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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 $B_T^T$?

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    i think this belongs in quant.stackexchange.com – 2017-02-02
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    ok. I can see that I'll have to wait 40 minutes but then I'll post it there. Thanks – 2017-02-03
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    Would you define $P$ and $B$ first ?It will help you. Indeed, obviously, we dont have $$B_t^T=e^{-\int\limits_t^Tr_udu}$$ Look at your formula of $P_t$, it must be $\mathcal{F_t}$-measurable, but the term $B_{t}^{T}$ is not . – 2017-03-09

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