Nelson–Siegel Yield Curve Derivation and Arbitrage-Free Extensions
Summary
The document raises two questions about the Nelson–Siegel yield curve model: how its functional form relates to a proposed differential equation, and why the model may permit arbitrage. It provides no derivation or arbitrage proof. The response instead points readers to a book on yield curve modeling that covers a dynamic extension of Nelson–Siegel and an arbitrage-free version.
As a result, the main practical lesson is a direction for further study: distinguish the original curve-fitting specification from dynamic and arbitrage-free term structure models. The document gives no equations beyond the proposed Nelson–Siegel form, assumptions, empirical evidence, or explanation of how arbitrage arises. Its value is therefore limited to identifying the topics and a reference for pursuing them, rather than establishing a method or result.
Key ideas
- The document asks how the Nelson–Siegel yield curve form is derived from a differential equation.
- It also asks why the model may be inconsistent with arbitrage-free pricing.
- The response recommends a reference that discusses dynamic and arbitrage-free extensions.
- No derivation, proof, assumptions, or empirical support are supplied in the document.
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# Derivation of the Nelson-Siegel model and proof of arbitrage
# 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?
## Answer by HookahBoy (score 3)
https://quant.stackexchange.com/a/9990
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
## Answer by brocode5253 (score -1)
https://quant.stackexchange.com/a/80255
I don't have enough reputation to comment, and I know this is 10 years late, but if you need access to this book you can get it on libgen for either pdf or epub.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.