Approximate Closed-Form Solutions for Pricing Zero-Coupon Bonds in the Zero Lower Bound Framework

After the 2007 financial crisis, many central banks adopted policies to lower their interest rates; the dynamics of these rates cannot be captured using classical models. Recently, Meucci and Loregian proposed an approach to estimate nonnegative interest rates using the inverse-call transformation. Despite the fact that their work is distinguished from others in the literature by their consideration of practical aspects, some technical difficulties still remain, such as the lack of analytic expression for the zero-coupon bond (ZCB) price. In this work, we propose novel approximate closed-form solutions 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. Further, empirical illustrations are given and discussed using (as input data) the interest rates of the AAA-rated bonds compiled by the European Central Bank ranging from 6 September 2004 to 21 June 2012 (a period that concerns the ZLB framework). Our approximate closed-form solution is able to show a good match between the actual and estimated yield-rate values for short and medium time-to-maturity values, whereas, for long time-to-maturity values, it is able to estimate the trend of the yield rates.