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Infinance, aT-forward measure is a pricing measure absolutely continuous with respect to arisk-neutral measure, but rather than using the money market asnumeraire, it uses a bond with maturityT. The use of the forward measure was pioneered byFarshid Jamshidian (1987), and later used as a means of calculating the price ofoptions on bonds.^[1]

Let^[2]

${\displaystyle B(T)=\exp \left({\int }_0^{T}r(u)du\right)}$

be the bank account or money market account numeraire and

${\displaystyle D(T)=1/B(T)=\exp \left(-{\int }_0^{T}r(u)du\right)}$

be the discount factor in the market at time 0 for maturityT. If${\displaystyle Q_{\ast }}$ is the risk neutral measure, then the forward measure${\displaystyle Q_T}$ is defined via theRadon–Nikodym derivative given by

${\displaystyle \frac{d{Q}_T}{d{Q}_{\ast }}=\frac{1}{B(T)E_{Q_{\ast }}[1/B(T)]}=\frac{D(T)}{E_{Q_{\ast }}[D(T)]}.}$

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 thechange of numeraire formula by changing the numeraire from the money market or bank accountB(t) to aT-maturity bondP(t,T). Indeed, if in general

${\displaystyle P(t,T)=E_{Q_{\ast }}\left[\frac{B(t)}{B(T)}|\mathcal{F}(t)\right]=E_{Q_{\ast }}\left[\frac{D(T)}{D(t)}|\mathcal{F}(t)\right]}$

is the price of a zero coupon bond at timet for maturityT, where${\displaystyle \mathcal{F}(t)}$ is the filtration denoting market information at timet, then we can write

${\displaystyle \frac{d{Q}_T}{d{Q}_{\ast }}=\frac{B(0)P(T,T)}{B(T)P(0,T)}}$

from which it is indeed clear that the forwardT measure is associated to theT-maturity zero coupon bond asnumeraire. For a more detailed discussion see Brigo and Mercurio (2001).

The name "forward measure" comes from the fact that under the forward measure,forward prices aremartingales, 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. Note that when interest rates are deterministic, this implies that forward prices and futures prices are the same.

For example, the discounted stock price is a martingale under the risk-neutral measure:

${\displaystyle S(t)D(t)=E_{Q_{\ast }}[D(T)S(T)|\mathcal{F}(t)].}$

The forward price is given by${\displaystyle F_S(t,T)=\frac{S(t)}{P(t,T)}}$. Thus, we have${\displaystyle F_S(T,T)=S(T)}$

${\displaystyle F_S(t,T)=\frac{E_{Q_{\ast }}[D(T)S(T)|\mathcal{F}(t)]}{D(t)P(t,T)}=E_{Q_T}[F_S(T,T)|\mathcal{F}(t)]\frac{E_{Q_{\ast }}[D(T)|\mathcal{F}(t)]}{D(t)P(t,T)}}$

by using the Radon-Nikodym derivative${\displaystyle \frac{d{Q}_T}{d{Q}_{\ast }}}$ and the equality${\displaystyle F_S(T,T)=S(T)}$. The last term is equal to unity by definition of the bond price so that we get

${\displaystyle F_S(t,T)=E_{Q_T}[F_S(T,T)|\mathcal{F}(t)].}$

• Damiano Brigo,Fabio Mercurio (2001).Interest Rate Models — Theory and Practice with Smile, Inflation and Credit (2nd ed. 2006 ed.). Springer Verlag.ISBN 978-3-540-22149-4.

• Risk-neutral measure
• Forward price

• Finance theories
• Mathematical finance
• Martingale theory

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