A Nonlinear Model of the Term Structure of Interest Rates

We present an economically motivated two–factor term structure model that generalizes existing stochastic mean term structure models. By allowing a certain parameter to acquire dynamical behavior we extend the two–factor model to obtain a nonlinear three–factor model that is shown, in a deterministic version, to be equivalent to the Lorenz system of differential equations. With reasonable parameter values the model exhibits chaotic behavior. It successfully emulates certain properties of interest rates including cyclical behavior on a business cycle time scale. Estimation and pricing issues are discussed. Standard PCA techniques used to estimate HJM type models are observed to be equivalent to dimensional estimates commonly applied to ‘spatial data’ in nonlinear systems analysis. It is concluded that techniques commonly used in the analysis of nonlinear systems may be directly applicable to interest rate models, offering new insights in the development of these models. Tests of nonlinearity in interest rate behavior may need to focus on long cycle times.