Do Bonds Span the Fixed Income Markets? Theory and Evidence for Unspanned Stochastic Volatility


  • Pierre Collin-Dufresne,

  • Robert S. Goldstein

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    • Pierre Collin-Dufresne is from the Graduate School of Industrial Administration, Carnegie Mellon University. Robert S. Goldstein is from Washington University, St. Louis. We thank Jesper Andreasen; Dave Backus; Dave Chapman; Darrell Duffie; Francis Longstaff; Claus Munk; Pedro Santa-Clara; Ken Singleton; Chris Telmer; Len Umantsev; Stan Zin; and seminar participants at the American Finance Association meetings in Atlanta 2002, the Texas Finance Festival, the 2001 Fixed Income Winter Conference at Stanford University, The European Finance Association Meetings in Barcelona 2001, The University of Connecticut, Columbia University, The University of Illinois, Penn State University, The University of Rochester, The University of Wisconsin, and Stanford University for helpful comments. All remaining errors are our own.


Most term structure models assume bond markets are complete, that is, that all fixed income derivatives can be perfectly replicated using solely bonds. How ever, 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.