Multi-factor models

Unfortunately, there exists no closed-form solution for the transform St a + i(j)). This directly implies that we need a new method for the approximation of the single exercise probabilities Tlj a [ ] assuming a multi-factor model with more than one payment date. On the other hand, the transform Et (n) can be solved analytically for nonnegative integer numbers n. This special solutions of Et z) can be used to compute the n-th moments of the underlying random variable V To Ti ) under the Ti forward measure. Then, by plugging these moments in the lEE scheme we are able to obtain an excellent approximation of the single exercise probabilities (see e.g. section (5.3.3) and (5.3.4)). 

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. 

Jamshidian [42] derived a closed-form solution for a coupon bond option assuming a 1-factor model. Nevertheless this solution is not applicable when we extend our analysis to a multi-factor model framework. 

Now, it is straightforward to extend this approach to a general multi-factor model. Following the last section we have seen that the moments of the random variable... 

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. 

For these reasons, practitioners may prefer to use an arbitrage-free model if one can be successfully implemented and calibrated. This is not always straightforward, and under certain conditions, it is easier to implement an equilibrium multi-factor model (which we discuss in the next section) than it is to implement a multi-factor arbitrage-free model. Under one particular set of circumstances, however, it is always preferable to use an equilibrium model, and that is when reliable market data is not available. If modelling the term stmcture in a developing or emerging bond market, it will be more efficient to use an equilibrium model. 

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 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. 

It is generally accepted that one-factor models can be used for most bond applications where multi-factor models are more appropriate may be in the following situations ... 

The optimum approach would appear to be a combination of a one-factor model and a multi-factor model to suit individual requirements. However, this may not be practical it might not be ideal to have different parts of a bank using different models (although this does happen, desks across the larger investment banks sometimes use different models) and valuing instruments using different models. The key factors to focus on are accessibility, accuracy, appropriateness and speed of computation. 

Wei JZ (2003) A multi-factor, credit migration model for sovereign and corporate debts. J Int Money Finance 22 709-735... 

Figure 13.4 illustrates some of the factors known to be involved in the development of AD many of the known and putative links between factors are also shown. It should also be noted that the patterns of inter-factor modulation may be either positive or negative. However, it is clear that no single factor or combination of factors can explain all AD cases. It is best to conceptually model AD as a broad end point that can be reached in numerous ways. Similar multi-factorial models have been proposed for schizophrenia and depression (Chapters 11 and 12) and almost certainly underlie every other complex psychobiological concept. 

In summary while conceptually appealing, the application of complex multi-solute models for Sr sorption to zeolite is in the early stages of development. While preliminary results are encouraging, additional work is required to develop more efficient computational methods and develop an improved database for parameter estimation. The remainder of this section focuses on the simpler retardation factor approach. 

Fig. 5 Chargeability factor A/can be predicted by a Fig. 6 Permeability prediction from electrical multi-linear model composed by different behaviour and structures parameters of porous parameters formation factor F, water porosity O, solids, k Katz and Thompson model Hg-specific surface Asp and water permeability k for kjsc Johnson, Schwartz and co-workers different textures. model.

Chargeability factor M depends on the brine/gas saturation of porous solids. Figure 3 gives the relationship between the chargeability and brine saturation for two samples. We noted that the M decreases hardly with the decrease of the brine saturation. The presence of vugs and karsts pore types (sample 9-LS8) seems to speed up the decrease of the M Chargeability factor M can be explained by a multi-linear model composed of different structures parameters such as the formation resistivity factor, water porosity, Hg-specific surface area and water permeability, e.g.. Fig. 5. 

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