In term structure models such as the Vasicek and CIR models, the starting point is the dynamics of the short-term interest rate. The drift and volatility are specified such that the short-term interest rate is Markovian. In the HJM framework the forward rate volatility and initial forward rate curve are used to characterise the term structure. Such a specification may give rise to non-Markovian short-term interest rate dynamics. In fact many HJM-based models cannot occur in a framework of Markovian short-term interest rates. The Markovian property of the short-term interest rate is desirable since it allows for simpler numerical valuation procedures of the term structure and interest rate contingent claims since:

The term structure at time t is a function of t, maturity T and the time t short-term interest rate.

The evolution of the short-term interest rate may be modelled using a recombining tree or lattice, which has significant implications for computational efficiency.

Carverhill andJeffrey characterise restrictions on the volatility structure of forward rates that lead to Markovian short-term interest rate dynamics.

Heath, Jarrow and Morton (HJM) present a unifying framework for term structure models. This framework introduces a formal elegance and generality to the interest rates modelling problem. It shows that the absence of arbitrage results in a link between the volatility of discount bonds and the drift of forward rates. In fact, in the risk-neutral world, the forward rate drift is completely determined by the specification of the discount bond volatility function. Previously developed models can be shown to be special cases of this general framework.

HJM specify an initial forward rate curve and a stochastic process describing its subsequent evolution. To ensure that the stochastic process is consistent with an arbitrage-free (and hence equilibrium) economy, it is chosen such that there exists an equivalent martingale probability measure.

The HJM methodology encompasses several new concepts:

• A stochastic structure is imposed on the evolution of the forward rate curve.
• Contingent claim prices are not dependent on the market prices of risk. This implies that inversion of the term structure to solve for these market prices of risk is not required.
• Evolution of the term structure is determined by the short-term interest rate, which follows a process influenced by a number of stochastic variables.

The derivation of the HJM model is rather technical in nature. It consists of a series of conditions to determine a restriction on the drift of the forward rate which ensures a risk-neutral and arbitrage-free pricing framework. In §11.2 we impose conditions to ensure well-behaved forward rate, money market and bond price processes. The relative or discounted bond price process is also defined. In §11.3 we examine necessary and sufficient conditions required to ensure the existence of a unique equivalent probability measure under which the discounted bond prices are martingales. This is equivalent to ensuring an arbitrage-free pricing framework. In §11.4 we specify a final condition ensuring a unique martingale measure across all bond maturities. It is here that the forward rate drift restriction is explicitly specified. In later sections I examine contingent claim pricing within the HJM framework, compare earlier term structure models to the HJM framework and examine conditions under which this framework gives rise to a Markovian short-term interest rate process.

The discrete time Black, Derman and Toy model, makes provision for two time-dependent factors: the mean short-term interest rate and the short-term interest rate volatility. The continuous time equivalent of the model clearly shows that the rate of mean reversion is a function of the volatility. This is equivalent to future short-term interest rate volatilities being fully determined by the observed volatility term structure. This dependence makes it impossible to specify these two factors independently.

Black and Karasinski (BK) develop a model, within a discrete time framework, where the target rate, mean reversion rate and local volatility are deterministic functions of time. The specification of three time-dependent factors allows the future short-term interest rate volatilities to be specified independently of the initial volatility term structure.

As in the BDT model, the short-term interest rate is assumed to have a lognormal distribution at any time horizon. The standard assumptions underlying perfect markets are also made.