Processus de Lévy en Finance - Laboratoire de Probabilités et ...

3.5. NUMERICAL SOLUTION 115
of X t under Q at the points u − i for real u:
ψ Q (u − i) = − A M−1
2 u(u − i) − (1 + iu) ∑
Introducing uniform grids
j=0
M−1
∑
(e x j
− 1)q j −
j=0
M−1
∑
q j +
j=0
e iux j
e x j
q j .
x j = x 0 + jd and u k = u 0 − k∆,
the last term becomes:
M−1
∑
j=0
e iu kx j
e x j
q j = e −ik∆x 0
M−1
∑
j=0
e −ikjd∆ e (iu 0+1)x j
q j
and comparing this to Equation (1.27), we see that if d∆ = 2π M , the last term of {ψQ (u k −i)} M−1
k=0
becomes a discrete Fourier transform (cf. Equation (1.27)):
ψ k := ψ Q (u k − i) = − A M−1
2 u ∑
k(u k − i) − (1 + iu k )
j=0
(e x j
− 1)q j −
M−1
∑
j=0
q j
+ e −ik∆x 0
DFT k [e (iu 0+1)x j
q j ]. (3.28)
This expression can therefore be computed using the fast Fourier transform algorithm. Note
that at this stage all computations are exact: there are no truncation or discretization errors.
For a given maturity date T we now need to compute the Fourier transform ˜ζ T
modified time value of options (see Equation (1.25)) at the points {u k } M−1
k=0 :
of the
˜ζ T (u k ) = e iu krT eT ψ k
− e − AT
2 (u k(u k −i))
iu k (1 + iu k )
. (3.29)
Option time values are then approximated using Equation (1.28) on the same grid of log-strikes
{x j } M−1
j=0
that was used to discretize the Lévy measure. 2 In the following equation and below
we denote approximated quantities by putting a hat over the corresponding variables.
ˆ˜z T (x j ) = ∆ M−1
∑
2π e−ix ju M−1
k=0
w k ˜ζT (u M−1−k )e −ix 0∆k e −2πijk/M
= ∆ 2π e−ix ju M−1
DFT j [w k ˜ζT (u M−1−k )e −ix 0∆k ], (3.30)
2 Actually, the grid of log-strikes may be shifted but we do not do this here to simplify the formulas.