On Quadratic Models of Yield in Risk-Neutral World

The paper presents a mathematically equivalent, but more compact description of the usually occurring quadratic model of yield. Equations for the functions of the term structure are obtained and general properties of their solutions are given. The main content of the paper is to consider the case when the probability properties of the model are subject to a risk-neutral setting, which allows to obtain the analytical solutions of the problem about the shape of the yield curves and forward curves in an explicit form. Equations of curves are presented in a compact form using hyperbolic functions. We have found families of yield curves and forward curves, which for a fixed short-term interest rate ensure the same limit long-term yield. The obtained results are illustrated by the numerical examples.The presented results illustrate the fact that quadratic models of the term structure of interest rate make it possible to obtain a richer spectrum of admissible yield curves and forward curves than affine models that allow only single curve to be obtained for a fixed starting interest rate, while quadratic models under conditions of fixed starting and long-term limiting rates and yields can give a fairly wide range of yield curves.Consequently, theoretically they have great opportunities for fitting theoretical curves to real market term structures. However, this will be realistically feasible if there is an opportunity to control the latent variables used in the quadratic model. So far, there is no description of these variables in the economic sense in the literature