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

1.4. PRICING EUROPEAN OPTIONS 51
Numerical Fourier inversion.
the inverse Fourier transform of ˜ζ T :
Option prices can be computed by evaluating numerically
˜z T (k) = 1
2π
∫ +∞
−∞
e −ivk ˜ζT (v)dv (1.26)
This integral can be efficiently computed for a range of strikes using the Fast Fourier transform,
an algorithm due to Cooley and Tukey [29] which allows to compute the discrete Fourier
transform DFT[f] n=0 N−1 , defined by,
DFT[f] n :=
using only O(N log N) operations.
N−1
∑
k=0
f k e −2πink/N , n = 0 . . . N − 1, (1.27)
To approximate option prices, we truncate and discretize the integral (1.26) as follows:
∫
1 ∞
e −ivk ˜ζT (v)dv = 1 ∫ L/2
e −ivk ˜ζT (v)dv + ε T
2π −∞
2π −L/2
=
N−1
L ∑
w m ˜ζT (v m )e −ikvm + ε T + ε D , (1.28)
2π(N − 1)
where ε T is the truncation error, ε D is the discretization error, v m = −L/2+m∆, ∆ = L/(N −1)
is the discretization step and w m are weights, corresponding to the chosen integration rule (for
instance, for the trapezoidal rule w 0 = w N−1 = 1/2 and all other weights are equal to 1).
Now, choosing k n = k 0 + 2πn
N∆
transform:
N−1
L
∑
2π(N − 1) eiknL/2 w m ˜ζT (k m )e −ik0m∆ e −2πinm/N
m=0
m=0
we see that the sum in the last term becomes a discrete Fourier
=
L
2π(N − 1) eiknL/2 DFT n [w m ˜ζT (k m )e −ik 0m∆ ]
Therefore, the FFT algorithm allows to compute ˜z T and option prices for the log strikes k n =
k 0 + 2πn
N∆
. The log strikes are thus equidistant with the step d satisfying
d∆ = 2π
N .
Error control.
We start with the truncation error.
Lemma 1.10. Let {X t } t≥0 be a Lévy process with characteristic triplet (A, ν, γ) and characteristic
function Φ T . Then