I have a question about the demonstration of the formula which states that: If we have an Hull & White Model for the short rate diffusion such that
Then the model is fully calibrated if and only if:
f^M is the market instantaious forward rate.
Now to demonstrate this formula I managed to arrive at the fact that the Zero Coupon bond price under the Hull and White Model for maturity T is given by
Now we only need to derive the bond prices in order to get the instantanious forward rate, so we get to the following:
My question is about the derivation of
Which is somehow equal to
How can we proove this? It seems to me that this derivation is incorrect because of the fact that the T variable is present both in the integral limits and in B(u,T).
Thanks in advance