Pricing Claims Under GARCH-Level Dependent Interest Rate Processes

This article establishes a family of models for pricing interest rate sensitive claims when the underlying interest rate is driven by a two state variable GARCH process. Analytical solutions are established for the case when the innovations in the short rate are combinations of a normal and chi-squared random variables and the volatility of rates takes on a special GARCH form. GARCH models that nest level dependent interest rate models, including the Cox, Ingersoll, and Ross model are also considered. Algorithms are provided that permit the e±cient pricing of American style interest rate claims under a rather broad array of dynamics, including GARCH and regime switching processes. The simple e±cient algorithms for pricing interest rate claims that we establish should permit empiricists to use term structure and option data to more fully evaluate alternative volatility dynamics in interest rate markets. Signi ̄cant research has been conducted on the class of single factor a±ne yield models. The empirical evidence suggests that these models are too restrictive to ̄t nominal interest rate behavior.1 As a result, researchers have considered more complex models. Examples include Longsta® and Schwartz (1992), who permit the short volatility to be stochastic, AitSahalia (1996), who allows mean reversion and volatility to be related to the level of interest rates in rather complex ways, Gray (1996), who permits regime switching, and Brenner, Harjes and Kroner (1996), hereafter BHK, who incorporate GARCH e®ects into the dynamics of the spot rate. The intent with these models is to add additional realism primarily through the volatility process. The recent empirical tests by Gray and BHK on the time series of interest rates highlight the restrictions of many of the common single factor models, and the importance of incorporating GARCH like features into the dynamics. BHK show that models which parameterize volatility only as a function of the level of interest rates over-emphasize the sensitivity of volatility to levels, and are unable to capture the serial correlation in conditional variances. They also show that simple GARCH models fail to capture the relationship between volatility and the level of rates. Their models, which incorporate both GARCH and Level e®ects, characterize the volatility process better than either Level or GARCH models alone. They conclude that there exists a strong need to establish theoretical option pricing models for interest sensitive claims that are driven by underlying GARCH-Level dependent processes. In this paper we develop such models. Our underlying interest rate is modeled by a two state variable GARCH process, the ̄rst variable capturing a mean reverting short term rate and a second state variable capturing volatility. Our models are capable of handling a wide array of dynamics for the volatility. In particular, its dynamics could depend on recent interest rate innovations, the level of interest rates and other known information. For the special case when the volatility process does not depend on the level of rates, the two state variable GARCH process leads to analytical solutions for bond prices and selected derivative instruments. In this case, one period rates have distributions that are combinations of normal and chi-squared distributions. When the chi-squared innovation is shut down, the model reduces to a GARCH extension of the discrete Vasicek model. In this model, rates are conditionally normal for the single period, but, due to the GARCH feature, over multiple periods, the rate is not normal, and its distribution can display signi ̄cant kurtosis and skewness that could be useful in explaining the volatility smile. Actually, since our analytical model also permits chi squared innovations in the riskless rate we are able to depart even further from normality with distributions that hopefully capture features that render the model a more realistic description of the process. Several other models exist which incorporate stochastic volatility for the short rate. For For an excellent discussion of these models and a summary of the empirical results see Chapter 11 of Campbell, Lo and MacKinlay (1997).