Modelling the dynamics of long-term bonds with Kalman filter

Abstract

We construct a time-consistent and arbitrage-free three-factor Vasicek model for long-term bonds. A new methodology based on a stochastic mean reversion rate which captures uncertainty in long-term bond yields is presented. To allow measurement errors to be accounted for in observed yields, the model is expressed in a state space form. Kalman filtering is then applied to filter uncertainty in the observed yields. An appropriate set of transition equations on state variables and measurement equations on observed yields are derived. Using historical market data from the US Treasury daily interest rates (March 2006 to June 2020), Germany Government bond yields (August 2000 to 15 January 2021) and Canada Government bond yields (16 January 2020 to 14 January 2021), parameters of one-, two- and three-factor models are estimated. The results indicate that the constructed Vasicek model can fit the US, Germany and Canada term structure of interest rates.