A Sinusoidal Hull-White Model for Interest Rate Dynamics: Capturing Long-Term Periodicity in U.S. Treasury Yields

Abstract

This study is motivated by empirical observations of periodic fluctuations in interest rates, notably long-term economic cycles spanning decades, which the conventional Hull-White short-rate model fails to adequately capture. To address this limitation, we propose an extension that incorporates a sinusoidal, time-varying mean reversion speed, allowing the model to reflect cyclic interest rate dynamics more effectively. The model is calibrated using a comprehensive dataset of daily U.S. Treasury yield curves obtained from the Federal Reserve Economic Data (FRED) database, covering the period from January 1990 to December 2022. The dataset includes tenors of 1, 2, 3, 5, 7, 10, 20, and 30 years, with the most recent yields ranging from 1.22% (1-year) to 2.36% (30-year). Calibration is performed using the Nelder-Mead optimization algorithm, and Monte Carlo simulations with 200 paths and a time step of 0.05 years. The resulting 30-year zero-coupon bond price under the proposed model is 0.43, compared to 0.47 under the standard Hull-White model. This corresponds to root mean squared errors of 0.12% and 0.14%, respectively, indicating a noticeable improvement in fit, particularly for longer maturities. These results highlight the model's enhanced capability to capture long-term yield dynamics and suggest significant implications for bond pricing, interest rate risk management, and the valuation of interest rate derivatives. The findings also open avenues for further research into stochastic periodicity and alternative interest rate modeling frameworks.