State-space of the Vasicek model for long-term bonds with Kalman filter

In this paper, a time-consistent and arbitrage-free state space for the one-, two-, and three-factor Vasicek models for long-term bonds is constructed. To account for the uncertainty in long-term bond yields, we propose a stochastic time-dependent mean-reversion model. The state-space model allows for the computation of measurement errors from observed yields. Appropriate state and measurement linear equations are derived to allow the use of the Kalman filter for model implementation. Based on weekly South African Government Bonds from February 2010 to February 2021, we give parameter estimates for the one-factor, two-factor, and three-factor models. The results from the study show that the developed model can fit the term structure of very long-term bonds.