Multi-Factor HJM models

In this section, we start from a simple multi-factor HJM term structure model and derive the drift term of the forward rate dynamics required to obtain an arbitrage-free model framework (see HJM [35]). Furthermore, we derive the equivalence between the HJM-firamework and a corresponding extended short rate model. Then, by applying our option pricing technique (see chapter (2)) we are able derive the well known closed-form solution for the price of an option on a discount bond (e.g. caplet or floorlet). 

On the contrary, there exists no closed-form solution for an option on a coupon bearing bond for multi-factor models. Furthermore, the characteristic function cannot be computed in closed-form and the Fourier inversion techniques are widely useless. Nevertheless, the moments of the underlying random variable can be computed and the lEE approach is applicable. 

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

For simplicity, we assume that the N sources of uncertainty are independent via 

as shown in HJM [35] the drift of the forward rate process must satisfy fi t, T) = r)cr (f, T) to obtain an arbitrage-free framework. This 

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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. 

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. 

The direct modeling of the term structure dynamics using a finite-factor HIM model (see chapter (5)) allows us to fit the initial term structure perfectly. Although the initial term structure is a model input, it does not permit consistency with the term structure fluctuations over time. Using e.g. a one-factor HJM-framework (see section (5.3.3)) implies that we are only able to model parallel shifts in the term structure innovations. When we relax this restriction through a multi-factor model this typically does not imply that we are able to capture aU possible fluctuations of the entire term structure. 

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. 

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