On the usage of the Lie group symmetries for term structure models with nonlinear drift and squared volatility functions

Abstract

One of the central tasks of financial analysis is the study of the behavior of the dynamics of interest rates. The most well­known affine models are not able to describe real yield curves with the necessary accuracy, so more and more often researchers are trying to build more complex and, it is believed, likelihood non­affinity models of the term structure of interest rate yields. One of the main problems of constructing such models is the solution of a parabolic differential equation in partial derivatives, which sets the cost of a zero­coupon bond – in order to study the properties of models it is convenient to have such a solution in an analytical form. In this paper, we consider a generalized model with nonlinear drift and squared volatility functions, which includes most of the already known models. To solve a parabolic equation associated with such a model, we use the theory of Lie groups, which makes it possible to systematize and completely algorithmize the approach to constructing solutions. On the basis of this approach, solutions are found for some particular cases of models, both new ones that have not been previously encountered by the author, and those that already known. Also for the non­affine Ana – Gao model, a more general solution is found in comparison with the original one. In the end, a numerical experiment was carried out using real data from the European Central Bank.