In finance, a T-forward measure is a pricing measure absolutely continuous with respect to a risk-neutral measure, but rather than using the money market as numeraire, it uses a bond with maturity T. The use of the forward measure was pioneered by Farshid Jamshidian (1987), and later used as a means of calculating the price of options on bonds.
[1] Let[2] be the bank account or money market account numeraire and be the discount factor in the market at time 0 for maturity T. If
is defined via the Radon–Nikodym derivative given by Note that this implies that the forward measure and the risk neutral measure coincide when interest rates are deterministic.
Also, this is a particular form of the change of numeraire formula by changing the numeraire from the money market or bank account B(t) to a T-maturity bond P(t,T).
Indeed, if in general is the price of a zero coupon bond at time t for maturity T, where
is the filtration denoting market information at time t, then we can write from which it is indeed clear that the forward T measure is associated to the T-maturity zero coupon bond as numeraire.
The name "forward measure" comes from the fact that under the forward measure, forward prices are martingales, a fact first observed by Geman (1989) (who is responsible for formally defining the measure).
[3] Compare with futures prices, which are martingales under the risk neutral measure.
The last term is equal to unity by definition of the bond price so that we get