2.12 More complicated model specifications

2.12 More complicated model specifications

The specific term structure model derived by CIR assumes the state of technology is represented by a single state variable, and randomness within the economy is explained by the stochastic dynamics of this variable. Hence bond prices of all maturities are determined by a single random variable, the shortterm interest rate. The model does allow some flexibility, since the term structure may assume a number of shapes , but the nature of single factor models implies that price changes in bonds of all maturities are perfectly correlated and independent of the path followed by the short-term interest rate to reach its current value.

Multi-factor models which allow a richer specification of the technology introduce more flexibility into the term structure, but often this is accompanied by an undesirable increase in complexity and lack of analytical tractability. The two models considered thus far always involve an explanatory variable that is not directly observable in the market. This is the market price of risk or factor risk premium. It is dependent on the utility function of individual investors which cannot be empirically determined. Multi-factor models will tend to have even more investor-specific, and hence unobservable, variables. At times, it may be possible to express these unobservable variables as functions of the endogenously determined prices (e.g. the risk-free rate of interest) and thereby eliminate them from the pricing model. This is the case with the Brennan and Schwartz (1979) model discussed in the next chapter.

2.13 Conclusion

The main characteristic of the CIR model is that current prices and stochastic properties of all contingent claims are derived endogenously. Since CIR use the rational asset pricing model to determine the term structure of interest rates, the following factors are all material in the derivation: investor anticipations, risk aversion, available investment alternatives and preferences with respect to timing of consumption. Equilibrium asset pricing principles are combined with appropriate models of stochastic processes describing the evolution of randomness in the economy, to derive consistent and possibly refutable theories .

The drawback of the CIR model is that it is only a general equilibrium model within their simplified and stylised economy. The investor-specific utility function always enters the model via the market price of risk; for calibration purposes it must be empirically estimated. Model risk will arise since reality is being forced into a simplified model. The following view may be taken of the CIR model:

  1. if the specification of the economic model is correct,

  2. if the stochastic process chosen for the short-term interest rate is in fact the ˜true' process describing its development,

  3. if the investor's utility function is fully specified,

then we may say that the model is fully specified and the endogenously derived term structure is the observed term structure. However, none of the above specifications is known with any certainty . In particular, the utility function is difficult to determine empirically. Hence, option prices and other factors derived from the CIR model cannot be seen as accurate quantifications of market characteristics, but rather as descriptive qualifications.

Later models assume that the general structure of the yield curve dynamics are known a priori . Details of various parameters, which are obtained from more fundamental factors [24] , are left unspecified. The current yield curve is used to fit the unspecified parameters. If the assumed structure of the model is an accurate description of reality then the unobservable quantities can be determined in this manner. Certain discrepancies will exist between observed market rates and those derived from the model. An analysis of these discrepancies will reveal their importance and effect as well as indicating whether they are the result of a poor model or a violation of the initial assumptions. Neither the Vasicek nor CIR models allows for complex yield curve patterns and hence tend to be poor representations of observed yield curves.

[24] Such as the utility function.