Interest-rate models fitting process

In Chapter 2, we introduced the concept of stochastic processes. Most but not all interest-rate models are essentially descriptions of the short-rate models in terms of stochastic process. Financial literature has tended to categorise models into one of up to six different types, but for our purposes we can generalise them into two types. Thus, we introduce some of the main models, according to their categorisation as equilibrium or arbitrage-free models. This chapter looks at the earlier models, including the first ever term structure model presented by Vasicek (1977). The next chapter considers what have been termed whole yield curve models, or the Heath-Jarrow-Morton family, while Chapter 5 reviews considerations in fitting the yield curve. 

The Hull-White (1990) model is an extension of the Vasicek model designed to produce a precise fit with the current term structure of rates. It is also known as the extended Vasicek model, with the interest rate following a process described by Equation (3.48) ... 

This chapter considers some of the techniques used to fit the model-derived term structure to the observed one. The Vasicek, Brennan-Schwartz, Cox-Ingersoll-Ross, and other models discussed in chapter 4 made various assumptions about the nature of the stochastic process that drives interest rates in defining the term structure. The zero-coupon curves derived by those models differ from those constructed from observed market rates or the spot rates implied by market yields. In general, market yield curves have more-variable shapes than those derived by term-structure models. The interest rate models described in chapter 4 must thus be calibrated to market yield curves. This is done in two ways either the model is calibrated to market instruments, such as money market products and interest rate swaps, which are used to construct a yield curve, or it is calibrated to a curve constructed from market-instrument rates. The latter approach may be implemented through a number of non-parametric methods. 

BDT, HW, and BK models extended the Ho-Lee model to match a term structure volatility curve (for example the cap prices) in addition to the term structure. The BK model is a generalization of the BDT model and it overcomes the problem of negative interest rates assuming that the short rate r is the exponential of an Ornstein-Uhlenbeck process having time-dependent coefficients. It is popular with practitioners because it fits the swaption volatility surface well. Nevertheless, it does not have closed formulae for bonds or options on bonds. 

Heidner et developed a kinetic model of the dissociation process and attempted to define a subset of critical rate constants by fitting to dissociation rate data. Unfortunately, several of the rate constants of interest could not be uniquely determined as they were strongly correlated. Consequently, two limiting rate constant sets were offered as constraints on a yet-to-be-determined set of final values. The limiting parameter sets (models 1... 

As with the kinetic analyses of homogeneous rate processes, quantitative comparisons are made between the experimentally measured data for a reaction of interest and the curve shapes of the various rate equations (Table 5.1) to identify the applicable kinetic model. This can be approached in several ways (29,105). One traditional method is to plot graphs of g(a) against time and decide which, from the available expressions (Table 5.1), provides the best (linear) representation, or fit. There is no general agreement on what criteria constitute a best or a satisfactory fit. The... 

The numerical model CoTReM was applied to investigate the depth dependent effects of respiration and redox processes related to CaCO dissolntion (Pfeifer et al. 2002 cf. Fig. 15.16 in chapter 15). Interestingly, if calculated until a steady-state situation is reached, the model-derived calcite dissolution and precipitation rates produce an almost perfect fit to the measured CaC03 profile in the sediment (Fig. 9.8), which suggests that 90 % of the CaC03 flux to the sea floor is redissolved in the sediment. 

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