The Longstaff – Schwartz Model of Yield Term Structure and its Expansion

The Longstaff – Schwartz model is considered both in the space of latent state variables and in the space of observable (or estimated) state variables. Analytical expressions for yield curves to maturity and forward curves are obtained in both cases. Based on the analysis of the Longstaff – Schwartz model, an extended version is proposed for an arbitrary number of state variables. Within the framework of this expansion, a method is proposed for obtaining analytical solutions of equations with respect to the functions of the yield term structure, when the initial equations for the dynamics of the short-term interest rate lead to non-linear systems of Riccati equations with respect to these functions, which do not allow an analytical solution to be obtained by known methods. This allows you to obtain analytical expressions for yield curves to maturity and forward interest rate curves. At the same time, the obvious principle is used: if the process of a short-term interest rate is specified, then the expressions for yield curves corresponding to this process do not depend on how the space of state variables of the financial market is described. It is shown that the application of this principle to the extension of the Longstaff – Schwartz model allows one to obtain analytical solutions of a system of nonlinear Riccati equations of arbitrary order, whose analytical methods are not described in the literature. The proposed extension of the Longstaff – Schwartz model is free from the drawback inherent in this model, which consists in the fact that in the transition from latent variables to real observable variables, the domain of definition of these real state variables is limited. In the expansion there is no such effect. Numerical examples are given