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Parameter estimation of stochastic interest rate models and applications to bond pricing Ananthanarayanan, A. L.


A partial equilibrium valuation model for a security, based on the idea of contingent claims analysis, was first developed by Black & Scholes. The model was considerably extended by Herton, who showed how the approach could be used to value liability instruments. Valuation models for default-free bonds, by treating them as contingent upon the value of the instantaneously riskfree interest rate, have been developed by Cox, Ingersoll & Ross, Brennan & Schwartz, Vasicek and Richards. There has, however, not been much attention directed towards the empirical testing of these valuation models of default-free bonds. This research is an attempt in that direction. Our attention is confined to retractable and extendible bonds. Central to arriving at any equilibrium model of bond valuation is the assumption about the instantaneously riskless interest rate process, since the bond value is treated as contingent upon it. These bond valuation models are partial equilibrium models, since the interest rate is assumed as exogenous to them. The choice of the interest rate process is made subject to some restrictions on its behaviour which are based on expected properties of interest rates. The interest rate process adopted in this study is a generalization of that used by Vasicek and Cox, Ingersoll & Ross., The properties of the chosen mathematical model are investigated to ascertain whether it conforms to those expected of an interest rate process based on economic reasoning. We go on to develop alternate estimation methods for the parameters of the interest rate process, using data on a realization of the process. One "exact" method and two others based on approximations are outlined. It is observed that the "exact" method is not available to the complete family of processes included in the continuous time stochastic specification assumed to model interest rates. The asymptotic properties of the estimators from the "exact" method are known from the existing literature. However, since we would have to adopt one of the approximate methods, we need to know something about the properties of the estimators based on these approaches. This could not be derived analytically and so a Monte Carlo study is conducted. The results seem to indicate that the properties of the estimators from the three methods are not very different. The yield to maturity on 91-day Canadian Federal Government Treasury bills, on the date of issue, is chosen as the proxy for the instantaneously riskfree interest rate. The impact of using such a proxy is briefly investigated and found to be negligible. The bond sample chosen is the complete issues of retractable and extendible bonds made by the Government of Canada. There were 20 issues between January 1959 and October 1975, and weekly prices on all these bonds are available in the Bank of Canada Review. To arrive at the final bond valuation equation, some assumptions are made about the term structure of interest rates. This study first assumes a form of the pure expectations hypothesis and it is shown that the performance of the model in predicting market price movements, is considerably improved when we assume a specific form of term/liquidity preference on the part of investors. Incorporating taxes into the model results in similar improvements. The hypothesis that the bond market is efficient to information contained in these models is tested and not rejected. Finally, an ad hoc regression based model is developed to serve as a bench mark for evaluating the performance of the partial equilibrium models. It is observed that these models perform atleast as well as the ad hoc model, and could be improved by relaxing some of the restrictive assumptions made.

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