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

102 CHAPTER 3. NUMERICAL IMPLEMENTATION
in the middle, but also the behavior of tails of the calibrated Lévy measure with uniform prior
is more erratic than in the case where Merton model was used as prior (see Figure 3.2, right
graph).
Comparison of Figures 3.1 and 3.2 shows that the exact values of parameters of the prior
model are not very important, but it is crucial to find the right shape of the prior.
8
10 2 Test A
7
6
5
Test A
10 1
10 0
4
10 −1
Test B
3
2
1
Test B
10 −2
10 −3
0
−1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6
10 −4
−0.5 −0.4 −0.3 −0.2 −0.1 0
Figure 3.2: Lévy measures calibrated to the same data set using two qualitatively different
priors. Solid curves correspond to calibrated measures and dotted ones depict the priors. Right
graph: zoom of the left tail of Lévy densities on log scale.
3.3 Choice of the regularization parameter
Theorem 2.17 of the preceding chapter suggests that for the regularization method to converge,
the parameter choice strategy α(δ) must satisfy the following limiting relations, when the data
error δ goes to zero:
• α(δ) → 0,
• If the exact data C M is attainable by the model, one must have
δ 2
α(δ)
δ
α(δ) → 0.
→ 0 and if not,