Group Classification of a General Bond-Option Pricing Equation of Mathematical Finance

and Applied Analysis 3 + λxu x ξ x − γλux q+1 τ x − βγλux q τ u u x + βγλux q τ x + λ 2 x 2 u x τ x + β 2 λ 2 u x τ x − 2βλ 2 xu x τ x − 2αx p τ u u x u tx − 2αx p τ x u tx + αξpx p−1 u xx − 2αx p ξ u u x u xx − 2αx p ξ x u xx + x p × (ατ t u xx − αγuτ u u xx x q − αλxτ x u xx +αβλτ x u xx + αu 2 x η uu + 2αu x η xu ) + αγuu 2 x τ uu x p+q − αλx p+1 u 3 x τ uu − αx p u 3 x ξ uu − 2αx p u 2 x ξ xu − αx p u x ξ xx + αx p η xx + 2αγuu x x p+q τ xu + α 2 x 2p u 2 x τ uu u xx − 2αλx p+1 u 2 x τ xu + αβλx p (u 3 x τ uu + 2u 2 x τ xu ) + αγuτ xx x p+q + α 2 x 2p u xx (2u x τ xu + τ xx ) − αλx p+1 u x τ xx + αβλx p u x τ xx = 0. (10) Separating (10) with respect to the derivatives of u, since the functions τ, ξ, and η do not depend on them, leads to the following linear PDEs:


Introduction
The theory of option pricing began in 1900 when the French mathematician Bachelier [1] deduced an option pricing formula based on the assumption that stock prices follow a Brownian motion. The Black-Scholes equation was introduced by Black and Scholes [2] as the general equilibrium theory of option pricing which is particularly attractive because the final formula is a function of observable variables. Merton [3] extended the Black-Scholes theory of option pricing by introducing more assumptions and found new explicit formulas for pricing both the call and put options as well as the warrants and the down-and-out options. The equation is mainly used to find the fair price of a financial instrument (option or derivative) and to find the implied volatility.
In 1881, Lie [21] was the first one to investigate the problem of group classification. In this regard, he studied a linear second-order partial differential equation with two independent variables. Suppose a differential equation contains an arbitrary element ( ). The main idea of group classification of this differential equation is to find the Lie point symmetries of the differential equation with arbitrary element ( ) and then find all possible forms of ( ) for which the principal Lie algebra can be extended.
Semi-invariants for the (1 + 1) linear parabolic equations with two independent variables and one dependent variable were derived by Johnpillai and Mahomed [22]. In addition, joint invariant equation was obtained for the linear parabolic equation and the (1 + 1) linear parabolic equation was reducible via a local equivalence transformation to the onedimensional heat equation. In [23], a necessary and sufficient condition for the parabolic equation to be reducible to the classical heat equation under the equivalence group was provided which improved the work done in [22].
Goard [24] found group invariant solutions of the bond pricing equation by the use of classical Lie method. The solutions obtained were shown to satisfy the condition for the bond price, that is, ( , ) = 1, where is the price of the bond. Here is the short-term interest rate which is governed by the stochastic differential equation and is time to maturity.
In [25], the fundamental solutions were obtained for a number of zero-coupon bond models by transforming the one-factor bond pricing equations corresponding to the bond models to the one-dimensional heat equation whose fundamental solution is well known. Subsequently, the transformations were used to construct the fundamental solutions for zero-coupon bond pricing equations.
Sinkala et al. [26] computed the zero-coupon bonds (group invariant solutions satisfying the terminal condition ( , ) = 1) using symmetry analysis for the Vasicek and Cox-Ingersoll-Ross (CIR) equations, respectively. In [27] an optimal system of one-dimensional subalgebras was derived and used to construct distinct families of special closed-form solutions of CIR equation. In [20], group classification of the linear second-order parabolic partial differential equation where , , , , and are constants, was carried out. Lie point symmetries and group invariant solutions were found for certain values of . Also the forms where equation (3) admitted the maximal seven Lie point symmetry algebra were transformed into the heat equation. Vasicek, CIR, and Longstaff models were recovered from group classification and some other equations were derived which had not been considered before in the literature. Furthermore, Mahomed et al. [28] used the invariant conditions developed in [23] to carry out group classification of [20] and some new cases were discovered. Dimas et al. [29] investigated some of the well-known equations that arise in mathematics of finance, such as Black-Scholes, Longtsaff, Vasicek, CIR, and heat equations. Lie point symmetries of these equations were found and their algebras were compared with those of the heat equation. The equations with seven symmetries were transformed to the heat equation.
In this paper, we study a general bond-option pricing equation. The partial differential equation which will be investigated is a generalisation of (1) and (2) and is given by where is time, is the stock (share or equity) price or instantaneous short-term interest rate at current time , and ( , ) is the current value of the option or bond. Here ≥ 0, ≥ 0, and , , , and are constants with , , ̸ = 0. When = 0, (4) is the option pricing equation and it is the bond pricing PDE when = 1.
This paper is structured as follows. In Section 2, we find two classifying equations on which group classification of (4) depends, one for ̸ = 2 and the other for = 2. Then we use the two equations to find possible values for arbitrary constants for which (4) admits nontrivial Lie point symmetry algebras. In Section 3, we obtain symmetry reductions and construct group-invariant solutions for Case 2.1(1) and finally in Section 4 we give conclusions.

Determination of Classifying
Equations of (4) The Lie point symmetries for (4) are given by the vector field: if and only if [2] ( + where [2] is the second prolongation of defined as Here 's are given by where the total derivatives and are defined as To perform the group classification of (4) it turns out that we need to consider two cases of separately: ̸ = 2 and = 2.

Classifying Equation of
(4) for ̸ = 2. Expanding the determining equation (6), we obtain Separating (10) with respect to the derivatives of , since the functions , , and do not depend on them, leads to the following linear PDEs: = 0, = 0, = 0, To solve the above system of equations, we first observe from (11) and (14) that does not depend on and , which means that is a function of . Thus Equation (12) implies that depends on both and but not on . Hence Integration of (13) with respect to twice gives where ( , ) and ( , ) are arbitrary functions of and .

Results of Group Classification
We note that our classifying equations (27) and (45) for some constants 1 and 2 . Thus using these values, for both cases, the coefficients of the infinitesimal operator are where ( , ) is any solution of (4).

Abstract and Applied Analysis
Case 0 ( , , , , , arbitrary). We obtain the following Lie point symmetries: where the symmetry associated with is the solution symmetry. Lie symmetries (49) generate what is called the principal Lie algebra. By equating the powers of in (27) and solving for we infer that possible extensions of the principal Lie algebra are possible for the following values of :  1 ( = 0). In this case, the principal Lie algebra extends by the following Lie point symmetries: Case 1.2 ( = 1). The principal Lie algebra extends by It should be noted that this case results in the Vasicek equation [26].
Case 3 ( = 2). We can conclude from (45) that can only take the value 0. Proceeding as before, we find that the principal Lie algebra extends for the case when = 0 by the following symmetry operators: This case gives us the Black-Scholes equation [30].

Symmetry Reductions and Group Invariant Solutions
We obtain symmetry reductions [31] and construct group invariant solutions of Case 2.1(1), that is, when = 1, = 0, ̸ = /(2 ), and ̸ = 3 /(2 ), while all other constants in (4) are arbitrary. Equation (4) is then given by Case 2.2 ( 3 ). The operator 3 gives the following two invariants: Hence, the invariant solution of (67) under 3 is given by where satisfies The solution of the above equation is where 1 and 2 are arbitrary constants. It should be noted that the operators 2 and do not provide invariant solutions.

Conclusions
In this paper we carried out group classification of the general bond-option pricing PDE (4) for = 0,1, and 2. The principal Lie algebra was found to be three-dimensional. These values of resulted in 16 cases, which extended the principal Lie algebra. We presented the Lie point symmetries for each case. Three cases gave us the option pricing equations, which were given by Cases 1.1, 2.1(1), and 3. In the last case, Black-Scholes equation was recovered. Seven bond pricing equations were obtained and these were Case 1.2 and Cases 2.3(1)-2.3(6). Cases 1.2 and 2.3(1) were found to be the Vasicek and CIR equations, respectively. Finally, symmetry reductions and construction of group invariant solutions for Case 2.1(1) were presented.