# The classification of term structure shapes in the two-factor Vasicek model -- a total positivity approach

## TL;DR

This paper fully classifies all possible shapes of interest rate term structures in a two-factor Vasicek model using total positivity theory, revealing how parameters influence shape attainability.

## Contribution

It provides a comprehensive classification of term structure shapes in the two-factor Vasicek model, introducing total positivity as a key analytical tool.

## Key findings

- Normal, inverse, humped, dipped, and hump-dip shapes are always attainable.
- Up to four additional shapes can occur depending on parameters.
- Correlation and mean-reversion speed differences are crucial in shape determination.

## Abstract

We provide a full classification of all attainable term structure shapes in the two-factor Vasicek model of interest rates. In particular, we show that the shapes normal, inverse, humped, dipped and hump-dip are always attainable. In certain parameter regimes up to four additional shapes can be produced. Our results apply to both forward and yield curves and show that the correlation and the difference in mean-reversion speeds of the two factor processes play a key role in determining the scope of attainable shapes. The key mathematical tool is the theory of total positivity, pioneered by Samuel Karlin and others in the 1950ies.