Webber, James

The background and mathematics to martingales can be found in Harrison and Kreps (1979) and Harrison and Pliska (1981) as well as Baxter and Rennie (1996). For a description of how, given that price processes are martingales, we are able to price derivative instruments, see James and Webber (2000, Chapter 3). 

James, J., Webber, N., 2000. Interest Rate Modelling. Wiley, Chichester, Chapters 3-5,7-9,15-16. Jamshidian, F., 1991. Bond and option valuation in the Gaussian interest rate model. Res. Finance 9, 131-170. 

An interest-rate model provides a description of the dynamic process by which rates change over time, in terms of a statistical construct, as well as a means by which interest-rate derivatives such as options can be priced. It is often the practical implementation of the model that dictates which type is used, rather than mathematical neatness or more realistic assumptions. An excellent categorisation is given in James and Webber (2000), who list models as being one of the following types ... 

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) ... 

There are also other types of models and we suggest that interested readers consult a specialist text James and Webber is an excellent start, which also contains detailed sections on implementing models as well as a comparison of the different models themselves. 

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). 

A landmark development in interest-rate modelling has been the specification of the dynamics of the complete term stracture. In this case, the volatility of the term structure is given by a specified functiOTi, which may be a function of time, term to maturity or zero-coupon rates. A simple approach is described in the Ho-Lee model, in which the volatility of the term structure is a parallel shift in the yield curve, the extent of which is independent of the current time and the level of current interest rates. The Ho-Lee model is not widely used, although it was the basis for the HJM model, which is widely used. The HJM model describes a process whereby the whole yield curve evolves simultaneously, in accordance with a set of volatility term structures. The model is usually described as being one that describes the evolution of the forward rate however, it can also be expressed in terms of the spot rate or of bond prices (see, e.g., James and Webber (1997), Chapter 8). For a more detailed description of the HJM framework refer to Baxter and Rennie (1996), Hull (1997), Rebonato (1998), Bjork (1996) and James and Webber (1997). Baxter and Reimie is very accessible, while Neftci (1996) is an excellent introduction to the mathematical background. 

The model is summarised here only readers interested in the derivation of the model are directed to the original paper or a discussion of it in Baxter and Rennie (1996), Hull (1997) or James and Webber (1997). To describe the model, we use the following notation ... 

James, J., Webber, N., 1997. Interest Rate Modelling. WUey, Chichester. 

For a good account of why this approach is not satisfactory see James and Webber (2000, Chapter 15). 

Curve-fitting techniques generally fit into two classes, as described, for example, in Chapter 15 of James and Webber (2000), parametric methods... 

A spline is a type of linear interpolation. It takes several forms. The spline function fitted using regression is the most straightforward and easiest to understand. Unfortunately, as illustrated in James and Webber (2000), section 15.3, when applied to yield-curve construction, this method can be overly sensitive to changes in parameters, causing curves to jump wildly. 

A good summary of the advantages and disadvantages of popular modeling methods can be found in James and Webber (2000), chapter 15-... 

The U.S. Federal Reserve uses an iterative technique to construct a term structure of expected inflation rates. First the nominal interest rate term structure is constructed using a version of the model described in Waggoner (1997) and discussed in James and Webber (2000). An initial assumed inflation term structure is then used to infer a term structure of real interest rates. This assumed inflation curve is usually set at a flat 3 or 5 percent. The real interest rate curve is then used to calculate an implied real interest rate forward curve. Next, the Fisher identity is applied at each point along the nominal and real interest rate forward... 

James, J., and N. Webber. 2000. Interest Rate Modeling. Chichester John Wiley Sons. 

Subsequently, James Sibley Watson, Jr. and Melville Webber created The Fall of the House of Usher (19Z8), choosing to adapt a work from Poe as a way to make their avant-garde filmmaking style more acceptable. 

James S. Webber Wadsworth Center, New York State Department of Health, Albany, New York... 

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