HJM model

Starting from a HJM-model for the dynamics of the forward rates, that is driven by N Brownian motions we have... 

We derive Brownian Field models in chapter (6). Furthermore, we extend the traditional HJM-models by an additional stochastic process for the volatility in (see chapter (7)). 

Note that we obtain the identical characteristic functions as in section (5.2.1). The new formula for the RF model differs from the corresponding HJM-model only in the variance... 

Similar to section (5.3.4), where we postulated a traditional multi-factor HJM-model, there exists no closed-form solution for the price of a coupon bond option assuming a multiple field term structure model. In the following, we show that the moments of the random variable V(7o, 7i ) can be computed in closed-form, even if the underlying random variable is driven by N admissible Random Fields. 

Now, comparing the option price postulating the T-differentiable RF with the corresponding price of a traditional HJM-model (see chapter (5.3)), we find that the difference in the option price is dominated hy two partially offsetting effects. 

The most commonly used models are the Hull-White type models which are relatively straightforward to implement, although HJM models are also more... 

A landmark development in interest-rate modelling has been the specification of the dynamics of the complete term stracture. In this case, the volatility of the term structure is given by a specified functiOTi, which may be a function of time, term to maturity or zero-coupon rates. A simple approach is described in the Ho-Lee model, in which the volatility of the term structure is a parallel shift in the yield curve, the extent of which is independent of the current time and the level of current interest rates. The Ho-Lee model is not widely used, although it was the basis for the HJM model, which is widely used. The HJM model describes a process whereby the whole yield curve evolves simultaneously, in accordance with a set of volatility term structures. The model is usually described as being one that describes the evolution of the forward rate however, it can also be expressed in terms of the spot rate or of bond prices (see, e.g., James and Webber (1997), Chapter 8). For a more detailed description of the HJM framework refer to Baxter and Rennie (1996), Hull (1997), Rebonato (1998), Bjork (1996) and James and Webber (1997). Baxter and Reimie is very accessible, while Neftci (1996) is an excellent introduction to the mathematical background. 

The general form of the HJM model is very complex, not surprisingly as it is a multi-factor model. We begin by describing the single-factor HJM model. This section is based on Chapter 5 of Baxter and Rennie, Financial Calculus, Cambridge University Press (1996), and follows their approach with permission. This work is an accessible and excellent text and is highly recommended. 

A default-free zero-coupon bond can be defined in terms of its current value imder an initial probability measure, which is the Wiener process that describes the forward rate dynamics, and its price or present value under this probability measure. This leads us to the HJM model, in that we are required to determine what is termed a change in probability measure , such that the dynamics of the zero-coupon bond price are transformed into a martingale. This is carried out using Ito s lemma and a transformatiOTi of the differential equation of the bmid price process. It can then be shown that in order to prevent arbitrage, there would have to be a relationship between drift rate of the forward rate and its volatility coefficient. 

The HJM model (1992) is a general approach which is a multi-factor whole yield curve model, where arbitrary changes in the entire term structure can be one of the factors. In practice, because of the mass of data that is required to derive the yield curve, the HJM model is usually implemented by means of Monte Carlo simulation, and requires powerful computing systems. The model is described in the next section. 

A multi-factor model of the whole yield curve has been presented by Heath et al. (1992). This is a seminal work and a ground-breaking piece of research. The approach models the forward curve as a process arising from the entire initial yield curve, rather than the short-rate only. The spot rate is a stochastic process and the derived yield curve is a function of a number of stochastic factors. The HJM model uses the current yield curve and forward rate curve, and then specifies a continuous time stochastic process to describe the evolution of the yield curve over a specified time period. 

The HJM model describes the evolution of forward rates given an initial forward rate curve, which is taken as given. For the period T e [0, t], the forward ratef(t, T) satisfies the Equation (4.28) ... 

The expression describes a stochastic process composed of n independent Wiener processes, from which the whole forward rate curve, from the initial curve at time 0, is derived. Each individual forward rate maturity is a function of a specific volatility coefficient. The volatility values ( t, t, T, w)) are not specified in the model and are dependent on historical Wiener processes. From Equation (4.28) following the HJM model, the spot rate stochastic process is given by Equation (4.29) ... 

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... 

Under the one-factor HJM model corresponding to the Ho-Lee model, a European option on a coupon bond can be valued as a portfolio of options contingent on zero discount bonds with maturities Ti,T2,...,Tm- Let Tq be the maturity of such a European option. 

Hence, the value at time 0 of a European call option with maturity Tq and strike price K on the coupon bearing bond, under the one-factor HJM model described above, is given by... 

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... 

The single-factor HJM model states that, given an initial forward-rate term structure/( , T) at time t, the forward rate for each maturity Tis given by (4.20), which is the integral of (4.19). 

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