Heath-Jarrow-Morton HJM Model

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. 

The single-factor HJM model captures the change in forward rates at time t, with a maturity at time T, using 

Consider a forward-rate term structure/JO, T) that is T-integrable— for which the integral can be computed in terms of the variable T. The dynamics of this structure may be described by the stochastic differential equation (4.17). 

In some texts the term k is used for a. a = the constant volatility level 

By applying Ito calculus, i.e., Ito s lemma, (4.17) can be transformed to solve for the price of an asset. Taking the integral of expression (4.17) results in equation (4.18), which derives the forward rate, 

Equation (4.18) assumes that the forward rate is normally distributed. Crucially, the forward rates with maturitiesy (0,l),/ (0,2).y (0, 7) are assumed to be perfectly correlated. The random element is the Brownian motion dz. The impact of this process is felt over time, rather than over different maturities. 

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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. 

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. 

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... 

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