sparse grid method in the libor market model. option valuation and the

At this point of the text all necessary mathematical tools have been covered to
proceed with derivation. It is the recursive application of multi-dimensional Girsanov’s
theorem and Change of Measure results that will help collect all the LIBOR rates under
the same measure. First, recall that according to Theorem (2.3.2), we can go from
one measure to another by introducing the drift into the original LIBOR process, as
in (2.26). In order to do so, we need to obtain a drift-correction term κ, which is at the
same time the volatility of a Radon-Nikodym derivative.
1. Working out drift-correction term
It was agreed that discount bonds D(t, T i ) serve as numeraire assets that impose
martingale measures on separate forward rates. Let Q i denote such numeraireborn
measure, therefore making dL i (t) a martingale. So, from (3.9):
Y t =
dQi
dQ i+1 = D(t, T i)/D(0, T i )
D(t, T i+1 )/D(0, T i+1 ) = D(0, T i+1)
D(0, T i ) (1 + α iL i (t)) (3.10)
At the same time, from (2.25):
( ∫ t
Y t =
dQi
dQ i+1 = exp κ(s)dW Qi+1 (s) − 1 2
0
∫ t
Note that Y t is a martingale process described by the SDE:
0
)
κ(s) 2 ds
(3.11)
dY t = κ(t)Y t dW i+1 (3.12)
The final expression for κ(t) now requires application of Itô’s Lemma to (3.10):
dY t = D(0, T i+1)
D(0, T i ) α idL i (t) = D(0, T i+1)
D(0, T i ) α iσ i (t)L i (t)dW i+1 (3.13)
The last equality comes directly from definition of a LIBOR process. Equating
(3.12) and (3.13):
κ(t)Y t dW i+1 = D(0, T i+1)
D(0, T i ) α iσ i (t)L i (t)dW i+1
D(0,
κ(t)
✟ ✟✟✟ T i+1 )
✟ D(0, T i ) (1 + α iL i (t))✘✘ dW i+1 ✘ D(0,
=
✟ ✟✟✟ T i+1 )
✟ D(0, T i ) α iσ i (t)L i (t)✘✘ dW i+1 ✘
κ(t) = α iσ i (t)L i (t)
1 + α i L i (t)
(3.14)
2. Adjusting the drift.
Now that we have a drift correction term κ(t) we can apply Girsanov’s theorem
to get a change of measure relation 3 :
dW i = dW i+1 − κ(t)ρ i,i+1 dt = dW i+1 − α iρ i,i+1 σ i (t)L i (t)
dt
1 + α i L i (t)
3 In case of a single underlying Brownian Motion, correlation term ρ i,j = 1
21