Forward measure

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.

Mathematical Definition

Let $D(T)\; =\; expleft(-int\_0^T\; r(u)\; du\; ight)$ be the discount factor in the market. If $Q\_*$ is the risk neutral measure, then the forward measure $Q\_T$ is defined via the Radon-Nikodym derivative given by

:$frac\{dQ\_T\}\{dQ\_*\}\; =\; frac\{D(T)\}\{E\_\{Q\_*\}\; [D(T)]\; \}$.

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 the change of numeraire formula by changing the numeraire from the money market to a bond.

Consequences

Under the forward measure, forward prices are martingales. 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:

:$S(t)\; D(t)\; =\; E\_\{Q\_*\}\; [D(T)S(T)\; |\; mathcal\{F\}(t)]$.

The forward price is given by $F\_S(t,T)\; =\; frac\{S(t)\}\{P(t,T)\}$ where $P(t,T)\; =\; E\_\{Q\_*\}\; [D(t)^\{-1\}D(T)|mathcal\{F\}(t)]$ is the T-zero coupon bond price at time t. Thus, we have $F\_S(T,T)$

:$F\_S(t,T)\; =\; frac\{E\_\{Q\_*\}\; [D(T)S(T)\; |\; mathcal\{F\}(t)]\; \}\{D(t)\; P(t,T)\}=\; E\_\{Q\_T\}\; [F\_S(T,T)\; |\; mathcal\{F\}(t)]\; frac\{E\_\{Q\_*\}\; [D(T)|mathcal\{F\}(t)]\; \}\{D(t)\; P(t,T)\}$

by the abstract Bayes' rule. The last term is equal to unity by definition of the bond price so that we get

:$F\_S(t,T)\; =\; E\_\{Q\_T\}\; [F\_S(T,T)|mathcal\{F\}(t)]$.

ee also

*Risk-neutral measure
*Forward price