Finance and Stochastics

, Volume 12, Issue 2, pp 149–172 | Cite as

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

  • Martin Keller-ResselEmail author
  • Thomas Steiner


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.


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

Mathematics Subject Classification (2000)

60J25 91B28 


E43 G12 


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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Vienna University of TechnologyWienAustria

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