Forward measure

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.

Let

be the bank account or money market account numeraire and

be the discount factor in the market at time 0 for maturity T. If ${\displaystyle Q_{*}}$ is the risk neutral measure, then the forward measure ${\displaystyle Q_{T}}$ 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

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