Modeling Multivariate Interest Rates using Time-Varying Copulas and Reducible Stochastic Differential Equations

We propose a new approach for modeling non-linear multivariate interest rate processes based on time-varying copulas and reducible stochastic differential equations (SDEs). In the modeling of the marginal processes, we consider a class of non-linear SDEs that are reducible to Ornstein-Uhlenbeck (OU) process or Cox, Ingersoll, and Ross (1985) (CIR) process. The reducibility is achieved via a non-linear transformation function. The main advantage of this approach is that these SDEs can account for non-linear features, observed in short-term interest rate series, while at the same time leading to exact discretisation and closed form likelihood functions. Although a rich set of specifications may be entertained, our exposition focuses on a couple of nonlinear constant elasticity volatility (CEV) processes, denoted OU-CEV and CIR-CEV, respectively. These two processes encompass a number of existing models that have closed form likelihood functions. The statistical properties of the two processes are investigated. In order to obtain more flexible functional form over time, we allow the transformation function to be time-varying. Results from our study of US and UK short term interest rates suggest that the new models outperform existing parametric models with closed form likelihood functions. We also find the time-varying effects in the transformation functions statistically significant. We study the conditional dependence structure of the two rates using Patton (2006a) time-varying Symmetrised JoeClayton copula. We find evidence of asymmetric dependence between the two rates, and that the level of dependence is positively related to the level of the two rates. ∗University of Liverpool. Corresponding author: Management School, Chatham Street, Liverpool, L69 7ZH, UK, Tel: +44-151-795-3122, Fax: +44-151-795-3005, Email: [email protected] (Ruijun Bu). †GREQAM, Marseille ‡Queen’s University, Belfast §GREQAM, Marseille and CNRS 1 ha ls hs -0 04 08 01 4, v er si on 1 15 N ov 2 00 9