Approximate Closed Formulas for Zero-Coupon Bonds in the Zero Lower Bound Framework

Abstract

Since the 2007 financial crisis, many central banks adopted policies to lower their interest rates, whose dynamics can not be captured using classical models. Recently, Meucci and Loregian (2016) proposed an approach to estimate nonnegative interest rates using the inverse-call transformation. Despite the fact that their work distinguishes from others in the literature by their consideration of practical aspects, some technical difficulties still remain, such as the lack of the analytic expression for the zero-coupon bond (ZCB) price. In this work, we provide approximate closed formulae for the ZCB price in the zero lower bound (ZLB) framework, when the underlying shadow rate is assumed to follow the classical one-factor Vasicek model. Then, a filtering procedure is performed using the Unscented Kalman Filter (UKF) to estimate the unobservable state variable (the shadow rate), and the model calibration is proceeded by estimating the model parameters using the Particle Swarm Optimization (PSO) algorithm. Lastly, empirical illustrations are given and discussed using (as input data) the interest rates of AAA-rated bonds compiled by the European Central Bank ranging from September 6, 2004 to June 21, 2012.