Zero-Coupon Treasury Rates and Returns using the Volatility Index

ArXiv ID: 2411.03699 “View on arXiv”

Authors: Unknown

Abstract

We study a multivariate autoregressive stochastic volatility model for the first 3 principal components (level, slope, curvature) of 10 series of zero-coupon Treasury bond rates with maturities from 1 to 10 years. We fit this model using monthly data from 1990. Unlike classic models with hidden stochastic volatility, here it is observed as VIX: the volatility index for the S&P 500 stock market index. Surprisingly, this stock index volatility works for Treasury bonds, too. Next, we prove long-term stability and the Law of Large Numbers. We express total returns of zero-coupon bonds using these principal components. We prove the Law of Large Numbers for these returns. All results are done for discrete and continuous time.

Keywords: Multivariate Stochastic Volatility, Principal Components Analysis (PCA), VIX, Treasury Bond Rates, Law of Large Numbers, Fixed Income

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail
  • Why: The paper uses advanced multivariate stochastic calculus and SDEs (Ornstein-Uhlenbeck processes, matrix norms, LLNs), qualifying for high math complexity, while it also includes backtest-ready empirical components like monthly data fitting, PCA, and a public GitHub repository with code and data.
  flowchart TD
    A["Research Goal"] -->|Investigate volatility in Treasury bond rates| B["Methodology"]
    B -->|1. Apply PCA to 10 zero-coupon rates (1-10Y)| C["Model & Data"]
    B -->|2. Model Principal Components with VAR Stochastic Volatility| C
    C -->|Input: Monthly data from 1990| D["Innovation: VIX as Observed Volatility"]
    C -->|Input: S&P 500 VIX| D
    D -->|Compute: Total Bond Returns via Principal Components| E["Computational Process"]
    E -->|Theoretical Proof: Law of Large Numbers| F["Key Findings/Outcomes"]
    E -->|Analysis: Discrete & Continuous Time Stability| F
    F -->|Result 1: VIX explains Treasury bond volatility| G((End))
    F -->|Result 2: Long-term stability proven| G
    F -->|Result 3: LLN proven for bond returns| G