An important problem in mathematics of finance is the mathematical modelling of optimal investment-consumption decisions under uncertainty conditions. The investment-consumption problem  has been extensively investigated in many works with various modifications and extensions. Cox  and Cox and Ross  have derived the well-known constant elasticity of variance (CEV) option pricing model and Schroder  afterwards broadened the model by stating the CEV option pricing formula in respect of noncentral Chi-square distribution. The CEV model is mostly used to investigate the option and asset pricing formula, as was investigated by Beckers , Davydov and Linetsky , Emanuel and Macbeth  and recently by Hsu et al. . Here, we reconsider the CEV model . Literature survey witness that there have been few studies [10,11,12,13,14,15,16,17] recently reported where its solutions are presented. Nonetheless, to the best of our knowledge, no work has been published on the closed-form solutions of the CEV model, which is the aim of the present work.
The classical Lie symmetry theory was discovered by the Norwegian mathematician Marius Sophus Lie (1842-1899) in the nineteenth century. This theory systematically unites the widely known ad hoc methods to find exact solutions for differential equations. After many years of discovery, the Lie’s theory was popularized by Ovsiannikov in Novosibirsk, Russia in the middle of twentieth century and by Birkhoff and Olver in the West. Lie’s theory is one of the most effective tools to find exact analytical solutions of nonlinear partial differential equations and is established on the analysis of the invariance under one-parameter group of point transformations. See for example, the references [18,19,20,21,22,23,24].
Recently Lie’s theory has been applied to partial differential equations (PDEs) of mathematical finance. One of the earliest study on the subject is , where the classical Black-Scholes equation was discussed. The bond-pricing equation via Lie group approach was investigated in [26, 27]. The invariant analysis of some well-known financial mathematics models was presented [28,29,30,31] . More recently the Lie’s theory has been applied to various PDEs of financial mathematics. See for example, Motsepa et al. , Lekalakala et al. , Nteumagne and Moitsheki , Bozhkov and Dimas , Caister et al. [36, 37], Naicker et al. , Lo , Taylor and Glasgow , Wang et al.  and Pooe et al. , are few important studies to mention.
In this paper, we discuss the optimal investment-consumption problem under the CEV model
with the terminal condition
from the viewpoint of Lie symmetry approach. The application of Lie’s theory, in general, reduces the partial differential equation in two independent variables to an ordinary differential equation and provides us with the group-invariant solution. Some nontrivial conservation laws are also constructed for the CEV model by employing a general theorem proved in .
2 Lie symmetries of (1)
In this section, we compute the Lie symmetries admitted by the model equation (1) and utilised them to obtain the closed-form group-invariant solution of the PDE (1) satisfying the terminal condition (2).
For detailed description on Lie symmetry method and its applications to various disciplines the reader is referred to Refs [18,19,20,21,22,23,24]. However, in this section we give detailed calculations on finding Lie point symmetries of (1).
In order to facilitate the calculations when employing the Lie group approach, we rewrite the PDE (1) in the form
is a Lie symmetry of PDE (3) if and only if
where X denotes the second prolongation of X which is defined by
and the total derivative operators
which are linear PDEs in τ, ξ and η. Solving the above system gives
where and C1, …, C4 are arbitrary constants. Since τ, ξ and η contain four constants, we conclude that the Lie algebra of infinitesimal symmetries of the PDE (3) is spanned by the four vector fields
3 Group invariant solution
Many researchers have developed various analytical methods for solving partial differential equations, such as inverse scattering transform , the Bäcklund transformation , the Hirota bilinear method , the Painlevé analysis method , the Bell polynomials , the homoclinic breather limit method .
We now obtain the closed-form group-invariant solution for the PDE (3) by making use of the Lie symmetry algebra calculated in the previous section. Firstly we calculate symmetry Lie algebra admitted by (3) that satisfies the terminal condition (2) [26, 28].
We consider the linear combination of the Lie point symmetries, namely
and use the terminal conditions
The condition (9) yields
while condition (10) gives
Splitting Eq. (12) on powers of x yields
Thus, the group-invariant solution of the PDE (3) is given by J2 = G(J1), which yields
Solving the reduced ODE (17) for G(z), we obtain
Substituting the value of G into Eq. (16), the solution for F(x, t) is written as
Finally, making use of the terminal condition F(x, t) = 1 in Eq. (18), we obtain
Therefore, the solution of (1+1) evolution PDE (3) satisfying the terminal condition is given by
4 Conservation laws
In this section we derive conservation laws for the (1+1) evolution partial differential equation (3). In classical physics, conservations laws are physical quantities which describe the conservation of energy, mass, linear momentum, angular momentum, and electric charge. One particularly important result concerning conservation laws is the celebrated Noether theorem, which gives us a sophisticated and useful way of constructing conservation laws when a Noether point symmetry connected to a Lagrangian is known for the corresponding Euler-Lagrange equation.
Recently, a theorem due to Ibragimov was proved, which gives a method to construct conservation laws irrespective of the existence of a Lagrangian. We start here by stating the definition of an adjoint equation.
be a second-order PDE with x, t as independent variables and f a dependent variable. Then, its adjoint equation is 
denoting the Euler–Lagrange operator and g is a new dependent variable. Here
are the total derivative operators.
The 2-tuple vector T = (Tt, Tx), is a conserved vector of (19) if Tt and Tx satisfy
We now state the following theorem.
 Any Lie point, Lie–Bäcklund or non-local symmetry
where w and 𝓛 are determined as follows:
We now apply the above theorem to our problem. The adjoint equation to the equation (3) is
We now apply the above theorem to each Lie point symmetry of Eq. (3). We start with X1 = ∂/∂t. Corresponding to symmetry X1 we obtain the conserved vector with components
The symmetry X2 provides us with the conserved vector whose components are
where . Likewise, the conserved vectors associated with the symmetries X3 and X4 are given by
5 Concluding remarks
The evolution (1+1) PDE (1) describing the optimal investment-consumption problem under the CEV model  satisfied the classical Black–Scholes–Merton equation with boundary condition which differ from those often used in the most common cases. It is well-known that the evolution (1+1) PDE (1) is related to the heat equation via the equivalence transformations and thus its general solution can be obtained. However, in this paper, for the first time, we have solved the PDE (1) subject to the terminal condition (2) by utilizing the Lie group method. This demonstrates the usefulness of Lie’s theory. We found a four-dimensional Lie symmetry algebra for evolution PDE (1). Using the nontrivial Lie point symmetry operator, we have shown that the governing PDE can be transformed into a second-order variable coefficient ODE. The reduced ODE is solved to obtain a new exact closed-form solution of the CEV model which also satisfy the terminal condition. Thus for the first time with the application of Lie’s theory closed-form solution of (1) is derived. Finally, we constructed conservation laws corresponding to the four Lie point symmetries by employing a general theorem on conservation laws. This is the first time that the evolution PDE (1) for optimal investment-consumption problem has been considered from the view point of group theoretical approach and the conservation laws have been derived in the literature.
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About the article
Published Online: 2018-03-08
Conflict of interestConflict of Interests: The authors declare that there is no conflict of interests regarding the publication of this paper.
Citation Information: Open Physics, Volume 16, Issue 1, Pages 31–36, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0006.
© 2018 T. Motsepa et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0