I have a problem figuring out some of the calculations in the book: Fixed Income modelling
In the chapter on forwards the author makes an argument that the forward is a martingale under the T-forward martingale measure.
I know that the forward is given by:
$$F_t^T=\frac{P_t}{B_t^T}\;(1)$$
And that the price of a security under the risk-neutral probability measure, with no payments in the given period is:
$$P_t=E_t^{Q}[e^{-\int\limits_t^Tr_udu}P_T]\;(2)$$
The T-forward martingale measure is: $E_t^{Q^T}$
And in the book we have:
$$P_t=B_t^TE_t^{Q^T}[P_T]\;(3)$$
First question: What is the difference between (2) and (3)? $B_t^T =e^{-\int\limits_t^Tr_udu}$, so how do they differ?
Next he says that with $B_t^T$ as a numeraire we have that:
$\frac{0}{B_t^T}=E_t^{Q^T}[\frac{P_T-F_t^T}{B_T^T}]$
How does he get that? More specifically. Why is B_T^T in the equation as that is equal to 0.
From (3) i have: $P_t=B_t^TE_t^{Q^T}[P_T]$
Subtracting $P_t$ on both sides: $0=B_t^TE_t^{Q^T}[P_T]-P_t$
Dividing by $B_t^T$ and using (1) I get:
$\frac{0}{B_t^T}=E_t^{Q^T}[P_T]-F_t^T$
So why the $B_T^T$?