Multidimensional Lévy Models - Computational Methods for Quantitative Finance - page 206

∞

( 2 π) − 2 det

1
2 s −

d

2 e − z − θs, Q − 1 (z − θs) /( 2 s) e −

s
ϑ (ϑs) − 1 d s d z

Q −

ν(B) =

B 0

2 e θ, Q − 1 + Q −

( 2 π) − 2 ϑ − 1 det

1

Q −

z

=

2

B

∞

1 e − z, Q − 1 z

− θ, Q − 1 θ
2 ϑ s d s d z

d
2 −

1

s −

+

×

2 s

0

∞

2 e θ, Q − 1 + Q −

( 2 π) − 2 ϑ − 1 det

1 d

2 −

1 e − β s − γs d s d z,

Q −

z

s − =

2

B 0

Q − 1 z Q − 1 θ

where β = z, / 2 and γ = θ, / 2 + 1 /ϑ . Integrating the second inte-

gral, we obtain the Lévy measure

β
γ

− d/ 4

K − d/ 2 2 βγ d z,
(14.12)

2 e θ, Q − 1 + Q −

2 ( 2 π) − 2 ϑ − 1 det

1

Q −

z

ν( d z) =

2

where K

d/ 2 (ξ ) is the modified Bessel function of the second kind. For small ξ ,we
have K − d/ 2 (ξ )

−

ξ − d/ 2 , and therefore ν( d z) Q − 1 z − d/ 2 d z | − d d z since ∼ ∼ z, ∼ | z

> 0. The marginal processes X i , i

Q = 1 ,...,d of X are vari ance gamma processes

ϑ − 1 e θ i /σ i z e − 2 /ϑ + θ i /σ i /σ i | z | | | − 1 d z .Weplot on R with Lévy measure ν i ( d z) = z

the density (10.9)for d = 2, θ = ( − 0 . 1 , − 0 . 2 ) , σ = ( 0 . 3 , 0 . 4 ) , ρ 12 = 0 . 5 and ϑ = 1

in Fig. 14.3 .

14.3.2 Lévy Copula Models

Lévy copulas F allow parametric constructions of multivariate jump densities from
univariate ones. Let U 1 ,...,U d be one-dimensional tail integrals with Lévy density
k 1 ,...,k d , and let F be a Lévy copula such that ∂ 1 ··· ∂ d F exists in the sense of
distributions. Then,
k(x 1 ,...,x d ) = ∂ 1 ··· ∂ d F | ξ 1 = U 1 (x 1 ),...,ξ d = U d (x d ) k 1 (x 1 ) ··· k d (x d ) (14.13)
is the jump density of a d -variate Lévy measure with marginal Lévy densities
k 1 ,...,k d . For example, we can use the Clayton Lévy copula (see Definition 14.6 )

2 2 − d d u i | − ϑ −

1
ϑ

η 1 { u 1 ··· u d ≥ 0 } −

η) 1 { u 1 ··· u d ≤ 0 } ,

1 |

F(u 1 ,...,u d ) = ( 1 −

i =

where ϑ> 0, η ∈[ 0 , 1 ] and consider α -stable marginal Lévy densities, k i (z) =

| z | − 1 − α i ,0 <α i < 2, i =

1 ,...,d . This leads to the d -dimensional Lévy density

1 d α i ϑ −

1
ϑ −

d

d

1

1 )ϑ α ϑ + 1
i

2 2 − d

α i ϑ −

α i

|

z i |

|

z i |

k(z) = + (i −

i = 1 i = 1

· η 1 { z 1 ··· z d ≥ 0 } + η) 1 { z 1 ··· z d ≤ 0 } .

( 1 − (14.14)