Interest rates are a function of time and term (time to maturity). As a function of time, rates behave as stochastic processes. As a function of term, interest rates on a given date form a yield curve. Term structure models describe the behavior of interest rates of different maturities as a joint stochastic process.

Term structure models are a necessary tool for valuation and risk management of interest rate contingent claims—that is, securities or transactions whose payoff depends on future values of interest rates, such as callable or putable bonds, swaps, swaptions, caps, and floors. For instance, a bond will be called if its value on the call date is greater than the call price. To determine the current value of the bond, it is necessary to know the subsequent behavior of interest rates. The same is true for all debt securities subject to prepayment, such as mortgages with refinancing options.

The immediate acceptance and application of term structure models in banking and investment practice is due to the fact that there are few financial instruments whose value is not in some degree dependent on future interest rates. Even stock options such as calls and puts depend on the development of interest rates. Interest rate models enter into valuation of firms and their liabilities. Besides valuation, term structure models are necessary for interest rate risk measurement, management, and hedging.

Interest rates of different maturities behave as a joint stochastic process. Not all joint processes, however, can describe interest rate behavior in an efficient market. For instance, suppose that a term structure model postulates that rates of all maturities change in time by equal amounts, that is, that yield curves move by parallel shifts (which, empirically, appears to be a reasonable first-order approximation). It can be shown that in this case a portfolio consisting of a long bond and a short bond would always outperform a medium-term bond with the same Macaulay duration. In an efficient market, supply and demand would drive the price of the medium maturity bond down and the prices of the long and short bonds up. As this would cause the yield on the medium bond to increase and the yields on the long and short bonds to decrease, the yield curves would not stay parallel. This model therefore cannot describe interest rate behavior.

In order that riskless arbitrage opportunities are absent, the joint process of interest rate behavior must satisfy some conditions. Determining these conditions and finding processes that satisfy them is the purpose of term structure theory. Term structure models are specific applications of term structure theory.

The joint stochastic process is driven by sources of uncertainty. For continuous processes, the sources of uncertainty are often specified as Wiener processes. If the evolution of the yield curve can be represented by Markovian state variables, these variables are called factors.

A general theory of one-factor term structure models is given in the 1977 paper “An Equilibrium Characterization of the Term Structure” (Chapter 6). It is proven that the term structure is fully described by the specification of the behavior of the short rate and the market price of risk. This relationship is expressed by the fundamental bond pricing equation (18) in that chapter, which gives the price of a bond as a function of the short rate and the market price of risk over the term of the bond.

The bond pricing equation was derived as the solution to a partial differential equation under certain assumptions, but it is valid generally for any arbitrage-free term structure model. The equation is valid even in the case of multiple factors or multiple risk sources, if the products in the equation are interpreted as scalar products of vectors. Every term structure model is either a direct application of that equation, or it assumes that the equation is true for bonds and uses it to price interest rate derivatives (as in the Heath, Jarrow, Morton model).

The 1977 paper gives an example of a term structure model in which the short rate follows a mean reverting random walk (the Ornstein-Uhlenbeck process), and the market price of risk is constant. In this example, which has become known as the “Vasicek model,” interest rates are Gaussian.

The difference between the forward rate and the expected spot rate has been traditionally called the liquidity premium. The unpublished memorandum written in 1979, “The Liquidity Premium” (Chapter 7), shows that the liquidity premium consists of two components: The first component, driven by the market price of risk, is equal to the expected integral over the span of the forward rate of the forward rate volatility multiplied by the market price of risk. The second component is equal to the negative of the expected aggregate over the forward rate span of the bond price volatility times the forward rate volatility. This component, which is present even if the market price of risk is zero, arises as a result of the nonlinear relationship between prices and rates.

A plot of bond yields on a given date as a function of term is called the yield curve. Since yield quotes are typically available only for selected maturities, it has been necessary to interpolate between these maturities, or differently stated, fitting a smooth curve to the discrete data. A favorite method for doing so had been the use of polynomial splines to the yields. The paper “Term Structure Modeling Using Exponential Splines” (Chapter 8), written in 1982 with H. Gifford Fong, proposes a different method, namely fitting exponential splines to the discount function. The advantages of using this type of splines are their desirable asymptotic properties, and their having both a sufficient flexibility to fit a wide variety of yield curves and a sufficient robustness to produce stable forward rate curves.

Heath, Jarrow, and Morton in their intricate 1992 paper proposed a framework for pricing interest rate derivatives based on the knowledge of the initial term structure and of the forward rate volatilities. By writing the dynamics of interest rates directly in terms of a process that is Wiener under the martingale measure, it is possible to price interest rate contingent claims without knowing the market price of risk. The 1994 memorandum “The Heath, Jarrow, Morton Model” (Chapter 9) provides a three-line derivation of the HJM model.