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!
1 Answer
$\begingroup$
$\endgroup$
0
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:
$y(0,t)+ y\prime (0,t) t=f(0,t)$
answered Oct 7, 2018 at 12:17