Forward measure

If a zero-coupon bond is chosen as the numeraire, the price of a caplet with period $\delta$ can be written as the bond price times the expected payoff. The price at time $t$ is $\delta\,Z_t^{T+\delta}\,E^{T+\delta}_t[(L(T,T) - K)^+]$, where $Z_t^{T+\delta}$ is the price of the bond maturing at $T+\delta$ and $E^{T+\delta}$ is the expected value under its forward measure. Written with another numeraire (the source's $B_t$) and a measure $Q$, the same price is $B_t$ times the expected value of the payoff divided by $B_{T+\delta}$¹.

Consider a hypothetical example of a caplet with $\delta = 0.5$. Let the price of the bond maturing at $T+\delta$ be 0.98, and let the expected payoff under the forward measure, $E^{T+\delta}[(L - K)^+]$, be 0.4%. The price of a caplet with a notional of 1 is $0.5 \times 0.98 \times 0.004 = 0.00196$, and with a notional of 10 billion yen it is 0.0196 billion yen (19.6 million yen).

Changing the measure also changes how the forward rates move. Under the forward measure of the last date $T_{M+1}$, the last forward rate $L_M$ is a martingale and its drift is 0. The other forward rates are not martingales and keep a drift². Another measure is the swap measure, used for swaptions³.

Notes

1. Martin Haugh, "Market Models" (Columbia University, Term Structure Models: IEOR E4710, 2010), page 2, "Black's Formula for Caplets". The text writes $E^{T+\delta}$ for the source's $P^{T+\delta}$. ↩

2. Haugh, "Market Models", page 8, around equation (23) (the bond price divided by $Z^{M+1}_t$ is a $P^{M+1}$-martingale), equations (24) and (25), and Remark 3. ↩

3. Haugh, "Market Models", page 6, "Numeraire-EMM Pairs". The spot measure (account $B^*_t$ as the reference), the forward measure $P^T$ (the bond maturing at $T$ as the reference), and the swap measure (used for analytical pricing of swaptions). ↩