Generalization of the Dybvig–Ingersoll–Ross Theorem and Asymptotic Minimality

Verena Goldammer and Uwe Schmock

Abstract: The long-term limit of zero-coupon rates with respect to the maturity does not always exist. In this case we use the limit superior and prove corresponding versions of the Dybvig–Ingersoll–Ross theorem, which says that long-term spot and forward rates can never fall in an arbitrage-free model. Extensions of popular interest rate models needing this generalization are presented. In addition, we discuss several definitions of arbitrage, prove asymptotic minimality of the limit superior of the spot rates, and illustrate our results by several continuous-time short-rate models.

Keywords: Dybvig–Ingersoll–Ross theorem, interest rate models, long-time forward rate, long-time zero-coupon rate, asymptotic monotonicity, asymptotic minimality

2010 Mathematics Subject Classification:

Journal: Mathematical Finance, Volume 22, Issue 1, pages 185–213, January 2012 (article first published online 22 Nov. 2010)

DOI: 10.1111/j.1467-9965.2010.00459.x

The preprint (27 pages, version October 19, 2010) is available in:


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