Shifted Forward Measure
The is a Martingale or driftless under its own measure . The solution to equation (3b) can be expressed as
(5)
where is the current (spot) forward rate. Under the volatility assumption described above, equation (5) can be further expressed as
(6)
Alternatively, we can reach the same Martingale conclusion by directly deriving the expectation of the forward rate (6); that is
(7)
where , are both Brownian motions with a normal distribution (0,t ) at time t , is the expectation conditional on the , and the variable substitution used for derivation is
(8)
This variable substitution that ensures that the distribution is centered on zero and symmetry is the key to achieve high accuracywhen we express the LMM in discrete finite form and use numerical integration to calculate the expectation. As a matter of fact, without this linear transformation, a lattice method in the LMM either does not exist or introduces too much error for longer maturities.
After applying this variable substitution (8), equation (6) can be expressed as
(9)
Since the LMM models the complete forward curve directly, it is essential to bring everything under a common measure. The terminal measure is a good choice for this purpose, although this is by no means the only choice. The forward rate dynamic under terminal measure is given by
(10)
The solution to equation (10) can be expressed as
(11a)
where the drift is given by
(11b)
where is the drift term.
Applying (8) to (11a), we have the forward rate dynamic under the shifted terminal measure as
(12)