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:


And that the price of a security under the risk-neutral probability measure, with no payments in the given period is:


The T-forward martingale measure is: $E_t^{Q^T}$

And in the book we have:


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:


How does he get that? More specifically. Why is $B_T^T$ in the equation as that is equal to 1.

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:


So where does the $B_T^T$ come from in (4)? It is equal to 1, so I know I can always divide by it. But why?

  • 1
    $\begingroup$ From (1), the forward price is a ratio of asset prices, with the price of a T-zero coupon bond in the denominator. Therefore the forward price is s martingale under the T-forward measure. $\endgroup$ – dm63 Feb 3 '17 at 11:20
  • $\begingroup$ Think about stochastic interest rates. $B_t^T$ is a number while $e^{-\int\limits_t^Tr_udu}$ is a random variable. The former is the conditionnel expectation under the risk-neutral measure $Q$ of the latter. $\endgroup$ – Quantuple Feb 3 '17 at 11:37
  • $\begingroup$ So there really isn't any difference between $E_t^{Q^T}$ and $E_t^Q$? @Quantuple $\endgroup$ – Anders Feb 3 '17 at 12:50
  • $\begingroup$ @Anders: $B_t^T = \Bbb{E}_t^Q \left[e^{-\int\limits_t^Tr_udu}\right]$, such that there is a difference between the measures $Q$ and $Q_T$ when interest rates are stochastic. $Q$ uses the $t$-value of the money market account as numéraire, $Q_T$ uses the price of a $T$-zero coupon bond. $\endgroup$ – Quantuple Feb 3 '17 at 14:03
  • $\begingroup$ thanks @Quantuple. That cleared a lot for me. I still don't get equation 4 though $\endgroup$ – Anders Feb 3 '17 at 17:10

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