### Abstract

In this paper, we consider factor models of the term structure based on a Brownian filtration. We show that the existence of a nondeterministic long rate in a factor model of the term structure implies, as a consequence of the

Dybvig–Ingersoll–Ross theorem, that the model has an equivalent representation in which one of the state variables is nondecreasing. For two-dimensional factor models, we prove moreover that if the long rate is nondeterministic, the yield curve flattens out, and the factor process is asymptotically nondeterministic, then the term structure is unbounded. Finally, we provide an explicit example of a three-dimensional affine factor model with a nondeterministic yet finite long rate in which the volatility of the factor process does not vanish over time.

Dybvig–Ingersoll–Ross theorem, that the model has an equivalent representation in which one of the state variables is nondecreasing. For two-dimensional factor models, we prove moreover that if the long rate is nondeterministic, the yield curve flattens out, and the factor process is asymptotically nondeterministic, then the term structure is unbounded. Finally, we provide an explicit example of a three-dimensional affine factor model with a nondeterministic yet finite long rate in which the volatility of the factor process does not vanish over time.

Original language | English |
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Pages (from-to) | 656-667 |

Journal | Mathematical Finance |

Volume | 28 |

Issue number | 2 |

DOIs | |

Publication status | Published - Apr 2018 |

### Keywords

- Dybvig–Ingersoll–Ross theorem
- factor model
- long rate
- term structure

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## Cite this

de Kort, J. (2018). A note on the long rate in factor models of the term structure.

*Mathematical Finance*,*28*(2), 656-667. https://doi.org/10.1111/mafi.12151