5.8 An overview of continuous-time term structure models
5.8.2 Some frequently applied models
Apparently, the first dynamic model of the term structure of interest rates was introduced by Merton (1970). His model is a diffusion model with a single state variable, which is the short-term interest rt itself. It makes good sense to use an interest rate as the state variable in models of bond prices and other interest rate derivatives. There is an obvious practical advantage in using the short rate as the state variable. As we have seen above it is necessary to specify how the short rate depends on the state variable chosen, which is evident when the short rate itself is the state variable. Furthermore, if another state variablexis used andrtis a monotonic function ofxt, we can express all relevant functions in terms of r and t instead of xand t. Therefore we might as well use r as the state variable from the beginning. Merton’s assumptions imply that the short rate follows a generalized Brownian motion under the risk-neutral probability measure, i.e.
(5.77) drt= ˆϕ dt+β dztQ,
where ˆϕandβ are constants.
Following Merton’s idea, many other one-factor diffusion models with r as the single state variable have been suggested in the literature. They all take a certain risk-neutral dynamics of the short rate of the form
5.8 An overview of continuous-time term structure models 137
where ˆαandβ are well-behaved functions. Either the functional form of ˆαis assumed directly or it is derived from assumptions on the real-world driftαand the market price of riskλ. In any case, it is necessary to knowαandλ, if the model should be used for more than just computing prices at a given date. The pricing differences between the models stem from differences in the specification of the functions ˆα and β. It turns out that models in which ˆαand β are affine functions of the current value of r are particularly tractable and allow many closed-form pricing equations to be derived.7 Such models are calledaffine models.
The two most famous term structure models are the one-factor affine diffusion models intro- duced by Vasicek (1977) and Cox, Ingersoll, and Ross (1985b). In the Vasicek model the basic assumption is that the short rate follows an Ornstein-Uhlenbeck process, cf. Section 3.8.2, and that the market price of risk is constant, i.e.
(5.78) drt=κ[θ−rt]dt+β dzt, λt=λconstant.
The risk-neutral drift of the short rate is thenκ[θ−rt]−βλ= (κθ−βλ)−κrt, which is affine inrt. The variance rate is simplyβ2, which is constant and hence a (degenerate) affine function ofr
t. In the Cox-Ingersoll-Ross (or CIR) model the assumption is that the short rate follows a square-root process, cf. Section 3.8.3, and that the market price of risk is proportional to the square-root of the short rate, i.e.
(5.79) drt=κ[θ−rt]dt+β√rtdzt, λt=λ√rt.
The risk-neutral drift of the short rate is thenκ[θ−rt]−βλrt=κθ−(κ+βλ)rt, which is affine in rt. The variance rate isβ2rt, which is also affine inrt. Chapter 7 provides a detailed analysis of these most important one-factor diffusion models and mentions a number of other, non-affine one-factor models.
A common problem for the one-factor models is that because all bond prices are assumed to be affected by a single exogenous variable, the price changes in any two bonds over an infinitesimal time period will be perfectly correlated, which conflicts with empirical evidence. Empirical studies by for example Litterman and Scheinkman (1991) and Stambaugh (1988) conclude that at least two, and perhaps three or four, state variables are necessary for the model to give a reasonable description of actual yield curve movements. This motivates the study of multi-factor diffusion models. In these models, the short rate is assumed to be a function of several state variables and each of these state variables are assumed to follow some diffusion process. Again it is common to distinguish between affine models and non-affine models, where an affine model is a model in which the risk-neutral drift rates and the variance and covariance rates of the state variables are affine functions of the current value of the state variables. Beaglehole and Tenney (1991) and Longstaff and Schwartz (1992a) have suggested two-factor affine models that extend the Vasicek model and the CIR model, respectively. We will review these and other multi-factor models in Chapter 8.
The one- and multi-factor diffusion models are absolute pricing models. They involve a small number of state variables and constant parameters. The derived prices and interest rates will also be functions of the state variables and these few parameters. Consequently, the resulting term structure of interest rates cannot typically fit the currently observed term structure perfectly. If 7A function of the formf(x) =a0+a1x is affine inx. The function is only linear in a strict mathematical
5.9 Exercises 138
the main application of the model is to price derivative securities, this mismatch is troublesome. If the model is not able to price the underlying securities (i.e. the zero-coupon bonds) correctly, why trust the model prices for derivative securities? To completely avoid this mismatch one must apply relative pricing models for the derivative securities.
We divide the relative pricing models of the term structure into three subclasses: calibrated diffusion models, Heath-Jarrow-Morton (HJM) models, and market models. The common starting point of all these models is to take the current term structure as given and then model the risk- neutral dynamics of the entire term structure. This is done very directly in the HJM models and the market models. The HJM models are based on assumptions about the dynamics of the entire curve of instantaneous, continuously compounded forward rates,T 7→fT
t . It turns out that only the volatility structure of the forward rate curve needs to specified in order to price term structure derivatives. We will discuss the general HJM model and various concrete models in Chapter 10. The market models are closely related to the HJM models, but focus on the pricing of money market products such as caps, floors, and swaptions. These products involve LIBOR rates that are set for specific periods, e.g. 3 months, 6 months, and 12 months, with a similar compounding period. The market models are all based on as assumption about a number of forward LIBOR rates or swap rates. Again, only the volatility structure of these rates needs to be specified. Market models are studied in Chapter 11. The third subclass of relative pricing models consists of so-called calibrated diffusion models. These models can be seen as extensions of absolute pricing models of the diffusion type. The basic idea is to replace one of the constant parameters in a diffusion model by a suitable deterministic function of time that will make the term structure of the model exactly match the currently observed term structure in the market. These calibrated diffusion models can be reformulated as HJM models, but since they are developed in a special way we treat them separately in Chapter 9.
5.9
Exercises
Chapter 6
The Economics of the Term Structure of
Interest Rates
6.1
Introduction
A bond is nothing but a standardized and transferable loan agreement between two parties. The issuer of the bond is borrowing money from the holder of the bond and promises to pay back the loan according to a predefined payment scheme. The presence of the bond market allows individuals to trade consumption opportunities at different points in time among each other. An individual who has a clear preference for current capital to finance investments or current consumption can borrow by issuing a bond to an individual who has a clear preference for future consumption opportunities. The price of a bond of a given maturity is, of course, set to align the demand and supply of that bond, and will consequently depend on the attractiveness of the real investment opportunities and on the individuals’ preferences for consumption over the maturity of the bond. The term structure of interest rates will reflect these dependencies. In Sections 6.2 and 6.3 we derive relations between equilibrium interest rates and aggregate consumption and production in settings with a representative agent. In Section 6.4 we give some examples of equilibrium term structure models that are derived from the basic relations between interest rates, consumption, and production.
Since agents are concerned with the number of units of goods they consume and not the dollar value of these goods, the relations found in those section apply to real interest rates. However, most traded bonds are nominal, i.e. they promise the delivery of certain dollar amounts, not the delivery of a certain number of consumption goods. The real value of a nominal bond depends on the evolution of the price of the consumption good. In Section 6.5 we explore the relations between real rates, nominal rates, and inflation. We consider both the case where money has no real effects on the economy and the case where money does affect the real economy.
The development of arbitrage-free dynamic models of the term structure was initiated in the 1970s. Until then, the discussions among economists about the shape of the term structure were based on some relatively loose hypotheses. The most well-known of these is the expectation hypothesis, which postulates a close relation between current interest rates or bond returns and expected future interest rates or bond returns. Many economists still seem to rely on the validity of this hypothesis, and a lot of man power has been spend on testing the hypothesis empirically. In