Multi-factor term structure 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). 

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. 

Note that the lEE approach again can be easily extended to multi-factor or multiple-Field term structure models. This property is directly linked to the fact that the moments of the underlying random variable can be computed in closed-form, regardless of the dimensionality of the underlying model structure. 

The traditional one-, two- and multi-factor equilibrium models, known as ajfine term structure models (see James and Webber, 2000 or Duffie, 1996, p. 136). These include Gaussian affine models such as Vasicek, Hull-White and Steeley, where the model describes a process with constant volatility and models that have a square-root volatility such as Cox-Ingersoll-Ross (CIR) ... 

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. 

In this chapter we consider multi-factor and whole yield curve models. As we noted in the previous chapter, short-rate models have certain drawbacks, which, though not necessarily limiting their usefulness, do leave room for further development. The drawback is that as the single short-rate is used to derive the complete term structure, in practice, this can be unsuitable for the calculation of bond yields. When this happens, it becomes difficult to visualise the actual dynamics of the yield curve, and the model no longer fits observed changes in the curve. This means that the accuracy of the model cannot be observed. Another drawback is that in certain equilibrium model cases, the model cannot be fitted precisely to the observed yield curve, as they have constant parameters. In these cases, calibration of the model is on a goodness of fit or best fit approach. 

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. 

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