Lévy Models - Computational Methods for Quantitative Finance - page 126

− =

Therefore, setting ψ( i) 0 and using the Lévy-Khinchine formula ( 10.5 ) yields

σ 2
2 +

(e z γ + − 1 − z)ν( d z) = 0 .

R

Note that ψ( − i) is well-defined due to [143, Theorem 25.17].

10.2 Lévy Models

Financial models with jumps fall into two categories: Jump-diffusion models have
a nonzero Gaussian component and a jump part which is a compound Poisson pro-
cess with finitely many jumps in every time interval. On the other hand, infinite
activity models have an infinite number of jumps in every interval of positive mea-
sure. A Brownian motion component is not necessary for infinite activity models
since the dynamics of the jumps is already rich enough to generate nontrivial small-
time behavior. If there is no Brownian motion component, the models are called
pure jump models.

10.2.1 Jump-Diffusion Models

A Lévy process of jump-diffusion has the following form

N t

X t = γ 0 t + σW t + Y i (10.6)

i = 1

where N is a Poisson process with intensity λ counting the jumps of X , and Y i are
the jump sizes which are modeled by i.i.d. random variables with distribution ν 0 .
The Lévy measure is given by ν

λν 0 . To define the parametric model completely,
we must specify the distribution ν 0 of the jump sizes.
In the Merton model [125], jumps in the log-price X areassumedtohaveaGaus-
sian distribution, i.e. Y i ∼ N(μ,δ 2 ) , and therefore,

=

1

√ 2 πδ 2 e − (z − μ) 2 /( 2 δ 2 ) d z.

ν 0 ( d z) = (10.7)

In the Kou model [107], the distribution of jump sizes is an asymmetric exponential
with a density of the form
ν 0 ( d z) = pβ + e − β + z 1

d z,

− p)β − e − β − | z | 1

} + ( 1

(10.8)

{ z> 0 { z< 0 }

with β + ,β − > 0 governing the decay of the tails for the distribution of the positive
and negative jump sizes and p

representing the probability of an upward
jump. The probability distribution of returns of this model has semi-heavy tails.

∈[ 0 , 1 ]