Most term structure models assume bond markets are complete, that is, that all fixed income derivatives can be perfectly replicated using solely bonds. However, we find that, in practice, swap rates have 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). While USV can be captured within an HJM framework, we demonstrate that bivariate models cannot exhibit USV. We determine necessary and sufficient conditions for trivariate Markov affine systems to exhibit USV. For such USV models, bonds alone may not be sufficient to identify all parameters. Rather, derivatives are needed.