Brennan-Schwartz model

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. 

In the model, the dynamics of the logarithm of the short rate are defined by equation (4.13). 

/ represents the relationship between the short rate, r, and the long-term rate, /. The short rate changes in response to moves in the level of the long rate, which follows the stochastic process described in equation (4.14). 

A = the risk premium associated with the long-term interest rate 

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The modified Brennan-Schwartz model is used in the markets which describes a realistic process for changes in the yield curve and is relatively straightforward to implement only two variables are required to model the entire term structure. 

Two-factor models were based on a second source of random shocks. Two factor models were developed by Brennan and Schwartz, Fong and Vasicek, and Longstaff and Schwartz. However, Hogan " proved that the solution to the Brennan and Schwartz model explodes, that is reaches infinity in a finite amount of time with positive probability. The Brennan and Schwartz model shows that adding more factors may cause unseen problems. More complex multifactor models are described by Rebonato, and by Brigo and Mercurio. 

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 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]). 

So-called consol models such as Brennan and Schwartz. 

Brennan, M., Schwartz, E., 1982. An equilibrium model of bond pricing and a test of market efficiency. J. Financ. Quant. Anal. 17 (3), 301-329. 

Michael J. Brennan and Eduardo Schwartz, An Equilibrium Model of Bond Prices and a Test of Market Efficiency, Journal of Financial and Quantitative Analysis 17 (1982), pp. 301-329. 

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