Lévy Models - Computational Methods for Quantitative Finance - page 128

The normal inverse Gaussian process (NIG) was proposed in [8] and has the
Lévy density

e βz K 1 (α δα

π

| z | )

ν( d z) = d z,

| z |

where K 1 denotes the modified Bessel function of the third kind with index 1 and
α> 0, α<β<α , δ> 0. The NIG model is a special case of the generalized

hyperbolic model [62].

−

10.2.3 Admissible Market Models

We make the following assumptions on our market models.

Assumption 10.2.3 Let X be a Lévy process with characteristic triplet (σ 2 ,ν,γ)
and Lévy density k(z) where ν( d z) = k(z) d z .

(i) There are constants β − > 0, β + > 1 and C> 0 such that

C e − β − | z | ,z< −

1 ,

k(z) ≤ (10.11)

e − β + z ,

z> 1 .

(ii) Furthermore, there exist constants 0 <α< 2 and C > 0 such that

+

1

k(z) ≤ C +

α ,

0 < | z | < 1 . (10.12)

1 +

| z |

(iii) If σ = 0, we assume additionally that there is a C − > 0 such that

1
2 (k(z)

1

+ k( − z)) ≥ C

+ α ,

0 < | z | < 1 . (10.13)

−

1

| z |

Note that due to the semi-heavy tails ( 10.11 ) the Lévy measure satisfies

and
| > 1 e z ν( d z) <

. All Lévy processes described before
satisfy these assumptions except for the variance gamma model. Here, α

> 1 |

z | ν( d z) < ∞ ∞

| z | z |

0in
( 10.13 ) which is not allowed. Nevertheless, we will show in the numerical example
that the finite element discretization still converges to the option value with optimal
rate.

=

10.3 Pricing Equation

As in the Black-Scholes case, we assume the risk-neutral dynamics of the underly-
ing asset price is given by

S 0 e rt + X t , S t =