The classical Cox–Ingersoll–Ross (CIR) bond-pricing model is based on the evolution space-time dependent partial differential equation (PDE) which represents the standard European interest rate derivatives. In general, such class of evolution partial differential equations (PDEs) has generally been resolved by classical methods of PDEs and by ansatz-based techniques which have been previously applied in a similar context. The author here shows the application of an invariant approach, a systematic method based on deductive group-theoretical analysis. The invariant technique reduces the scalar linear space-time dependent parabolic PDE to one of the four classical Lie canonical forms. This method leads us to exactly solve the scalar linear space-time dependent parabolic PDE representing the CIR model. It was found that CIR PDE is transformed into the first canonical form, which is the heat equation. Under the proper choice of emerging parameters of the model, the CIR equation is also reduced to the second Lie canonical form. The equivalence transformations which map the CIR PDE into the different canonical forms are deduced. With the use of these equivalence transformations, the invariant solutions of the underlying model are found by using some well-known results of the heat equation and the second Lie canonical form. Furthermore, the Cauchy initial-value model of the CIR problem along with the terminal condition is discussed and closed-form solutions are deduced. Finally, the conservation laws associated with the CIR equation are derived by using the general conservation theorem.
The bond-pricing theory began in the 19th century when Bachelier [
For the Vasicek model, the instantaneous rate
The mean-reversion feature of these bond-pricing models is consistent with economic phenomenon. However, the main disadvantage of the Vasicek model is that it allows the negative interest rates. On the contrary, the CIR model provides improvement in the Vasicek model by ensuring that the interest rate can never be negative. The CIR model, therefore, incorporates the number of important characteristics of interest rate models.
For Vasicek model, the price bond
In the CIR model, the price of a zero-coupon
For both the above PDEs, the value
A mapping has been constructed in [
It is important to note that the general PDE in which both Vasicek equation (
The above equation models the price of a zero-coupon bond when the spot rate is determined by the general stochastic process:
The function
The applications of Lie groups to problems in mathematical finance are not new, and it is certainly young enough, with a possibility to expand. Generally, the nonlinear DEs that arise in mathematical finance are usually tackled with either approximate or numerical methods. Whereas, the Lie symmetry approach allows us to find complete local one-parameter transformation groups that can be used to find exact solutions or to reduce nonlinear DEs to linear DEs. Concerning the relationship between Lie symmetry methods and financial mathematics and economic models, we would like to note the presence of several recent studies [
Lie [
From the definition point of view, a general scalar
Together with equivalence transformations of independent variables,
The first form is the classical heat PDE which contains six nontrivial symmetry generators. The second canonical equation has four nontrivial symmetry generators, the third generally has two symmetry generators, and the fourth has one nontrivial symmetry generator. The following theorems give the algorithm on the invariant characterization of scalar
The scalar
The scalar linear
The parabolic PDE (
The functions
The derivation of conservation laws plays an important role in understanding the mathematical models described by DEs. For several mathematical and physical models, the conservation laws are used to study the existence and uniqueness of solutions of nonlinear DEs. The conservation laws are applied in the case of PDEs to search for possible potential symmetries and related nonclassical symmetries. Consequently, it is necessary to derive the conservation laws of the CIR equation, arguably one of the most popular mathematical finance models. There are many approaches available to the construction of conservation laws for PDEs [
Motivated by the facts stated above, the purpose of this paper is to describe the systematic application of invariant criterion to precisely solve the traditional CIR model from mathematical finance. The majority of financial interest rate problems are modeled using linear parabolic (1 + 1) PDEs. Because many of the DEs in the literature addressing financial models are highly rich in symmetry, the Lie group theory of transformation groups is widely useful. As a result, the remainder of the study demonstrates the algebraic features of the CIR model in the context of an invariant criterion based on Lie groups. The closed-form solutions will be derived for the model problem under a suitable choice of terminals condition. By not using the ansatz method or relying on adhoc approach, this will be achieved through an algorithmic technique: the invariant criterion.
The fundamental solutions of CIR bond-pricing equation under the implication of invariant criterion are constructed in this section. Two separate cases will be considered for the reduction of CIR equation (
First, find the equivalence transformations which reduce equation (
With the use of Theorem
It is clear that
From the above equation, one obtains the following parametric conditions:
For specific values of the parameter
Now, the transformation relations will be computed which give reduction of the CIR equation (
With the use of the coefficients as defined in Theorem
In order to seek for the fundamental solutions of the PDE (
It is well known that the fundamental solution of the heat is given in barred coordinates [
To look for the fundamental solution of CIR equation (
Finally, using the values of
Under the same set of equivalence transformations for reduction into the heat equation, one can find another closed-form solution for PDE (
Substituting the values of
We look for the case when the CIR PDE (
Under the same procedure used previously, the equivalence transformations that transform equation (
Now, to deduce the closed-form solution such that the CIR equation is transformed into the second Lie canonical equation subject to the equivalence transformations given in equation (
In order to find the exact solution of CIR equation, one first needs to seek the solution of PDE (
Let us assume that the PDE (
Using equation (
The solutions of ODEs (
In order to obtain the solution of PDE (
Substituting the values of
Here, the solutions of the ODEs (
The solution of CIR PDE (
Incorporating the values of
This section is devoted to discuss the fundamental solution of the Cauchy initial-value problem for the CIR model subject to the terminal condition. Since the transformation relations are already deduced for PDE (
The standard Cauchy initial-value problem for the heat equation is given by [
One needs to transform solution (
Keeping in mind condition (
Finally, by substituting of the values of
The value of the integral in the above solution could be found in terms of the Kummer confluent hypergeometric functions [
The conservation laws for the CIR equation (
The formal second-order Lagrangian for the CIR equation (
The Lagrangian defined in (
Every Lie symmetry generator of the CIR equation of the form
The symmetries of the CIR equation (
Similarly, if one uses the symmetry operator
The purpose of this work is to demonstrate the utility of an algorithmic method, an invariant approach based on the theory of Lie groups. Utilizing this invariant technique, the valuation parabolic PDE of interest rate derivative in the CIR model framework is studied in detail. The classification of the CIR equation was based on the classical results of Lie’s classification theory for the parabolic PDEs. Hence, the invariant theory helped in the computation of equivalence transformations which transform the underlying model to different Lie canonical equations. It was found that, for proper choice of parametric values, the CIR equation was transformed to the first and the second Lie canonical equations. For each case, the transformation relations are derived which reduced the CIR equation into the first and the second Lie canonical form. These transformations were further employed to find the invariant solutions of the CIR equation by using some well-known fundamental solutions of the first and the second Lie canonical equations. Moreover, the Cauchy initial-value problem of the CIR model, with a suitable choice of a terminal condition, is also studied in detail. The solution technique was based on translating the Cauchy initial-value problem for the CIR model into the classical heat equation Cauchy problem and then implementing the well-known Cauchy solutions of the heat equation.
The conservation laws corresponding to the two Lie point symmetries of the CIR equation are constructed by using a general theorem on conservation laws. The notion of conservation laws and associated Lie point symmetries can be used to formulate exact solutions of such type of complicated equations arising in the study of financial mathematics models.
Lie’s group theory provides a systematic way to the resolution of the various DEs which arise in financial mathematics. It should come as no surprise that many evolution PDEs in the field of mathematical finance have a rich symmetry group structure. These symmetry structures help to explain the ease with which ansatz-based techniques can yield solutions. However, such nonalgorithmic adhoc procedures have no place, especially when dealing with real-world problems that are described in terms of complicated PDEs. It is important to note that the invariant analysis performed on the CIR equation can be carried out on any other class of parabolic models which transform not only to the heat equation but also to other canonical equations.
No data were used to support the findings of the study.
The author declares that there are no conflicts of interest regarding the publication of this paper.