Classifying Yield Curve Shapes in the Two-Factor Vasicek Model

Summary

The paper classifies the term structure shapes that can arise in the two-factor Vasicek interest rate model. It identifies normal, inverse, humped, dipped, and hump-dip curves as always attainable, and finds that some parameter regimes allow as many as four further shapes. The results cover both yield curves and forward curves, making the classification relevant to how model assumptions translate into observed interest rate curve forms.

The analysis highlights the correlation between the two factors and the difference in their mean-reversion speeds as key determinants of which shapes are possible. It uses total positivity, a mathematical theory associated with Karlin, as its main tool. This is a result about attainable shapes within a particular model, rather than evidence that a given shape will occur in market data. The brief account does not enumerate the additional shapes or provide parameter thresholds, so practical application requires consulting the full analysis and calibrating the model.

Key ideas

• The two-factor Vasicek model can generate several standard yield and forward curve shapes.
• Normal, inverse, humped, dipped, and hump-dip shapes are always attainable in the model.
• Some parameter settings permit up to four additional shapes.
• Factor correlation and differences in mean-reversion speeds affect the range of possible shapes.
• Total positivity provides the mathematical framework for the classification.

Tags

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


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









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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.