On the Resolution of a Remarkable Bond Pricing Model from Financial Mathematics: Application of the Deductive Group Theoretical Technique

Taha Aziz,Kamal Shah

doi:10.1155/2021/9974073

Abstract

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.

Highlights

E pioneering work of Merton [2] and later by Black and Scholes [3] gave a new dimension to mathematical formulation of finance problems. e actual mathematical formulation of these models was initially given in the form of stochastic differential equations (DEs), but after incorporating certain assumptions, these models were derived in terms of parabolic partial differential equation (PDE) with variable coefficients. e fundamental model of bond-pricing theory was first proposed by Vasicek [4], which was modified by Cox et al in [5], famous in literature as the CIR bond-pricing model
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. e majority of financial interest rate problems are modeled using linear parabolic (1 + 1) PDEs
We look for the case when the CIR PDE (4) is transformed to the second Lie canonical equation

Summary

Taha Aziz

E invariant technique reduces the scalar linear space-time dependent parabolic PDE to one of the four classical Lie canonical forms. 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 [12,13,14,15,16,17,18,19,20,21,22], which represent the applications and utility of well-known concepts of Lie theory to several problems from mathematical finance.

PDE is written as

Every Lie symmetry generator of the CIR equation of the form