Unspanned Stochastic Volatility: Empirical Evidence and Affine Representation

Most models of the term structure are restrictive in that they assume the bond market forms a complete market. That is, they assume all sources of risk affecting fixed income derivatives can be completely hedged by a portfolio consisting solely of bonds. Below, we demonstrate that this prediction fails in practice. In particular, we find that changes in swap rates have very limited explanatory power for returns on at-the-money straddles -- portfolios mainly exposed to volatility risk. We term this empirical feature `unspanned' stochastic volatility (USV). We demonstrate that bivariate Markov (affine such as Fong and Vasicek (1991) and Longstaff and Schwartz (1992), or not) models cannot exhibit USV. Then, we determine necessary (and apparently sufficient) parameter restrictions for trivariate Markov affine systems to exhibit USV. Finally, we show that USV occurs naturally within the Heath-Jarrow-Morton framework.