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

Abstract

Abstract. We consider a model for interest rates, where the short rate is given under the risk-neutral measure by a time-homogenous, 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 rt. We give conditions under which the short rate process will converge 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.

What Should You Pay to Cap your ARM?—A Note on Capped Adjustable Rate Mortgages

“…In conclusion, the boundary provides a build-in cap on the adjustable rate of the formalized ARM. The conclusion relies on an assumption of a one-factor ATS, and is a direct consequence of existing results provided by Keller-Ressel and Steiner [1] regarding the yield curve shapes of ATSs. The two mortgage products are formalized following the work of Nordfang and Steffensen [2].…”

“…In this section, the main result from [1] is stated. The theorems and definitions summarized here are presented in a version which only applies to the simple affine short rate model without jumps presented in Definition 1.…”

“…The theorems and definitions summarized here are presented in a version which only applies to the simple affine short rate model without jumps presented in Definition 1. When relevant, we refer to the corresponding original definitions and theorems from [1]. The original versions cover a general class of ATSs.…”

“…Finally, the main result from [1] is stated in the specific case of the simple short rate model. Theorem 1 (Theorem 3.9 in [1]).…”

“…In Section 3, we introduce the formalized ARM and FRM originally presented in [2]. In Section 4, we summarize the results from ATS theory provided by Keller-Ressel and Steiner [1]. The results are presented only in the special case of the simple interest rate model of Section 2.…”