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

Paper: arXiv 2506.06317

Authors: Amit Kumar Jha

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.

Complexity vs Empirical Score

• Math Complexity: 7.0/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces a novel extension to the Hull-White model with a sinusoidal time-varying parameter, involving advanced SDEs, Fourier analysis, and Monte Carlo simulations, indicating high mathematical density. It is supported by comprehensive real-world data from FRED, calibration using Nelder-Mead optimization, and backtest-ready metrics like RMSE, demonstrating strong empirical implementation.

Research Flowchart

  flowchart TD
  A["Research Goal: <br>Model Long-Term Periodicity in Interest Rates"] --> B{"Methodology: <br>Extended Hull-White Model"}
  B --> C["Data Inputs: <br>U.S. Treasury Yields (1990-2022)"]
  C --> D["Calibration: <br>Nelder-Mead Optimization"]
  D --> E["Computational Process: <br>Monte Carlo Simulation"]
  E --> F{"Key Outcomes"}
  F --> G["RMSE: 0.12% <br>(Improved Fit vs. 0.14%)"]
  F --> H["Bond Price: $0.43 <br>(30-Year Zero-Coupon)"]
  F --> I["Implications: <br>Bond Pricing & Risk Management"]

Cited References (19)

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3. Options, Futures, and Other Derivatives (2006)
4. The Econometrics of Financial Markets (1997)
5. Interest Rates under Falling Stars (2018) — link
6. U.S. Treasury Yields (2022)
7. Numerical Procedures for Implementing Term Structure Models II: Two-Factor Models (1994) — link
8. Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation (1992) — link

0 of 19 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.