Term structure models: a perspective from the long rate

Term structure models resulted from dynamic asset pricing theory are discussed by taking a perspective from the long rate. This paper attempts to answer two questions about the long rate: in frictionless markets having no arbitrage, what should the behavior of the long rate be; and, in existing dynamic term structure models, what can the behavior of the long rate be. In frictionless markets having no arbitrage, the yields of all maturities should be positive and the long rate should be finite and non-decreasing. The yield curve should level out as term to maturity increases and slopes with large absolute values occur only in the early maturities. In a continuous-time framework, the longer the maturity of the yield is, the less volatile it shall be. Furthermore, the long rate in continuous-time factor models with a non-singular volatility matrix should be a non-decreasing deterministic function of time. In the Black-Derman-Toy model and factor models with the short rate having the mean reversion property, the long rate is finite. The long rate in Duffie-Kan models with the mean reversion property is a constant. The long rate in a Heath-Jarrow-Morton model can be infinite or a non-decreasing process. Examples with the long rate being increasing are given in this paper. A model with the long rate and short rate as two state variables is then obtained.