### Abstract

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.

Language | English |
---|---|

Pages | 656-667 |

Journal | Mathematical Finance |

Volume | 28 |

Issue number | 2 |

DOIs | |

Publication status | Published - Apr 2018 |

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### Keywords

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

### Cite this

*Mathematical Finance*,

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

}

*Mathematical Finance*, vol. 28, no. 2, pp. 656-667. https://doi.org/10.1111/mafi.12151

**A note on the long rate in factor models of the term structure.** / de Kort, Jan.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - A note on the long rate in factor models of the term structure

AU - de Kort, Jan

PY - 2018/4

Y1 - 2018/4

N2 - 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 theDybvig–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.

AB - 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 theDybvig–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.

KW - Dybvig–Ingersoll–Ross theorem

KW - factor model

KW - long rate

KW - term structure

U2 - 10.1111/mafi.12151

DO - 10.1111/mafi.12151

M3 - Article

VL - 28

SP - 656

EP - 667

JO - Mathematical Finance

T2 - Mathematical Finance

JF - Mathematical Finance

SN - 0960-1627

IS - 2

ER -