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

3.5. NUMERICAL SOLUTION 117
since
∣
∣˜z ′′
T (k) ∣ ∣ = 1
2π
∣
∫ ∞
−∞
v 2 e −ivk ˜ζT (v)dv
∣
≤ 1 ∫ ∞
|Φ T (v − i)|dv + 1
2π −∞
2π
∫ ∞
−∞
√
√
|Φ A
2
T (v − i)|dv ≤ πAT ,
where we have used Equation (1.25) and Lemma 1.10. Combining the above two expressions,
one obtains:
|ε I | ≤ 1 L 2 √
π 3
2AT .
Taking L and M sufficiently large so that L/M becomes small, one can make the total error
ε = ε T + ε D + ε I arbitrarily small. In practice, the parameters M and L of the discretization
scheme should be chosen such that the total numerical error is of the same order as the noise
level in the data. The continuity result (Theorem 2.16) then guarantees that the numerical
approximation of the calibrated measure will be close to the true solution.
3.5.2 Computing the gradient of the calibration functional
We emphasize that for the optimization algorithm to work correctly, we must compute the exact
gradient of the approximate functional (3.26), rather than an approximation of the gradient of
the exact functional. The gradient of the approximate calibration functional is computed as
follows:
∂Ĵα(q 0 , . . . , q M−1 )
∂q k
=
N∑
w i (ĈQ (T i , K i ) − C M (T i , K i )) ∂ĈQ (T i , K i )
∂q k
⎛
⎞
+ α(ex k
− 1)
⎝ A M−1
2A 2 + ∑
bP + (e x j
− 1)q j
⎠ + α log(q k /p k ).
i=1
The nontrivial part is therefore to compute the gradient of the approximate option price
Ĉ Q (T i , K i ) and for this, due to the linear structure of the interpolating formula (3.31) it suffices
to know the gradient of ˆ˜z T at the points {x i }
i=0 M−1 . From Equation (3.30),
[
∂ˆ˜z T (x j )
= ∆ ∂q m 2π e−ix ju M−1
∂
DFT j w ˜ζ
]
T (u M−1−k )
k e −ix 0∆k
,
∂q m
j=0