Bivariate Unspanned Stochastic Volatility Models

Abstract

As unspanned stochastic volatility (USV) models gain popularity in the literature, bivariate USV models (with one spanned and one unspanned factor) become a fundamental model class: they are the simplest USV models that are potentially useful, and (like general one-factor models) they are helpful in introducing and comparing higher-order models when simple ones are insufficient. We present a fundamental theorem of these models, which shows that they exist (contradicting a claim in Collin- Dufresne and Goldstein [2002]); that they share a particular affine form for the bond pricing function; and that they are attained, uniquely, by a specific class of diffusions. We then propose a specific, parametric bivariate USV model – the LADQ(1,1) model. The specification is analysed, and, finally, is implemented in two brief empirical investigations.

Keywords: Term Structure Models, Unspanned Stochastic Volatility, Market Completeness, Short-rate Models, Bivariate Models.