## Approaches to term structure modelling

The modelling of the term structure of interest rates in continuous time lends itself to various approaches. The most widely used approach has been to assume the short-term interest rate follows a diffusion process ^{ [1] } . Bond prices are then determined as solutions to a partial differential equation which places restrictions on the relationship of risk premia of bonds of varying maturities. Unfortunately it is particularly difficult and cumbersome to fit the observed term structure of interest rates and volatilities within this simple diffusion model.

** One factor models. ** One of the first models to make a significant impact on interest rate modelling was by Vasicek [ ** 50 ** ]. Although his paper is entitled * "An Equilibrium Characterisation of the Term Structure" * he does not make any assumptions about equilibrium within an underlying economy, nor does he make use of an equilibrium argument in the derivation. Instead, his derivation relies on an arbitrage argument, much like the one used by Black and Scholes in the derivation of their option pricing model. Vasicek makes assumptions about the stochastic evolution of interest rates by exogenously specifying the process describing the short-term interest rate.

A later approach utilised by Cox, Ingersoll and Ross (CIR) [ ** 18 ** ] begins with a rigorous specification of an equilibrium economy which becomes the foundation for the model specifications. Assumptions are made about the stochastic evolution of exogenous state variables and about investor preferences. The form of the short-term interest rate process and hence the prices of contingent claims are endogenously derived from within the equilibrium economy. The CIR model is a complete equilibrium model, since bond prices are derived from exogenous specifications of the economy, that is: production opportunities, investors' tastes and beliefs about future states of the world.

Most models, including the Vasicek model, are partial equilibrium theories , since they take as input beliefs about future realisations of the short-term interest rate (depicted within the functional form of the short-term interest rate process) and make assumptions about investors' preferences (specified by the market prices of risk). The resulting discount bond yields are based on these assumptions.

The equilibrium approach has the advantage that the term structure, its dynamics and the functional form of the market prices of risk are endogenously determined by means of the imposed equilibrium. CIR [ ** 18 ** ] criticise the partial equilibrium approach, since it applies an arbitrage argument to exogenously specified interest rate dynamics and allows an arbitrary choice of the form of the market prices of risk which may lead to internal inconsistencies.

The assumption implicit within one-factor models is that all information about future interest rates is contained in the current instantaneous short- term interest rate and hence the prices of all default-free bonds may be represented as functions of this instantaneous rate and time only. Also, within a one-factor framework the instantaneous returns on bonds of all maturities are perfectly correlated. These characteristics are inconsistent with reality and motivate the development of multi-factor models.

** Multi-factor models. ** Brennan and Schwartz [ ** 10 ** ] propose an interest rate model based on the assumption that the whole term structure can be expressed as a function of the yields of the longest and shortest maturity default-free bonds. Longstaff and Schwartz [ ** 38 ** ] develop a two factor model of the term structure based on the framework of Cox, Ingersoll and Ross [ ** 18 ** ]. The two factors are the short-term interest rate and the instantaneous variance of changes in this short-term interest rate (volatility of the short-term interest rate). Therefore the prices of contingent claims reflect the current levels of the interest rate and its volatility. Langetieg [ ** 36 ** ] develops a general framework where the short-term interest rate is expressed as the sum of a number of underlying stochastic factors. The model is essentially an extension of Vasicek's approach where the evolution of the short-term interest rate is subject to multiple sources of uncertainty.

The use of two or more factors improves the explanatory power of the models, but increases the degree of numerical complexity. Identifying additional factors is quite difficult and cumbersome numerical procedures need to be used. Although not always explicitly stated, the multi-factor models rely on the assumption of market completeness. This means there must be at least as many tradable assets as there are sources of uncertainty. If this is not the case, the market is incomplete and there are stochastic fluctuations in the Brownian motions which are not picked up as price changes in some asset, and hence cannot be hedged.

** An attempt at preference-free pricing. ** In the above models the underlying stochastic state variables are interest rates or other non-tradable securities. This means that the resulting valuation formulae for contingent claims depend on investor preferences, and empirical approximations must be used to estimate the investor-specific variables.

In an attempt to avoid this problem, Ball and Torous [ ** 4 ** ]proposeamodel where the underlying state variable is the bond directly. The price of a risk- free zero coupon bond is assumed to follow a Brownian bridge process. The specification of this process ensures that the price of the bond converges to its face value at maturity. Also, since this underlying state variable is a tradable security, a preference-free closed-form valuation formula for European options may be derived. However, this model has shortcomings which make it unsuitable.

** Fitting the initial term structure. ** The above models attempt to model interest rates so as to produce a realistic future yield curve. No explicit attempts are made to match the current observed term structure. CIR [ ** 18 ** ] mention a possible extension to their model that allows a time-dependent drift term and point out that information contained within the initial term structure could be used to determine the drift without placing any restrictions on its functional form. This point was taken up much later by Hull and White (HW) [ ** 29 ** ] in their extension of the Vasicek and CIR models. Hull and White propose an extension to these models allowing time-dependent drift and volatility parameters. The extended Vasicek model allows analytical solutions for bonds and bond options. Model parameters, including those involving the market price of risk, are determined in terms of the initial term structure. This approach allows an exact fit to the initial term structure of interest rates and possibly also interest rate volatilities.

Black, Derman and Toy (BDT) [ ** 6 ** ] developed a one-factor discrete time model of the term structure. A binary tree of one period interest rates is constructed in such a way that the rate and transition probabilities at each node match an initial observed term structure of interest rates and volatilities. Here the one-period rate is the analogue of the short-term interest rate within a continuous time setting. This model is in fact a time-discretisation of a diffusion model where the short-term interest rate is lognormal.

** Modelling the forward rate. ** Ho and Lee [ ** 27 ** ] introduced a new approach to term structure modelling. Instead of modelling the short-term interest rate, they developed a discrete time model of the evolution of the whole yield curve. The short-term interest rate is a single point on the yield curve, which, in one-factor models, is assumed to be the only factor determining the entire yield curve. The Ho and Lee model admits an arbitrary specification of the initial yield curve, so it may be calibrated to the observed initial yield curve.

Heath, Jarrow and Morton (HJM) [ ** 25 ** ] developed a general framework of interest rate dynamics allowing an arbitrary specification of the initial term structure. They approached the problem by exogenously specifying the dynamics of instantaneous, continuously compounded forward rates. Rather than a traditional no arbitrage argument (as used, for example, in the derivation of the Vasicek model) they use a change of probability measure technique initially formulated by Harrison and Kreps [ ** 23 ** ] and Harrison and Pliska [ ** 24 ** ]. This involves transforming to the risk-neutral measure under which all asset prices have the same drift, that is the risk-free rate of interest. Within a complete market, without profitable arbitrage opportunities, this risk-neutral measure is unique and allows all discounted ^{ [2] } asset prices to be martingales. Default-free zero coupon bonds and derivative securities may now be valued by taking the expectation, under the risk-neutral measure, of the discounted terminal (maturity) value.

The Ho and Lee model is a special case of the HJM framework and may be viewed as its discrete time predecessor.

^{ [1] } A diffusion process is a Markovian process for which all realisations or sample functions { * X _{ t } ,t * ˆ [0, ˆ )} are continuous. A Markovian process has the characteristic that given the value of

*X*, the values of

_{ t }*X*do not depend on the values of

_{ s },s > t*X*. Brownian motion is a diffusion process. [

_{ u },u < t**34**]

^{ [2] } Here, ˜discounted' implies that the asset prices are expressed as a ratio of the money market account.