Quadratic Models of Yield in a Risk-Neutral World

A quadratic modelQuadratic model of the interest rate of the term structure of interest rates A quadratic modelQuadratic model of the interest rate of the term structure of interest rates for the yield of a zero-coupon bondZero-coupon bond is considered when the short-term rateShort-term rate has a quadratic dependence on the state variablesState variable . This chapter presents a mathematically equivalent, but more compact, description of the usual quadratic model of yield. Equations for the functions of the term structure are obtained and general properties of their solutions are given. The main focus of the chapter is on the case when the probability properties of the model are subject to a risk-neutral setting, which allows us to obtain analytical solutions to the problem of determining the shape of the yield curvesYield curve and forward curvesForward curve in an explicit form. The equations of the 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 numerical examples. The analysis is also carried out for the case when the state variables form a vector with independent components, and the term structure is determined for a risk-neutral setting. It is shown that in this case the short-term rateShort-term rate process has a gamma distributionGamma distribution , the same as for the affine Duffie–Kan modelDuffie–Kan model . The term structures of the yield of this affine model are compared with the quadratic model of yield. It is shown that the limiting (short-term and long-term) yields of these models completely coincide, although the term structures themselves differ. It is shown that the components of the state vector do not influence the form of the quadratic term structure considered, but it depends only on the value of the starting interest rate. The comparative properties of the affine Duffie–Kan model and the quadratic model of yield are illustrated by a numerical example.