# Yield Curve Shapes and the Asymptotic Short Rate Distribution in Affine One-Factor Models

## TL;DR

This paper analyzes the shapes of yield curves and the long-term distribution of the short rate in one-factor affine interest rate models, providing conditions for convergence and characterizing the limit distribution.

## Contribution

It characterizes all possible yield curve shapes and describes the asymptotic distribution of the short rate in affine models, including jump-diffusion variants.

## Key findings

- Yield curves are only normal, inverse, or humped.
- Conditions for short rate convergence to a limit distribution.
- Explicit description of the limit distribution in affine models.

## Abstract

We consider a model for interest rates, where the short rate is given by a time-homogenous, one-dimensional affine process in the sense of Duffie, Filipovic 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. We give conditions under which the short rate process will converge to a limit distribution and describe the limit distribution in terms of its cumulant generating function. We apply our results to the Vasicek model, the CIR model, a CIR model with added jumps and a model of Ornstein-Uhlenbeck type.