Optimizing Nelson–Siegel Yield-Curve Parameters
Summary
The document asks how to estimate the four parameters of a Nelson–Siegel yield curve by minimizing squared differences between model-implied bond prices and observed market prices. Each model price discounts the bond’s cash flows using a rate determined by the Nelson–Siegel function, and the objective sums pricing errors across the bonds. The question raises whether Newton’s method is appropriate for this nonlinear calibration problem.
The replies suggest derivative-free optimization methods. One reports that Differential Evolution worked well in prior use and notes that repeated gradient searches from random starting values can also work. Another reports using Nelder–Mead for yield-curve calibration because it does not require first or second derivatives. These are practitioner recommendations rather than a controlled comparison, and the discussion does not specify parameter bounds, starting values, objective scaling, or validation procedures. The best method may therefore depend on the calibration setup and should be assessed for stability and fit on the particular bond data.
Key ideas
- Calibrate Nelson–Siegel parameters by minimizing squared bond-pricing errors.
- The model prices each bond by discounting its cash flows at the fitted curve rates.
- Differential Evolution and Nelder–Mead are suggested as derivative-free methods.
- Repeated gradient searches with varied initial values are another proposed approach.
- The recommendations do not compare methods under a shared dataset or setup.
Tags
Full text
# How to minimize Nelson-Siegel parametric form
# How to minimize Nelson-Siegel parametric form
Problem I am given the following function to minimize (w.r.t. $\theta$) $$f= \sum_{k=1}^5 \Big [ \sum_{i=1}^{N_k} CF_{k, i} \cdot e^{-r(t_{k, i}, \theta)\cdot t_{k, i}} - P_k^* \Big]^2$$ where $\theta = (\beta_0, \beta_1, \beta_2, \lambda)$ and $$ r(t, \theta) = \beta_0 + \beta_1 \big(\frac{1-e^{-\frac{t}{\lambda}}}{\frac{t}{\lambda}} \big) + \beta_2 \big(\frac{1-e^{-\frac{t}{\lambda}}}{\frac{t}{\lambda}} - e^{-\frac{t}{\lambda}} \big)$$
Context We are given 5 bonds, their cashflows, $CF_{k, i}$, and market price, $P_k^*$. All these values are given to us as numbers.
My attempt I tried to apply Newton's Method using python. However, I am pretty sure that this method is not applicable here, and I need another minimization algorithm.
Could anyone suggest which algorithm is the best for such a function? ?
## Answer by Enrico Schumann (score 5)
https://quant.stackexchange.com/a/68207
When we worked with that model several years go, we used Differential Evolution and it worked very well. See Calibrating the Nelson-Siegel-Svensson Model. At least in the standard version, a best-of-many gradient searches (with random initial values) also worked well. See A Note on 'Good Starting Values' in Numerical Optimisation. If you were willing to use R as well, there are many code examples in the NMOF package documentation.
## Answer by T123 (score 0)
https://quant.stackexchange.com/a/68220
I built something like you attempted here to estimate yield curves for various sectors using Bbg bond quotes years ago in VBA. For the calibration of the parameters I used Nelder Mead. It doesn't need the first and second derivative or estimators of it such as Newton-Ralphson and such.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.