Multidimensional Lévy Models - Computational Methods for Quantitative Finance - page 202

Fig. 14.1 Clayton copula ( 14.6 )in d = 2for ϑ = 0 . 5( top )and ϑ = 1 . 5( bottom )

(iii) Clayton Lévy copulas

2 2 − d d u i | − ϑ −

1
ϑ

η 1 ,

F(u 1 ,...,u d ) =

1 |

} −

( 1 − η) 1

{ u 1 ··· u d ≥ 0 { u 1 ··· u d ≤ 0 }

i =

(14.6)

where ϑ> 0 and η ∈[ 0 , 1 ] .For η = 1 and ϑ → 0, F converges to the indepen-

dence Lévy copula, for η to the complete dependence Lévy

copula. In Fig. 14.1 , the Clayton copula in d

= 1 and ϑ →∞

= 2for ϑ = 0 . 5 , 1 . 5 and η = 1is

plotted. We include the upper bound min {| u 1 | , | u 2 |} and additionally give the

corresponding contour plot.

An important class of Lévy copulas are the so-called 1-homogeneous copulas.

Definition 14.2.10 A Lévy copula is called 1-homogeneous if for any r> 0the
following holds:

F(ru 1 ,...,ru d ) = rF(u 1 ,...,u d ),

for all (u 1 ,...,u d ) ∈ R

d .