Zero-Coupon Treasury Rates and Returns using the Volatility Index

TL;DR

This paper models zero-coupon Treasury bond rates using a multivariate autoregressive stochastic volatility model driven by the VIX, demonstrating stability and deriving return laws in both discrete and continuous time.

Contribution

It introduces a novel model linking Treasury bond volatilities to the VIX, showing its effectiveness and proving stability and return laws.

Findings

VIX-based volatility model fits Treasury bond data well

Proves long-term stability of the model

Establishes the Law of Large Numbers for bond returns

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.