Calibration of Nelson-Siegel (1987) and Svensson (1994) yield curve models on real Bundesbank spot rate data. Pure scipy/numpy — no QuantLib, no sklearn.

$$y(\tau) = \beta_0 + \beta_1 \cdot \frac{1 - e^{-\tau/\lambda}}{\tau/\lambda} + \beta_2 \cdot \left(\frac{1 - e^{-\tau/\lambda}}{\tau/\lambda} - e^{-\tau/\lambda}\right)$$

Forward rates are derived analytically as f(τ) = d/dτ [τ · y(τ)].

Extends Nelson-Siegel with a second independent hump term:

$$y(\tau) = \beta_0 + \beta_1 \cdot L(\tau, \lambda_1) + \beta_2 \cdot H(\tau, \lambda_1) + \beta_3 \cdot H(\tau, \lambda_2)$$

where L(τ,λ) = (1 − e^{−τ/λ}) / (τ/λ) and H(τ,λ) = L(τ,λ) − e^{−τ/λ}.

Non-convex problem (due to λ) solved in two stages:

1. Grid search over a (λ₁, λ₂) grid — for each pair, solve for (β₀…β₃) via OLS → collect top-k starting points
2. L-BFGS-B (scipy) from those starting points → fine optimization over all 6 parameters

The Bundesbank itself publishes its calibrated Svensson parameters, enabling direct validation against the official values.

pip install numpy scipy pandas matplotlib

Download the spot rate time series from the Bundesbank statistics portal:

Zeitreihendatenbank → Geld- und Kapitalmärkte → Zinssätze und Renditen → Zinsstruktur am Rentenmarkt – Schätzwerte → Zinsstrukturkurve für börsennotierte Bundeswertpapiere (Tageswerte)

Save the CSV as data/bundesbank.csv.

python calibration.py

Prints the calibrated parameters and displays a plot of the fitted curve vs. observed spot rates.