Chapter 11: The Heath, Jarrow and Morton Model


Chapter 11: The Heath, Jarrow and Morton Model

Overview

Heath, Jarrow and Morton (HJM) [ 25 ] 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:

  1. A stochastic structure is imposed on the evolution of the forward rate curve.

  2. 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.

  3. 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.



11.1 Initial specifications

HJM develop their model within a continuous trading economy, with trading interval [0, ], > 0 fixed. Uncertainty within the economy is represented by the probability space ( , F,Q ), where represents the state space, F the ƒ -algebra representing all measurable events and Q the probability measure. Information becomes available over the trading period according to the filtration { F t : t ˆˆ [0, ]} which is generated by n independent Brownian motions { z 1 ( t ), , z n ( t ): t ˆˆ [0, ]} with n 1.

Assume there exist default-free zero coupon bonds with maturities on each trading day T , T ˆˆ [0, ]. If P ( t, T ) represents the time t price of a T -maturity bond, where T ˆˆ [0, ] and t ˆˆ [0, T ], then the following must be true:

Define the time t instantaneous forward rate for time T , T > t as:

Solving this differential equation for the bond price yields:

The short- term interest rate at time t is the instantaneous forward rate for time t , hence:

Alternatively, expressed in terms of the bond price [1] :

Hence the short-term interest rate may be interpreted as the rate of return on an instantaneously maturing bond.

[1] Here, make use of the Taylor series expansion of the natural logarithm of a number:

Consider:

By definition h is small, so P ( t, T ) is only slightly greater than P ( t, T + h ), and is only slightly larger than 1; hence:

Therefore applying this expansion:

since the higher order terms are negligibly small by the definition of h .