Modeling Interest-Rate Derivatives

The aim of this dissertation is to give an overview of the various techniques that can be used to consistently price interest-rate derivatives.

First, the traditional term structure of interest rates models are described (also called equilibrium models), which were developed by Vasicek, Cox, Ingersoll, and Ross, and Longstaff and Schwartz. The assumptions underlying these models and, more importantly, their ability to replicate the real world are examined. Moreover, it is shown how these models price various types of interest-rate derivative securities. Finally, some numerical examples are provided.

Afterwards, the more modern no-arbitrage models are dealt with. A comparison between these models and equilibrium models is made, focusing on their different approach to the absence of arbitrage: the former adopt the martingale framework developed by Harrison and Pliska; the latter use the approach of Vasicek. Subsequently, specific models are addressed, namely the Ho and Lee, the Black, Derman, and Toy, and the Hull and White models. These models have been chosen amongst many others because they are the models most used by practitioners in the real world. For each, a brief review is provided, in which the corresponding benefits and limitations are stressed.

A major problem with the short-rate models considered so far lies in insuring that they produce a realistic volatility function for zero-coupon bond yields. Therefore, the general approach of Heath, Jarrow, and Morton is considered, which models interest rates more realistically than the earlier short-rate models. A brief history of the approach is presented and then its structure is explored. Moreover, it is shown how the Heath, Jarrow, and Morton (HJM) framework is limited in practice, at least in its full generality, because of its intensive computational requirements.

Chapter 5 illustrates how derivatives can be priced within the HJM framework by means of Monte Carlo simulation. This provides a viable numerical method for derivative pricing, especially if techniques such as martingale variance reduction, which results in a notable improvement in efficiency compared to standard Monte Carlo simulation, are employed. In the second part of this chapter an empirical investigation of the Italian government bond market is undertaken, and explores the extension of the number of volatility factors in the HJM framework from one to two or, possibly, three. This investigation employs factor analysis of the covariance matrix of the historical changes in zero-coupon bond yields.