Using non-linear/chaotic dynamics for interest rate determination
A new class of interest rate models is proposed where the main driving terms for the large scale dynamics of the system are deterministic. As an example, an economically motivated two factor model of the term structure is presented that generalises existing stochastic mean term structure models. By allowing a certain parameter to acquire dynamical behaviour the model is extended to three factors. It is shown, that in a deterministic version, the model is equivalent to the Lorenz system of differential equations. With reasonable parameter values the model exhibits chaotic behaviour. It successfully emulates certain properties of interest rates including regime switching and behaviour of a business cycle nature. Pricing and term structure 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 non-linear systems analysis. An empirical investigation uncovers surprising structure consistent with the existence of a low dimensional attractor. Issues of control of the chaotic system with reference to the underlying economic model are discussed. A heuristic approach is made to estimating the three factor model. Exploiting properties of the term structure, the existence of noise, and the geometry of the system allows a variety of methods for uncovering the parameters of the model. A better approach is found in the application of the Kalman filter to the estimation problem. Lack of explicit solutions motivates an investigation into the use of approximated forms for the term structure. The traditional Kalman filter is seen to be unstable when applied to the chaotic three factor model. A stable variant, from the class known as 'square-root' filters, is adopted. A new method is created for finding the analytical derivatives of the log-likelihood function such that it is consistent with the 'square-root' filter. Estimates for the empirical estimation of the models developed earlier in the thesis are given. It is concluded that there is much scope for expanding the literature within the new class of models proposed. The particular three factor model developed has been shown to have realistic properties and be amenable to bond and contingent claim pricing. The chaotic nature of the model, underpinned by an economic derivation, opens up new methods for authorities to control/stabilise the economy. An analysis of the underlying dynamical structure of UK money market rates is consistent with a low dimensional deterministic driving force. Heuristic methods employed to estimate the parameters of the model allow for an insight into exploiting the geometry of the system. Application of the Kalman filter to estimation of non-linear models is found to be problematic due to the linearisations/approximations that are necessary. An outline of areas for future research is given, providing ideas for extending the economic formulation, further investigating control of chaotic interest rate systems, testing empirical data for evidence of chaotic invariants and methods for quantifying and improving the Kalman filtering procedures for better handling non-linear models.