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

5.1. PARAMETRIC FAMILIES 171
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−2.5
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
2.5
−2.5
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
Figure 5.1: Contour plots of Lévy density with 1-stable margins and Lévy copula (5.4) with
θ = 5 and different values of η. Top left: η = 0. Top right: η = 1. Bottom: η = 0.5. On each
curve the Lévy density ν(x, y) is constant and it increases from outer curves to inner curves.
defines a two-parameter family of Lévy copulas (F is in fact a Lévy copula for all θ > 0 and
η ∈ [0, 1]). The role of parameters is most easy to analyze in the case d = 2, when (5.3) becomes
F (u, v) = (|u| −θ + |v| −θ ) −1/θ (η1 uv≥0 − (1 − η)1 uv