One-Factor Term-Structure Models

Again, we start with a one-factor term structure model with USV and show that the exponential affine guess... 

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) ... 

In this chapter, we have considered both equilibrium and arbitrage-free interest-rate models. These are one-factor Gaussian models of the term structure of interest rates. We saw that in order to specify a term structure model, the respective authors described the dynamics of the price process, and that this was then used to price a zero-coupon bond. The short-rate that is modelled is assumed to be a risk-free interest rate, and once this is modelled, we can derive the forward rate and the yield of a zero-coupon bond, as well as its price. So, it is possible to model the entire forward rate curve as a function of the current short-rate only, in the Vasicek and Cox-Ingersoll-Ross models, among others. Both the Vasicek and Merton models assume constant parameters, and because of equal probabilities of forward rates and the assumption of a normal distribution, they can, xmder certain conditions relating to the level of the standard deviation, produce negative forward rates. 

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... 

The Vasicek model was the first term-structure model described in the academic literature, in Vasicek (1977). It is a yield-based, one-factor equilibrium model that assumes the short-rate process follows a normal distribution and incorporates mean reversion. The model is popular with many practitioners as well as academics because it is analytically tractable—that is, it is easily implemented to compute yield curves. Although it has a constant volatility element, the mean reversion feature removes the certainty of a negative interest rate over the long term. Nevertheless, some practitioners do not favor the model because it is not necessarily arbitrage-free with respect to the prices of actual bonds in the market. 

Equation (5.15) describes one structure factor in terms of diffractive contributions from all atoms in the unit cell. Equation (5.16) describes one structure factor in terms of diffractive contributions from all volume elements of electron density in the unit cell. These equations suggest that we can calculate all of the structure factors either from an atomic model of the protein or from an electron density function. In short, if we know the structure, we can calculate the diffraction pattern, including the phases of all reflections. This computation, of course, appears to go in just the opposite direction that the crystallographer desires. It turns out, however, that computing structure factors from a model of the unit cell (back-transforming the model) is an essential part of crystallography, for several reasons. 

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. 

So, now we have determined that a short-rate model is related to the dynamics of bond yields and therefore may be used to derive a complete term structure. We also said that in the same way the model can be used to value bonds of any maturity. The original models were one-factor models, which describe the process for the short-rate r in terms of one source of uncertainty. This is used to capture the short-rate in the following form ... 

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. 

Kim, J., 1993. A Discrete-Time Approximation of a ONE-FACTOR MARKOV MODEL of the Term Structure of Interest Rates. Graduate School of Industrial Administration, Camegie-Mellon University, Pittsburgh, PA. 

CIR wrote arguably the first of several papers developing one-factor models of the term structure of interest rates. Other papers which were... 

Wolfgang M. Schmidt, On a General Class of One-Factor Models for the Term Structure of Interest Rates, Finance and Stochastics 1 (1997), pp. 3-24. 

Thus, the theoretical probabilistic structural model makes it possible to express the probabilities of the distributions of shifts of A ions of the distributions of shifts of A ions in the crystal lattice, both in magnitudes and directions, in terms of only one positional fitting parameter and calculate the contribution of these spontaneous shifts to the anisotropic factor W. Within these models, consideration of the anisotropy of the factor W does not lead to an increase in the number of fitting parameters of the theory. 

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