# Zero Coupon Bond prices in One Factor Hull White model

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

The Zero Coupon Bond Prices from this yield curve are:

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

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

(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:

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