Fundamental Properties of Bond Prices in Models of the Short-Term Rate

Abstract

This article develops restrictions that arbitrage-constrained bond prices impose on the short-term rate process in order to be consistent with given dynamic properties of the term structure of interest rates. The central focus is the relationship between bond prices and the short-term rate volatility. In both scalar and multidimensional diffusion settings, typical relationships between bond prices and volatility are generated by joint restrictions on the risk-neutralized drift functions of the state variables and convexity of bond prices with respect to the short-term rate. The theory is illustrated by several examples and is partially extended to accommodate the occurrence of jumps and default. A standard approach to modeling the term structure of interest rates is to derive sets of arbitrage-free bond prices using as an input an exogenously given short-term rate process [see, e.g., Vasicek (1977) and Cox, Ingersoll, and Ross (1985)]. While this approach is widely used, there is no theoretically sound work establishing systematic answers to such fundamental questions as: When are bond prices a decreasing function of the shortterm rate? When are bond prices a convex function of the short-term rate? Are bond prices a decreasing function of the short-term rate volatility? This article demonstrates that it is possible to develop results addressing these questions in relation to all sets of economically admissible (i.e., noarbitrage) bond prices.