About One Family of Nonaffine Models of Yield Term Structure

The equation of term structure for the price of a zero-coupon bond is considered, the solution of which in analytical form is known, basically, for the simplest models and has an affine structure with respect to the short-term rate. In this paper, we construct solutions of this equation for a family of term structure models that are based on short-term rate processes, in which the square of volatility is proportional to the third power of the short-term rate in stochastic differential equations. The solution of the equation is sought in the form of a definite functional series and, as a result, is reduced to a confluent hypergeometric function. Three versions of the underlying stochastic differential equations for short-term rate processes are considered: with zero drift, linear drift, and quadratic drift. Numerical examples are given for the yield curve and the forward rate curve for these versions. Some conditions for the existence of nontrivial solutions of the equation of term structure in the family of processes under consideration are formulated. Unfortunately, models that admit such solutions are few and, in particular, include some well-known models: the CIR(1980) model and the Ahn-Gao model. The requirements for the structure of the short-term interest rate model, which would allow the receipt of a term structure of the bond price in the form considered in the article, are reduced to the following. 1. To obtain a non-trivial solution, it is necessary that the degrees of polynomials determing the drift and volatility of the short-term interest rate satisfy certain constraints. 2. Another necessary condition is connected with the fact that the functional series is power-law with respect to a function that does not depend on the index of summation of the series. 3. In addition, it is necessary that the coefficients of the functional series do not depend on the maturity of the bond. Simultaneous fulfillment of these necessary conditions significantly narrows the family of models for which the solution of the time-structure equation has the form considered in the paper.