Multidimensional Lévy Models - Computational Methods for Quantitative Finance - page 216

( left ) and compressed matrix A J Stiffness matrix A J Fig. 14.6 ( right )forlevel L = 8

14.7 Application: Impact of Approximations of Small Jumps

In this section, we consider a regularization of the (multivariate) Lévy measure
where small jumps are either neglected or approximated by an artificial Brownian
motion. This Gaussian approximation is often proposed to simulate Lévy processes
or to price options using finite differences. Applying the methods developed in the
previous chapters gives accurate numerical schemes for either model. We use our
scheme to study and compare the error of diffusion approximations of small jumps
in multivariate Lévy models via accurate numerical solutions of the corresponding
PIDEs for various types of contracts.

14.7.1 Gaussian Approximation

Let X be a d -dimensional Lévy process with the characteristic exponent

1 ν( d z),

− e i ξ,z + i ξ,z ψ(ξ) =− i γ,ξ +

R

d

whereweassume

> 1 |

|

z ν( d z) < ∞ .For ε> 0let ν ε be a measure such that

| z |

ν ε is a finite measure and

ν ε

2 ν ε ( d z) <

= ν − | z | ∞ . Then, the characteristic

d

R

exponent can be decomposed into two parts

1

e i ξ,z ν ε ( d z)

1 ν ε ( d z)

γ ε ,ξ e i ξ,z + ψ(ξ) =− i + − + − i ξ,z ,

d d

R R

ψ ε (ξ) ψ ε (ξ)

(14.28)

γ i − where γ i z i ν i ( d z i ) , i 1 ,...,d . Correspondingly, we can decompose

X into its small and large jump parts
X t = γ ε t + N t

= =

R

+ X ε,t = X t + X ε,t , (14.29)