Unspanned Stochastic Volatility in the Multi-factor CIR Model

TL;DR

This paper investigates whether multi-factor CIR models can exhibit unspanned stochastic volatility (USV), providing necessary and sufficient conditions, and constructing examples of three-factor models that do, thus addressing an open question in bond market modeling.

Contribution

It establishes conditions under which multi-factor CIR models can exhibit USV and constructs explicit three-factor models demonstrating this capability.

Findings

Multi-factor CIR models with diagonal drift cannot exhibit USV.

Necessary and sufficient conditions for USV in multi-factor CIR models.

Construction of three-factor CIR models that exhibit USV.

Abstract

Empirical evidence suggests that fixed income markets exhibit unspanned stochastic volatility (USV), that is, that one cannot fully hedge volatility risk solely using a portfolio of bonds. While [1] showed that no two-factor Cox-Ingersoll-Ross (CIR) model can exhibit USV, it has been unknown to date whether CIR models with more than two factors can exhibit USV or not. We formally review USV and relate it to bond market incompleteness. We provide necessary and sufficient conditions for a multi-factor CIR model to exhibit USV. We then construct a class of three-factor CIR models that exhibit USV. This answers in the affirmative the above previously open question. We also show that multi-factor CIR models with diagonal drift matrix cannot exhibit USV.