Heath-Jarrow-Morton

In Chapter 2, we introduced the concept of stochastic processes. Most but not all interest-rate models are essentially descriptions of the short-rate models in terms of stochastic process. Financial literature has tended to categorise models into one of up to six different types, but for our purposes we can generalise them into two types. Thus, we introduce some of the main models, according to their categorisation as equilibrium or arbitrage-free models. This chapter looks at the earlier models, including the first ever term structure model presented by Vasicek (1977). The next chapter considers what have been termed whole yield curve models, or the Heath-Jarrow-Morton family, while Chapter 5 reviews considerations in fitting the yield curve. 

Whole yield curve models such as Heath-Jarrow-Morton ... 

Cheyette, O., 1992. Term structure dynamics and mortgage valuation. J. Fixed Income 1 (4), 28 1. Das, S., 1997. A Direct Discrete-Time Approach to Poisson-Gaussian Bond Option Pricing in the Heath-Jarrow-Morton Model Working Paper. Harvard Business School, Boston, MA, pp. 1 14. Das, S., Foresi, S., 1996. Exact solutions for bond and option prices with systematic jump risk. Rev. Deriv. Res. 1, 1-24. 

The approach described in Heath-Jarrow-Morton (1992) represents a radical departure from earlier interest rate models. The previous models take the short rate as the single or (in two- and multifactor models) key state variable in describing interest rate dynamics. The specification of the state variables is the fundamental issue in applying multifactor models. In the HJM model, the entire term structure and not just the short rate is taken to be the state variable. Chapter 3 explained that the term structure can be defined in terms of default-free zero-coupon bond prices or yields, spot rates, or forward rates. The HJM approach uses forward rates. 

Some of the newer models refer to parameters that are difficult to observe or measure direcdy. In practice, this limits their application much as B-S is limited. Usually the problem has to do with calibratii the model properly, which is crucial to implementing it. Galibration entails inputtii actual market data to create the parameters for calculating prices. A model for calculating the prices of options in the U.S. market, for example, would use U.S. dollar money market, futures, and swap rates to build the zero-coupon yield curve. Multifactor models in the mold of Heath-Jarrow-Morton employ the correlation coefficients between forward rates and the term structure to calculate the volatility inputs for their price calculations. 

Practitioners increasingly model credit risk as they do interest rates and use spread models to price associated derivatives. One such model is the Heath-Jarrow-Morton (HJM) model described in chapter 4. This analyzes interest rate risk, default risk, and recovery risk—that is, the rate of recovery on a defaulted loan, which is always assumed to retain some residual value. 

The first generation of term structure models started with a finite factor modeling of the process dynamics with constant coefficients (e.g. Vasicek [73], Brennan and Schwartz [10], Cox, Ingersoll, and Ross [22]). Due to the fact that this type of models are inconsistent with the current term structure, the second generation of models exhibits time dependent coefficients (e.g. Hull and White [41]). A completely different approaeh starts from the direct modeling of the forward rate dynamies, by using the initial term strueture as an input (e.g. Ho, and Lee [39], Heath, Jarrow, and Morton [35]). 

Heath, Jarrow, and Morton (HJM) derived one-factor and multifactor models for movements of the forward rates of interest. The models were complex enough to match the current observable term structure of forward rate and by equivalence the spot rates. Ritchken and Sankara-subramanian provide necessary and sufficient conditions for the HJM models with one source of error and two-state variables such that the ex post forward premium and the integrated variance factor are sufficient... 

The original interest rate models describe the dynamics of the short rate later ones—known as HJM, after Heath, Jarrow, and Morton, who created the first whole yield-curve model—focus on the forward rate. 

Heath D, Jarrow R, Morton A (1992) Bond Pricing and the Term Structure of Interest Rates A New Methodology for Contingent Claims Evaluation. Econometrica 60 77-105. 

A landmark development in the longstanding research into yield ciuve modelling was presented by David Heath, Robert Jarrow and Andrew Morton in their 1989 paper, which formally appeared in volume 60 of Econometrica (1992). The paper considered interest-rate modelling as a stochastic process, but applied to the entire term structure rather than only the short-rate. The importance of the HJM presentation is this in a market that permits no arbitrage, where interest... 

Heath, D., Jarrow, R., Morton, A., 1990. Bond pricing and the term structure of interest rates a discrete time approximation. J. Financ. Quant. Anal. 25 (4), 419M40. 

David Heath, Robert Jarrow, and Andrew Morton, Bond Pricing and the Term Structure of Interest Rates A Discrete Time Approximation, Journal of Financial and Quantitative Analysis 25 (1990), pp. 419—440 Contingent Claim Valuation with a Random Evolution of Interest Rates, Review of Futures Markets 9 (1990), pp. 54-76 Bond Pricing and the Term Structure of Interest Rates, Econometrica 60, no. 1 (1992), pp. 77-105. 

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