Which of the two calculations below, is wrong? Why?
$dF = \sigma F dW$
First:
$dF^2 = (F^2)' dF + \frac{1}{2}(F^2)''dF.dF$
$dF^2 = 2F dF + dF.dF$
$dF^2 = 2 \sigma F^2 dW + \sigma^2 F^2 dt$
$\frac{dF^2}{F^2}=2\sigma dW + \sigma^2 dt$
$d \ln F^2=2\sigma dW + \sigma^2 dt$
OR
Second:
$d \ln F^2 = (\ln F^2)' dF + \frac{1}{2}(\ln F^2)'' dF.dF$
$d \ln F^2 = \frac{2F}{F^2} dF + \frac{1}{2} (\frac{2F}{F^2})' dF.dF$
$d \ln F^2 = \frac{2}{F} dF +\frac{1}{2} (\frac{2}{F})' dF.dF$
$d \ln F^2 = \frac{2}{F} dF +\frac{1}{2} \frac{-2}{F^2} dF.dF$
$d \ln F^2 = \frac{2}{F} dF - \frac{1}{F^2} dF.dF$
$d \ln F^2 = 2\sigma dW - \sigma^2 dt$
Just for context, I am trying to understand the calculation of In-Arears Swap pricing. So, need to compute expectation of "Square of Forward rate".
The book (Brigo Mercurio) says $E(F(T)^2)=F(0)^2 e^{\sigma^2 T}$
My computations above (Second) tells me that $E(F(T)^2)=F(0)^2 e^{-\sigma^2 T}$
I am stuck and unable to explain the "-" sign before $\sigma^2 dt$ which I see.
Ref: Brigo and Mercurio "Interest Rates Models - Theory and Practice" Second Edition, Part V, Chapter 13, Equation 13.3