Term-structure modeling Hull-White model

The first generation of term structure models started with a finite factor modeling of the process dynamics with constant coefficients (e.g. Vasicek [73], Brennan and Schwartz [10], Cox, Ingersoll, and Ross [22]). Due to the fact that this type of models are inconsistent with the current term structure, the second generation of models exhibits time dependent coefficients (e.g. Hull and White [41]). A completely different approaeh starts from the direct modeling of the forward rate dynamies, by using the initial term strueture as an input (e.g. Ho, and Lee [39], Heath, Jarrow, and Morton [35]). 

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) ... 

Hull, J., White, A., 1994. Numerical procedures for implementing term structure models II two-factor models. J. Deriv. 2 (2), 37 8. 

John C. Hull and Alan White, Numerical Procedures for Implementing Term Structure Models I Single-Factor Models, of Derivatives 2, no. 1 (1994), pp. 7-16. 

The academic literature and market participants have proposed a large number of alternatives to the Vasicek term-structure model and models, such as the Hull-White model, that are based on it. Like those they seek to replace, each of the alternatives has advantages and disadvant es. 

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) ... 

The Hull-White model can be fitted to an initial term structure, and also a volatility term structure. A comprehensive analysis is given in Pelsser (1996) as well as James and Webber (2000). 

The problem with some of the models just discussed is that they generate their own term structures that, in the absence of adjustment, do not match the term structure observed in the market. A category of arbitrage-free models proposed by Ho and Lee, Hull and White, and Black, Herman, and Toy seek to eliminate this problem. For example, the Black, Herman and Toy model enjoys a degree of popularity among market practitioners because (1) it takes account of and matches the term structure observed in the market, (2) it eliminates the possibility of... 

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