Brennan-Schwartz interest rate model

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. 

The model described in Brennan and Schwartz (1979) uses the short rate and the long-term interest rate to specify the term structure. The long-term rate is defined as the market yield on an irredeemable, or perpetual, bond, also known as an undated or consol bond. Both interest rates are assumed to follow a Gaussian-Markov process. A Gaussian process is one whose marginal distribution, where parameters are random variables, displays normal distribution behavior a Markov process is one whose future behavior is conditional on its present behavior only, and independent of its past. A later study, Longstaff and Schwartz (1992), found that Brennan-Schwartz modeled market bond yields accurately. 

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