# Derivation of the Nelson-Siegel model and proof of arbitrage

1. I am looking for a derivation of the Nelson-Siegel model

$y(m)=a+b\left( \frac{1-e^{-\lambda m}}{\lambda m}\right)+c\left( \frac{1-e^{-\lambda m}}{\lambda m} -e^{-\lambda m} \right)$

It is supposed to follow the Differential Equation

$y''(m)+uy'(m)+vy(m)=0$

However, taking the derivative of $y$ and plugging that into the Differential Equation does not work. Where can I find a mathematical derivation and the assumptions beeing made? Just saying that the term structure follows aboves process is not enough.

2. Why does the Nelson-Siegel model allow arbitrage? And how do I proof it?

Ive already looked at this post

Why isn't the Nelson-Siegel model arbitrage-free?

without finding it very helpful. The paper of Bjoerk and Christensen shows inconsistency of the NS-model with Hull-White and Ho-Lee but I do not understand why arbitrage opportunities follow from this.

Is there any mathematical proof of arbitrage opportunities within the NS model?

• Regarding your second question: Did you also look at the Filipovic paper? – Bob Jansen Jan 18 '14 at 19:31
• I did but I did not understand it completely. How I understand it is that Nelson-Siegel is not consistent with any Ito-process and hence not arbitrage free. That would mean that only Ito-process consistent models can be arbitrage free which I dont understand. – user7015 Jan 19 '14 at 10:30
• Your misunderstanding regarding part one is that the short rate of interest follows that differential equation. See page 474 to 475 – user25064 Feb 18 '14 at 16:22

## 1 Answer

The book by Francis Diebold & Glenn Rudebusch "Yield Curve Modeling and Forecasting" addresses both a dynamic extension of Nelson-Siegel and an arbitrage-free version - may be helpful for what you are looking for. Link below:

http://www.amazon.com/Yield-Curve-Modeling-Forecasting-Nelson-Siegel/dp/0691146802/ref=sr_1_2?ie=UTF8&qid=1390136363&sr=8-2&keywords=diebold+and+rudebusch

• Thanks but I dont have access to this book. – user7015 Jan 20 '14 at 17:28