Forward Prices as Martingales Under the T-Forward Measure

Summary

The document asks why a security’s price can be expressed both as a risk-neutral expectation of discounted terminal value and as a T-forward-measure expectation scaled by the price of a zero-coupon bond. It then asks how the forward-price martingale relation can be written using the bond as numeraire, including why the terminal bond value appears in the expression even though it equals one at maturity.

The discussion gives the relevant pricing identities and shows an algebraic route from the T-forward expectation to the forward price. It is framed as a question rather than a resolved explanation: no answer is provided to clarify the change of measure or the numeraire argument. The useful takeaway is the distinction the question raises between discounting under the risk-neutral measure and taking expectations under the forward measure. The document does not provide a derivation, numerical example, or conditions beyond the stated no-payment setting.

Key ideas

• Risk-neutral pricing discounts terminal value using the short-rate process.
• Under the T-forward measure, the zero-coupon bond price scales the expected terminal security price.
• The forward price is defined as the security price divided by the maturity-matched zero-coupon bond price.
• The document asks why a terminal bond numeraire appears in the martingale expression, but does not answer it.

Tags

# Forward price - T-forward martingale


# Forward price - T-forward martingale












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:

$$\frac{0}{B_t^T}=E_t^{Q^T}[\frac{P_T-F_t^T}{B_T^T}];(4)$$

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:

$\frac{0}{B_t^T}=E_t^{Q^T}[P_T]-F_t^T$

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?

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