The Heath-Jarrow-Morton Interest Rate Model

Bill Burns, Ph.D.
Vice President, Quantitative Research

Ying Shen, Ph.D.
Vice President, Quantitative Research

In this article, we briefly review the best-known term structure models with a focus on the Heath-Jarrow-Morton (HJM) model. The BondEdge term structure model implemented in BondEdge is a member of the HJM class.

Background 

All of the well-known interest rate models in the market are either short-rate models or forward-rate models. The Ho-Lee model is an early example of arbitrage-free modeling of the forward-rate dynamics. A generalized framework for forward-rate modeling was subsequently developed in the celebrated work of Heath-Jarrow-Morton. A short-rate model posits a short-rate random process, ,and a risk-neutral probability measure . The short-rate process is typically Markovian so there is a partial differential equation associated with each model. Following a suggestion by Cox, Ingersoll, and Ross, we can fit the endogenous term structure and volatility structure of short rates to the observed term structure. The class of short-rate models includes the Rendleman-Bartter model, the Vasicek model, the Cox-Ingersoll-Ross (CIR) model, the Hull-White (HW) model, the Black-Derman-Toy (BDT) model, the Black-Karasinski (BK) model, the Jamshidian model, and the Sandmann-Sondermann model.

There are many issues one has to consider when selecting a model for a valuation system, and we highlight a few of these issues here. First, one should examine the effectiveness of the hedging performance of a model. Models that exhibit better hedging performance are more likely to predict accurate future market prices of vanilla derivatives such as at-the-money caps and swaptions. These models enable risk managers to correctly measure risk exposure. Secondly, calibration is an integral part of any interest rate model. For a given class of securities, the ease and robustness of parameter estimation procedures often lead to a significantly narrow list of choices among the known models. Thirdly, the choice of a particular model should reflect the specific features of the securities being analyzed. For example, some short-rate models such as BDT and BK model are not suitable for valuing a portfolio of mortgage-backed securities since a long-term Treasury par yield such as the 10-year yield is a major factor that drives the mortgage market and these short-rate models do not explicitly provide information on future yield curves as needed to model MBS prepayments.

There are few studies on the empirical performance of term structure models for the valuation of interest rate derivatives. Using data from the German market for interest-rate warrants for the four-year period from 1990 through 1993, Buler, Uhrig-Homburg, Walter and Weber investigated which models are better suited in supporting interest-rate risk management. They tested seven one- and two-factor models in both the forward- and short-rate classes. One remarkable conclusion of their study is that two-factor models may not necessary outperform one-factor models in each class. In fact, by imposing some additional risk management criteria, they conclude that a one-factor Heath-Jarrow-Morton (HJM) model with linear proportional volatility outperforms all other models.

   (1)

where the drift and the volatility can depend on the history of the Brownian motion and on the forward-rates up to time t . In particular, the forward rate process can be either normal or lognormal.

Equation (1) is equivalent to

   (1')

From equation (1), we know that the HJM model is actually a class of forward-rate models. All existing forward-rate models in the market belong to that class. In fact, one can show that there is a mathematical transformation that makes every short-rate model an HJM model. Conversely, it is easy to see that HJM models are short-rate models since

The starting point of the HJM approach is the observed yield curve, determined either by zero coupon bond prices , or by the instantaneous forward rates . Note that zero coupon bond prices and forward rates are equivalent because of the following equations

  (2)

     (3) 

Therefore, either the zero coupon bonds or the forward rates can be taken as the equivalent building block for the HJM model. If we proceed to derive the HJM model by using equation (1), there exists a martingale measure equivalent to such that the following constraint condition holds for arbitrary   such that

     (4)

(If you are interested in learning more about the details of the HJM model, please contact Interactive Data Fixed Income Analytics Client Services at (800) 228-9715 for a Interactive Data Fixed Income Analytics white paper on the subject.)

Model Implications

Equation (4) states that the drift term, and hence the interest-rate term structure, are determined by the entire volatility term structure of interest rates. One of the reasons the HJM model has encountered great popularity in the market is because the analytic tractability of the stochastic process of interest rates is guaranteed by equation (4). Various numerical techniques are readily available to practitioners for pricing and hedging options by using equation (4). We remark that a general HJM model is non-Markovian, i.e., an ‘up’ move for the yield curve followed by a ‘down’ move does not lead to the same result as a ‘down’ move followed by an ‘up’ move. As a result, a general HJM process cannot be mapped to a recombining tree. This makes HJM model a more computationally intensively than some Markovian models such as the Ho-Lee and the Hull-White models.

In order to implement the HJM model, it is necessary to derive the discrete versions of equation (1) and equation (4). One may assume that trades occur every units of time and divide into sub-periods . This implementation procedure is a straightforward calculus exercise. Details of implementation algorithms can be found in the research papers of Das and Amin-Bodurtha. A lattice approach or Monte Carlo approach can be used depending on the specification of the interest-rate options. For example, Monte Carlo techniques are generally quite effective for handling path-dependent securities such as MBS, whereas, lattice approaches are more effective for callable bonds. Since the HJM model is a class of general arbitrage-free interest-rate term structures, we also need to specify the volatility structure in the implementation. Different functional forms of the volatility lead to different interest-rate models. In practice, it is often convenient to assume that the time component of the volatility functional form consists only of time to maturity; such a functional form is said to be homogeneous in time to maturity.

We remark that very few comparison studies have been performed on HJM models with different volatility term structures. Amin and Morton conducted some empirical studies in which they tested the one-factor HJM model with parsimonious parameterizations (one and two parameters) of the volatility functional form. They found that the number of parameters has a stronger impact on the performance of the model than does the form the model used. They also confirm the general perception that two-parameter models tend to fit prices better. However, one-parameter models result in implied parameter estimates which are more stable and consistently outperform the other models studied. They found that the model with constant volatility is preferable among all one-parameter models.

Conclusion

HJM is one of the most elegant and popular models in the market place. The BondEdge term structure model currently implemented in BondEdge also belongs to the HJM class. It has the benefits that it is arbitrage-free and it is analytically tractable.

The HJM model allows for a wide class of volatility term structures when determining the interest rate process. It is capable of handling a wide range of interest-rate sensitive derivatives using market observables. The one-factor HJM framework can easily be extended to a multi-factor model to handle other exotic derivative products. Interactive Data Fixed Income Analytics will continue its innovative research on the one- and multi-factor HJM models and their applications for practical fixed-income portfolio management.