I'm reading the book, "Derivatives Analytics with Python" by Yves Hilpisch. In an application of calibration of CIR85 process for the short-term interest rate. I found some codes which can be interpreted as which follows. The forward rate = spot rate + first order derivative of spot yield curve w.r.t. time horizon * time horizon. The full python script can ben found in the following link, http://www.riskreversal.net/calibration-of-cir85-model-to-euribor-rates/, and the formula in question is located in line 35. Could anyone give some additional explanation about the logic of the formula or indicate some reference regarding the issue. Thanks a lot!


You know the zero coupon bond price can be written in terms of spot rate, say y, or forward rates:

$B(0,t)=e^{-y(0,t) \, t}=e^{-\int_0^t{f(0,u) \, du}}$

Which means:

$y(0,t) \, t=\int_0^t{f(0,u) \, du}$

Differentiating with respect to t, you get your answer:

$y(0,t)+ y\prime (0,t) t=f(0,t)$

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