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I implemented the one factor Hull White model for educational purposes and I calibrated the model from a given (made up!) yield curve:

Yield curve

The Zero Coupon Bond Prices from this yield curve are:

Zero curve

Taking the log of the bond prices and use cubic splines for interpolation gives:

enter image description here

Calculating the instantaneous forward rates from the curve above using

$$ f^M(t) = -\frac{\partial \operatorname{log}P(t)}{\partial t} $$

where i use the first derivative of the cubic spline at time $t$ to calculate $\frac{\partial \operatorname{log}P(t)}{\partial t}$ results in

enter image description here

(blue are the forward rates, orange is the original yield curve)

When I calculate the Bond prices from the model I get the following result:

Model yield curve

The orange line are the bond prices from the model, the blue dots are the original bond prices.

My questions:

  • The forward curve has quite a swing. Is there a problem / fault in my approach?
  • Is it plausible that the model prices (last image) differ that much from the data I used for calibration?

The whole jupyter notebook is available here: https://nbviewer.jupyter.org/gist/wpla/435437ddc5bcb1f6bdcae274117725e7

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The derivative of the bond prices is very sensitive to the interpolation mode. actually, if you use a linear interpolation mode, you will have some cases for which the right derivative is different from the left derivative at a given point.

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