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Yield curve shapes and the asymptotic short rate distribution in affine one-factor models


We consider a model for interest rates where the short rate is given under the risk-neutral measure by a time-homogeneous one-dimensional affine process in the sense of Duffie, Filipović, and Schachermayer. We show that in such a model yield curves can only be normal, inverse, or humped (i.e., endowed with a single local maximum). Each case can be characterized by simple conditions on the present short rate r t . We give conditions under which the short rate process converges to a limit distribution and describe the risk-neutral limit distribution in terms of its cumulant generating function. We apply our results to the Vasiček model, the CIR model, a CIR model with added jumps, and a model of Ornstein–Uhlenbeck type.

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Correspondence to Martin Keller-Ressel.

Additional information

Supported by the Austrian Science Fund (FWF) through project P18022 and the START program Y328.

Supported by the module M5 “Modeling of Fixed Income Markets” of the PRisMa Lab, financed by Bank Austria and the Republic of Austria through the Christian Doppler Research Association.

Both authors would like to thank Josef Teichmann for most valuable discussions and encouragement. We also thank various proofreaders at FAM and the anonymous referee for their comments.

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Keller-Ressel, M., Steiner, T. Yield curve shapes and the asymptotic short rate distribution in affine one-factor models. Finance Stoch 12, 149–172 (2008).

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  • Affine process
  • Term structure of interest rates
  • Ornstein–Uhlenbeck process
  • Yield curve

Mathematics Subject Classification (2000)

  • 60J25
  • 91B28


  • E43
  • G12