𝑠=𝑠(𝑥,𝑡) is an arbitrary solution to (3).

From the above, we observe that 𝐺₁ is a time translation and 𝐺₂ and 𝐺₃ are trivial scaling transformations, while 𝐺_𝑖 (𝑖=4,5,6) are nontrivial local groups of transformations. Their appearances are far from obvious from basic physical principles, but they are important for us to investigate the exact solutions to PDEs (see, e.g., [3–5, 10]).

3. Symmetry Reductions and Exact Solutions to the Bond Pricing Types of Equations

In the preceding section, we obtained the symmetries and symmetry groups of (2) and (3). Now, we deal with the symmetry reductions and exact solutions to the equations.

3.1. The Exponentiated Solutions

Since each 𝐺_𝑖 (𝑖=1,…,6,𝑠) is a symmetry group, it implies that if 𝑢=𝑓(𝑥,𝑡) is a solution to (2), then 𝑢^(𝑖)(𝑖=1,…,6,𝑠) are all solutions to the following equation as well: 𝑢⁽¹⁾=𝑓(𝑥,𝑡−𝜖),𝑢⁽²⁾=𝑓𝑒^−𝜖𝑥,𝑡,𝑢⁽³⁾=𝑒^𝜖𝑓(𝑥,𝑡),𝑢⁽⁴⁾=𝑥^𝜖exp𝛼−𝛽𝜖𝑡−𝛼𝜖²𝑡𝑓𝑒^{−2𝛼𝜖𝑡}𝑥,𝑡,𝑢⁽⁵⁾=exp𝛼−𝛽2𝛼1−𝛿⁻¹+log𝑥𝛼−𝛽²−4𝛼𝛾4𝛼1−𝛿⁻²𝑡×𝑓𝑥^1/𝛿,𝛿⁻²𝑡,𝑢⁽⁶⁾=1√1+4𝛼𝜖𝑡exp𝛼−𝛽12𝛼log𝑥+4𝛼𝑡log²𝑥+𝛼−𝛽²−4𝛼𝛾𝑡4𝛼4𝛼𝜖𝑡𝑥1+4𝛼𝜖𝑡×𝑓^{1/(1+4𝛼𝜖𝑡)},𝑡,1+4𝛼𝜖𝑡𝑢^(𝑠)=𝑓(𝑥,𝑡)+𝜖𝑠, (13a)(13b)(13c)(13d)(13e)(13f)13g

where 𝛿=𝑒^2𝛼𝜖, 𝜖 is an arbitrary real number, and the function 𝑠=𝑠(𝑥,𝑡) satisfies (2).

For (3), the exponentiated solutions are 𝑢^(𝑖) (𝑖=1,3,𝑠) as above while 𝑠=𝑠(𝑥,𝑡) satisfies (3).

Such exponentiated solutions are one of group-invariant types of solutions to the PDEs, which are generated from the one-parameter groups and are of importance for studying the exact solutions and investigating the properties of solutions (see Remark 2).

Next, we investigate the symmetry reductions and exact explicit solutions to the two bond pricing equations. Firstly, we consider (2).

3.2. Similarity Solution for 𝑉₁

For the generator 𝑉₁, we have the following reduced ordinary differential equation (ODE): 𝛼𝜉²𝑓^′′+𝛽𝜉𝑓^′+𝛾𝑓=0, (14) where 𝑓^′=𝑑𝑓/𝑑𝜉. This is an Euler equation; the corresponding characteristic equation is 𝛼𝐾²−(𝛼−𝛽)𝐾+𝛾=0. Solving this equation, we have 𝐾=((𝛼−𝛽)±√Δ)/2𝛼, where Δ=(𝛼−𝛽)²−4𝛼𝛾.

When Δ>0, (14) has the general solution 𝑓=𝑐₁𝜉^𝐾1+𝑐₂𝜉^𝐾2. Thus, we obtain the exact solution to (2) as follows: 𝑢(𝑥,𝑡)=𝑐₁𝑥^𝐾1+𝑐₂𝑥^𝐾2, (15) where 𝑐₁ and 𝑐₂ are arbitrary constants and 𝐾_1,2=((𝛼−𝛽)±√Δ)/2𝛼 are two real roots to the characteristic equation, respectively.

When Δ=0, (14) has the general solution 𝑓=𝜉^𝐾(𝑐₁+𝑐₂log𝜉). Thus, we obtain the exact solution to (2) as follows: 𝑢(𝑥,𝑡)=𝑥^𝐾𝑐₁+𝑐₂log𝑥, (16) where 𝑐₁ and 𝑐₂ are arbitrary constants, 𝐾=(𝛼−𝛽)/2𝛼 are the real root to the characteristic equation.

When Δ<0, (14) has the general solution 𝑓=𝜉^𝐾(𝑐₁cos(√−Δ/2𝛼)log𝜉+𝑐₂sin(√−Δ/2𝛼)log𝜉). Thus, we obtain the exact solution to (2) as follows: 𝑢(𝑥,𝑡)=𝑥^𝐾𝑐₁√cos−Δ2𝛼log𝑥+𝑐₂√sin−Δ,2𝛼log𝑥 (17) where 𝑐₁ and 𝑐₂ are arbitrary constants, 𝐾=(𝛼−𝛽)/2𝛼.

3.3. Similarity Solution for 𝑉₂

For the generator 𝑉₂, we have the following reduced ODE: 𝑓^′+𝛾𝑓=0, (18) where 𝑓^′=𝑑𝑓/𝑑𝜉. Solving this equation, we have 𝑓=𝑐𝑒^−𝛾𝜉. Thus, we obtain the exact solution to (2) as follows: 𝑢(𝑥,𝑡)=𝑐𝑒^−𝛾𝑡, (19) where 𝑐 is an arbitrary constant.

3.4. Similarity Solution for 𝑉₄

For the generator 𝑉₄, we have the following similarity transformation: 1𝜉=𝑡,𝜔=log𝑢−4𝛼𝑡log𝑥+𝑎𝑡², (20) and the similarity solution is 𝜔=𝑓(𝜉); that is, 𝑓(𝑡)+1𝑢=exp4𝛼𝑡log𝑥+𝑎𝑡². (21) Substituting (21) into (2), we reduce the bond pricing equation to the following ODE: 2𝜉𝑓^′+2𝛾𝜉+1=0, (22) where 𝑓^′=𝑑𝑓/𝑑𝜉. It implies that if 𝜔=𝑓(𝜉) is a solution to (22), then (21) is a solution to (2). Solving (22), we get 𝑓(𝜉)=−(1/2)log𝜉−𝛾𝜉+𝑐₁. Thus, we obtain the solution to (2) as follows: 𝑢()1𝑥,𝑡=𝑐exp4𝛼𝑡log𝑥+𝑎𝑡²−12log𝑡−𝛾𝑡, (23) where 𝑐 is an arbitrary constant.

3.5. Similarity Solution for 𝑉₅

For the generator 𝑉₅, we have the following similarity transformation: 𝜉=𝑡^−1/2log𝑥,𝜔=log𝑢−𝛼−𝛽2𝛼log𝑥−𝛼−𝛽²−4𝛼𝛾4𝛼𝑡, (24) and the similarity solution is 𝜔=𝑓(𝜉); that is, 𝑢=exp𝑓𝑡^−1/2log𝑥+𝛼−𝛽2𝛼log𝑥+𝛼−𝛽²−4𝛼𝛾𝑡.4𝛼 (25) Substituting (25) into (2), we reduce the bond pricing equation to the following ODE: 𝛼𝑓^′′+𝛼𝑓^′2−12𝜉𝑓^′=0, (26) where 𝑓^′=𝑑𝑓/𝑑𝜉.

Letting 𝑓^′=𝑦, we get the Bernoulli equation 𝑑𝑦=1𝑑𝜉2𝛼𝜉𝑦−𝛼𝑦². (27) Clearly, 𝑦=0; that is, 𝑓=𝑐 is a solution to (26). Thus, we get a solution to (2) as follows: 𝑢(𝑥,𝑡)=exp𝛼−𝛽2𝛼log𝑥+𝛼−𝛽²−4𝛼𝛾,4𝛼𝑡+𝑐 (28) for an arbitrary constant number 𝑐.

When 𝑦≠0, solving the Bernoulli equation, we get 𝑦=𝑒^{(1/4𝛼)𝜉2}/(∫𝑒^{(1/4𝛼)𝜉2}𝑑𝜉+𝑐₁). Thus, we obtain the solution to (26) as follows: 𝑓(𝜉)=𝑒^{(1/4𝛼)𝜉2}∫𝑒^{(1/4𝛼)𝜉2}𝑑𝜉+𝑐₁𝑑𝜉+𝑐₂, (29) where 𝑐₁ and 𝑐₂ are constants of integration. Substituting (29) into (25), we obtain the exact solution to (2) immediately.

3.6. Similarity Solution for 𝑉₆

For the generator 𝑉₆, we have the following similarity transformation: 𝜉=𝑡⁻¹1log𝑥,𝜔=log𝑢+2log𝑡−𝛼−𝛽−2𝛼log𝑥𝛼−𝛽²−4𝛼𝛾14𝛼𝑡−𝑡4𝛼⁻¹log²𝑥, (30) and the similarity solution is 𝜔=𝑓(𝜉); that is, 𝑢=exp𝑓𝑡⁻¹−1log𝑥2log𝑡+𝛼−𝛽+2𝛼log𝑥𝛼−𝛽²−4𝛼𝛾14𝛼𝑡+𝑡4𝛼⁻¹log²𝑥. (31) Substituting (31) into (2), we reduce the bond pricing equation to the following ODE: 𝑓^′′+𝑓^′2=0, (32) where 𝑓^′=𝑑𝑓/𝑑𝜉.

Solving (32), we get 𝑓(𝜉)=log|𝜉+𝑐₁|+𝑐₃. Thus, we obtain the solution to (2) as follows: 𝑢(𝑥,𝑡)=𝑐₂1𝑡log𝑥+𝑐₁×exp𝛼−𝛽2𝛼log𝑥+𝛼−𝛽²−4𝛼𝛾𝑡+14𝛼𝑡4𝛼⁻¹log²1𝑥−2log𝑡, (33) where 𝑐₁, 𝑐₂ are arbitrary constants.

3.7. Similarity Solution for 𝑉₁+𝑣𝑉₂

For the linear combination 𝑉=𝑉₁+𝑣𝑉₂ (𝑣≠0 is an arbitrary constant), we have the following similarity transformation: 𝜉=log𝑥−𝑣𝑡,𝜔=𝑢, (34) and the similarity solution is 𝜔=𝑓(𝜉); that is, 𝑢=𝑓log𝑥−𝑣𝑡. (35) Substituting (35) into (2), we reduce the bond pricing equation to the following ODE: 𝛼𝑓^′′−𝑣+𝛼−𝛽𝑓^′+𝛾𝑓=0, (36) where 𝑓^′=𝑑𝑓/𝑑𝜉.

This is a second-order linear ODE; the corresponding characteristic equation is 𝛼𝜆²−(𝑣+𝛼−𝛽)𝜆+𝛾=0. Solving the algebraic equation, we have 𝜆₁=(𝑣+𝛼−𝛽+√Δ)/2𝛼, 𝜆₂=(𝑣+𝛼−𝛽−√Δ)/2𝛼, where Δ=(𝑣+𝛼−𝛽)²−4𝛼𝛾.

When Δ>0, (36) has the solution 𝑓(𝜉)=𝑐₁𝑒^𝜆1𝜉+𝑐₂𝑒^𝜆2𝜉. Thus, we obtain the solution to (2) as follows: 𝑢(𝑥,𝑡)=𝑐₁𝑥^𝜆1𝑒^{−𝜆1𝑣𝑡}+𝑐₂𝑥^𝜆2𝑒^{−𝜆2𝑣𝑡}, (37) where 𝑐₁, 𝑐₂ are arbitrary constants.

When Δ=0, (36) has the solution 𝑓(𝜉)=(𝑐₁+𝑐₂𝜉)𝑒^𝜆𝜉, where 𝜆=(𝑣+𝛼−𝛽)/2𝛼. Thus, we obtain the solution to (2) as follows: 𝑢(𝑥,𝑡)=𝑐₁+𝑐₂log𝑥−𝑣𝑡𝑥^𝜆𝑒^{−𝜆𝑣𝑡}, (38) where 𝑐₁, 𝑐₂ are arbitrary constants.

When Δ<0, (36) has the solution 𝑓(𝜉)=(𝑐₁cos(√−Δ/2𝛼)𝜉+𝑐₂sin(√−Δ/2𝛼)𝜉)𝑒^{((𝑣+𝛼−𝛽)/2𝛼)𝜉}. Thus, we obtain the solution to (2) as follows: 𝑢(𝑥,𝑡)=𝑥^{(𝑣+𝛼−𝛽)/2𝛼}𝑒^{−(𝑣(𝑣+𝛼−𝛽)/2𝛼)𝑡}×𝑐₁√cos−Δ2𝛼log𝑥−𝑣𝑡+𝑐₂√sin−Δ,2𝛼log𝑥−𝑣𝑡 (39) where 𝑐₁, 𝑐₂ are arbitrary constants.

3.8. Similarity Reduction for 𝑉₁+𝑣𝑉₃

For the linear combination 𝑉=𝑉₁+𝑣𝑉₃ (𝑣≠0 is an arbitrary constant), we have the following similarity transformation: 𝜉=𝑥,𝜔=log𝑢−𝑣𝑡, (40) and the similarity solution is 𝜔=𝑓(𝜉); that is, 𝑢=exp𝑓(𝑥)+𝑣𝑡. (41) Substituting (41) into (2), we reduce the bond pricing equation to the following ODE: 𝛼𝜉²𝑓^′′+𝛼𝜉²𝑓^′2+𝛽𝜉𝑓^′+𝑣+𝛾=0, (42) where 𝑓^′=𝑑𝑓/𝑑𝜉. This is a nonlinear second-order ODE. In the next section, we will deal with such an equation by the special transformation technique.

3.9. Similarity Solution for 𝑉₂+𝑣𝑉₃

For the linear combination 𝑉=𝑉₂+𝑣𝑉₃ (𝑣≠0 is an arbitrary constant), we have the following similarity transformation: 𝜉=𝑡,𝜔=𝑥^−𝑣𝑢, (43) and the similarity solution is 𝜔=𝑓(𝜉); that is, 𝑢=𝑥^𝑣𝑓(𝑡). (44) Substituting (44) into (2), we reduce the bond pricing equation to the following ODE: 𝑓^′+𝛼𝑣(𝑣−1)𝑓+𝛽𝑣𝑓+𝛾𝑓=0, (45) where 𝑓^′=𝑑𝑓/𝑑𝜉.

Solving (45), we get 𝑓=𝑐exp{−[𝛼𝑣²−(𝛼−𝛽)𝑣+𝛾]𝜉}. Thus, we obtain the solution to (2) as follows 𝑢(𝑥,𝑡)=𝑐𝑥^𝑣exp−𝛼𝑣²−𝛼−𝛽𝑣+𝛾𝑡, (46) where 𝑐 is an arbitrary constant.

3.10. Similarity Reduction for 𝑉₁+𝑣𝑉₄

For the linear combination 𝑉=𝑉₁+𝑣𝑉₄ (𝑣≠0 is an arbitrary constant), we have the following similarity transformation: 𝜉=log𝑥−𝑣𝛼𝑡²,2𝜔=log𝑢−𝑣𝑡log𝑥+3𝛼𝑣²𝑡³−12𝛼−𝛽𝑣𝑡², (47) and the similarity solution is 𝜔=𝑓(𝜉); that is, 𝑢=exp𝑓log𝑥−𝑣𝛼𝑡²2+𝑣𝑡log𝑥−3𝛼𝑣²𝑡³+12𝛼−𝛽𝑣𝑡². (48) Substituting (48) into (2), we reduce the bond pricing equation to the following ODE: 𝛼𝑓^′′+𝛼𝑓^′2−𝛼−𝛽𝑓^′+𝑣𝜉+𝛾=0, (49) where 𝑓^′=𝑑𝑓/𝑑𝜉. This is a nonlinear second-order ODE also. In the next section, we will deal with the exact solutions to such equations.

Secondly, we consider (3). In fact, for this equation, we have the nontrivial cases as follows only.

3.11. Similarity Reduction for 𝑉₁ of (3)

For the generator 𝑉₁, we have the following reduced ordinary differential equation (ODE): 𝛼𝜉𝑓^′′+𝛽𝑓^′+𝛾𝑓=0, (50) where 𝑓^′=𝑑𝑓/𝑑𝜉. This is a nonlinear second-order ODE as well; there is no general method for tackling it yet. In Section 4, we will deal with such equations by the power series method.

3.12. Similarity Reduction for 𝑉₁+𝑣𝑉₂ of (3)

For the linear combination 𝑉=𝑉₁+𝑣𝑉₂ (𝑣≠0 is an arbitrary constant), we have the following similarity transformation: 𝜉=𝑥,𝜔=log𝑢−𝑣𝑡, (51) and the similarity solution is 𝜔=𝑓(𝜉); that is, 𝑢=exp𝑓(𝑥)+𝑣𝑡. (52) Substituting (52) into (3), we reduce the second bond pricing equation to the following ODE: 𝛼𝜉²𝑓^′′+𝛼𝜉²𝑓^′2+𝛽𝜉𝑓^′+𝛾𝜉+𝑣=0, (53)

where 𝑓^′=𝑑𝑓/𝑑𝜉. This is a nonlinear second-order ODE also. Similar to the above equations, we will deal with such equations by the generalized power series method in the next section.

4. Exact Analytic Solutions in terms of the Generalized Power Series Method

In Section 3, we considered the symmetry reductions and exact solutions to the bond pricing types of (2) and (3). In this section, we will deal with the nonlinear ODEs (42), (49), (50), and (53) by the special transformation technique and generalized power series method. Thus, the exact analytic solutions to (2) and (3) are obtained.

4.1. Exact Solution to (2)

Firstly, we consider the ODE (42). Letting 𝑓^′=𝑦, we get the Riccati equation 𝑑𝑦𝑑𝜉=−𝑦²−𝛽𝛼𝜉𝑦−𝑣+𝛾𝛼𝜉². (54) Now, we solve the equation by the transformation technique directly. Suppose that (54) has the solution of the form 𝑦=𝑝𝜉⁻¹, (55) where 𝑝 is a constant to be determined. Substituting (55) into (54), we have 𝛼𝑝²−(𝛼−𝛽)𝑝+𝑣+𝛾=0. Solving the algebraic equation, we get ±√𝑝=𝛼−𝛽Δ,2𝛼 (56) where Δ=(𝛼−𝛽)²−4𝛼(𝑣+𝛾).

Setting 𝑦=𝑧+𝑝𝜉⁻¹ and plugging it into (54), we get 𝑑𝑧𝑑𝜉=−𝑧²𝑧−𝑞𝜉𝛽,𝑞=2𝑝+𝛼. (57) This is a Bernoulli equation. Solving the equation, we have the following results.

When 𝑞=1, we get 𝑓(𝜉)=𝑝log𝜉+log(log𝜉+𝑐₁)+𝑐₂. Thus, the exact solution to (2) is 𝑢(𝑥,𝑡)=𝑐₂𝑥^𝑝log𝑥+𝑐₁𝑒^𝑣𝑡, (58) where 𝑐₁ and 𝑐₂ are arbitrary constants; 𝑝 and 𝑞 are given by (56) and (57).

When 𝑞≠1, we get 𝑓(𝜉)=𝑝log𝜉+(1−𝑞)∫𝑑𝜉/(𝜉+𝑐₁𝜉^𝑞)+𝑐₂. Thus, the exact solution to (2) is 𝑢(𝑥,𝑡)=𝑐₂𝑥^𝑝exp1−𝑞𝑑𝑥𝑥+𝑐₁𝑥^𝑞+𝑣𝑡, (59) where 𝑐₁ and 𝑐₂ are arbitrary constants and 𝑝 and 𝑞 are given by (56) and (57).

4.2. Exact Analytic Solution to (2)

Through the transformation technique, we solve the Riccati equation (54), so the exact solutions to (2) are obtained. But for the other equations such as (49), (50), and (53), we cannot get the exact solutions by such special transformation technique. However, we know that the power series can be used to solve nonlinear ODEs, including many complicated differential equations with nonconstant coefficients [10–13, 19, 20]. Now, we consider the power series solution to the reduced equation (49). Letting 𝑓^′=𝑦, we get the following Riccati equation: 𝛼𝑦^′+𝛼𝑦²−𝛼−𝛽𝑦+𝑣𝜉+𝛾=0. (60)

We will seek a solution of (60) in a power series of the form 𝑦=^∞_𝑛=0𝑐_𝑛𝜉^𝑛=𝑝+^∞_𝑛=1𝑐_𝑛𝜉^𝑛,𝑝=𝑐₀, (61) where the coefficients 𝑐_𝑛 (𝑛=0,1,2,…) are constants to be determined.

Substituting (61) into (60) and comparing coefficients, we obtain 𝑐₁=−𝑝²+𝛼−𝛽𝛼𝛾𝑝−𝛼,𝑐₂=−𝑝𝑐₁+𝛼−𝛽𝑐2𝛼¹−𝑣.2𝛼 (62) Generally, for 𝑛≥2, we have 𝑐_𝑛+1=1()𝛼𝑐𝑛+1𝛼−𝛽_𝑛−𝛼^𝑛_𝑘=0𝑐_𝑘𝑐_𝑛−𝑘. (63)

Thus, for arbitrarily choosing the parameter 𝑐₀, from (62), we can get 𝑐₁ and 𝑐₂. Furthermore, in view of (63), we have 𝑐₃=𝛼−𝛽𝑐3𝛼²−132𝑝𝑐₂+𝑐²₁,𝑐₄=𝛼−𝛽𝑐4𝛼³−12𝑝𝑐₃+𝑐₁𝑐₂, (64) and so on.

Therefore, the other terms of the sequence {𝑐_𝑛}^∞_𝑛=0 can be determined successively from (63) in a unique manner. This implies that for (60) there exists a power series solution (61) with the coefficients given by (62) and (63). Furthermore, we can show the convergence of the power series solution (61) with the coefficients given by (62) and (63) (see, e.g., [10, 12, 13, 19]); the details are omitted here. So, this solution (61) to (60) is an exact analytic solution.

Hence, the exact power series solution to (49) can be written as follows: 𝑓(𝜉)1=𝑐+𝑝𝜉+2𝑐₁𝜉²+13𝑐₂𝜉³+^∞_𝑛=21𝑐𝑛+2^𝑛+1𝜉^𝑛+2. (65)

Substituting (65) into (39), we obtain the exact analytic solution to (2) as follows: 𝑢(𝑥,𝑡)=𝑞exp𝑝log𝑥−𝛼𝑣𝑡²+12𝑐₁log𝑥−𝛼𝑣𝑡²²+13𝑐₂log𝑥−𝛼𝑣𝑡²³+^∞_𝑛=21𝑐𝑛+2^𝑛+1log𝑥−𝛼𝑣𝑡^2𝑛+2+12𝛼−𝛽𝑣𝑡²−23𝛼𝑣²𝑡³+𝑣𝑡log𝑥, (66) where 𝑝=𝑐₀ and 𝑞 are arbitrary constants and the other coefficients 𝑐_𝑛 (𝑛=1,2,…) are given by (62) and (63) successively.

Similarly, we can give the exact power series solution to (50) in the power series form (61). So, the exact analytic solution to (3) is obtained. The details are omitted here.

4.3. Exact Analytic Solution to (3)

In Section 4.2, we construct the exact analytic solution to (49) by the power series method and obtain the exact analytic solution to (2). Now, we consider (53). Firstly, let 𝑓^′=𝑦; then we get the following Riccati type of equation: 𝛼𝜉²𝑦^′+𝛼𝜉²𝑦²+𝛽𝜉𝑦+𝛾𝜉+𝑣=0. (67)

We will seek a solution of (67) in a generalized power series of the form 𝑦=𝐴𝜉⁻¹+^∞_𝑛=0𝑐_𝑛𝜉^𝑛, (68) where the parameters 𝐴 and 𝑐_𝑛 (𝑛=0,1,2,…) are constants to be determined.

Substituting (68) into (67) and comparing coefficients, we obtain ±√𝐴=𝛼−𝛽Δ,2𝛼 (69) where Δ=(𝛼−𝛽)²−4𝛼𝑣, and 𝑐₀=−𝛾2𝛼𝐴+𝛽,2𝛼𝐴+𝛽≠0. (70)

Generally, for 𝑛≥0, we have 𝑐_𝑛+1=−𝛼(𝑛+1)𝛼+2𝛼𝐴+𝛽^𝑛_𝑘=0𝑐_𝑘𝑐_𝑛−𝑘,𝑛=0,1,2,…. (71)

Thus, from (69) and (70), we can get 𝐴 and 𝑐₀. Furthermore, in view of (71), we have 𝑐₁=−𝛼𝑐²₀𝛼+2𝛼𝐴+𝛽,𝑐₂=−2𝛼𝑐₀𝑐₁,𝑐2𝛼+2𝛼𝐴+𝛽₃=−𝛼2𝑐₀𝑐₂+𝑐²₁,3𝛼+2𝛼𝐴+𝛽 (72) and so on (see Remark 3).

Therefore, the other terms of the sequence {𝑐_𝑛}^∞_𝑛=0 can be determined successively from (71) in a unique manner. This implies that for (67) there exists a generalized power series solution (68) with the coefficients given by (69)–(71). The convergence of the generalized power series solution (68) to (67) is similar to that in Section 4.2; we omit it in this paper. Thus, the power series solution (68) to (67) is also an exact analytic solution.

Hence, the power series solution of (53) can be written as follows: 𝑓(𝜉)=𝑐+𝐴log𝜉+𝑐₀1𝜉+2𝑐₁𝜉²+^∞_𝑛=11𝑐𝑛+2^𝑛+1𝜉^𝑛+2. (73)

Substituting (73) into (52), we get the exact analytic solution to (3) as follows: 𝑢(𝑥,𝑡)=𝑐𝑥^𝐴exp𝑐₀1𝑥+2𝑐₁𝑥²+^∞_𝑛=11𝑐𝑛+2^𝑛+1𝑥^𝑛+2+𝑣𝑡, (74) where 𝑐 is an arbitrary constant and 𝐴 and 𝑐_𝑛 (𝑛=0,1,2,…) are given by (69)–(71) successively.

Remark 1. We note that the generalized power series solution (68) differs from the regular form (61) since 𝐴≠0 in (68). In other words, there is no exact power series solution of the form (61) for (67). In particular, the determination of parameter 𝐴 depends on the equation greatly (cf. [10, 11] for details).

5. Further Discussion about the General Bond Pricing Type of (1)

In the above sections, we considered the symmetries, symmetry reductions, and exact solutions to the general bond pricing type of equation for the cases 𝜈=0 and 𝜈=1, which are the common forms in many practical applications, such as in financial mathematics. In this section, we discuss the generalized bond pricing type of equation of the form 𝑢_𝑡+𝛼𝑥²𝑢_𝑥𝑥+𝛽𝑥𝑢_𝑥+𝛾𝑥^𝜈𝑢=0, (75) where 𝜈≠0,1 is an arbitrary positive number. Firstly, by the group classification method, we get the geometric vector field of (75) as follows: 𝑉₁=𝜕_𝑡,𝑉₂=𝑢𝜕_𝑢,𝑉_𝑠=𝑠𝜕_𝑢, (76) where the function 𝑠=𝑠(𝑥,𝑡) satisfies (75).

Moreover, through the similarity transformation (42), we can reduce this equation to the following equation (ODE): 𝛼𝜉²𝑓^′′+𝛼𝜉²𝑓^′2+𝛽𝜉𝑓^′+𝛾𝜉^𝜈+𝑣=0, (77) where 𝑓^′=𝑑𝑓/𝑑𝜉. Similarly, we can consider the symmetry reductions and exact solutions to the equation. Now, as an example, we study the special case 𝜈=2. In this case, we have 𝑢_𝑡+𝛼𝑥²𝑢_𝑥𝑥+𝛽𝑥𝑢_𝑥+𝛾𝑥²𝑢=0. (78) Referring to (77) and setting 𝑓^′=𝑦, then we get the following reduced ODE of (78): 𝛼𝜉²𝑦^′+𝛼𝜉²𝑦²+𝛽𝜉𝑦+𝛾𝜉²+𝑣=0. (79)

Suppose that (79) has the power series solution of the generalized form (68). Then, substituting (68) into (79) and comparing coefficients, we obtain ±√𝐴=𝛼−𝛽Δ,2𝛼 (80) where Δ=(𝛼−𝛽)²−4𝛼𝑣, 2𝛼𝐴+𝛽𝑐₀=0,𝑐₁=−𝛼𝑐²₀−𝛾.𝛼+2𝛼𝐴+𝛽 (81)(82)

Generally, for 𝑛≥1, we have 𝑐_𝑛+1=−𝛼(𝑛+1)𝛼+2𝛼𝐴+𝛽^𝑛_𝑘=0𝑐_𝑘𝑐_𝑛−𝑘,𝑛=1,2,…. (83)

In view of (81), we have two special cases as follows.

When 2𝛼𝐴+𝛽≠0, from (81), we have 𝑐₀=0. Furthermore, from (82) and (83), we have 𝑐₁=−𝛾𝛼+2𝛼𝐴+𝛽,𝑐₂𝑐=0,₃=−𝛼𝑐²₁3𝛼+2𝛼𝐴+𝛽,𝑐₄=0, (84) and so on. In this case, by induction method, we have 𝑐_2𝑛=0,𝑛=0,1,2,…. (85)

When 2𝛼𝐴+𝛽=0, from (81), we get that 𝑐₀ is an arbitrary constant. Furthermore, from (82) and (83), we have 𝑐₁=−𝛼𝑐²₀−𝛾𝛼+2𝛼𝐴+𝛽,𝑐₂=−2𝛼𝑐₀𝑐₁,𝑐2𝛼+2𝛼𝐴+𝛽₃=−𝛼2𝑐₀𝑐₂+𝑐²₁3𝛼+2𝛼𝐴+𝛽,𝑐₄=−2𝛼𝑐₀𝑐₃+𝑐₁𝑐₂,4𝛼+2𝛼𝐴+𝛽 (86) and so on (see Remark 3).

Thus, the exact power series solutions to (79) are obtained. In view of (42), the exact analytic solutions to (78) are provided in power series form, respectively. More generally, for 𝜈 is an arbitrary positive integer, the exact power series solutions to (75) can be considered similarly by the generalized power series method; the details are omitted here.

6. Conclusion and Remarks

In this paper, we investigate the symmetry classifications and exact solutions to the bond pricing types of equations by the combination of Lie symmetry analysis and the generalized power series method; all of the exponentiated solutions and similarity solutions are obtained explicitly for the first time in the literature. Furthermore, for the generalized bond pricing type of equation, the vector field and exact solutions are provided simultaneously. These similarity solutions possess significant features in both financial problems and physical applications. On the other hand, it is known that tackling exact solutions to the vc-PDEs is a difficult problem; from the above discussion, we can see that the combination of Lie symmetry analysis and generalized power series method is a feasible approach and is worthy of further study.

Remark 2. Since there is no space translation (𝑥,𝑡,𝑢)→(𝑥+𝜖,𝑡,𝑢), the bond pricing equations have no traveling wave solutions. However, based on the exponentiated solutions, we can consider the other types of solutions, such as the fundamental solutions and sometimes iterative solutions [3–5, 10].

Remark 3. In general, we cannot get the exact explicit solutions to the nonlinear equations such as (49), (53), and (79) by the classical analysis method. To tackle these equations, the generalized power series method and special techniques are necessary sometimes. For getting the exact analytic solutions in Sections 4.3 and 5, the condition (𝑛+1)𝛼+2𝛼𝐴+𝛽≠0 is necessary for 𝑛=0,1,2,….

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant no. 11171041, the NSAF under Grant no. 11076015, and the Doctorial Foundation of Liaocheng University.