Processus de Lévy en Finance - Laboratoire de Probabilités et ...

Chapter 1
Lévy processes and exp-Lévy models
This introductory chapter serves essentially two purposes. First, in Section 1.1, we give an
overview of the probabilistic properties of Lévy processes that will be used in the sequel. Second,
we define exponential Lévy models (Section 1.2), review various parametrizations of the Lévy
measure, proposed by other authors (Section 1.3) and discuss a method for option pricing in
these models, based on Fourier transform (Section 1.4). This option pricing method is later used
for the numerical solution of the calibration problem. While the results of the first three sections
can be found in the literature, Section 1.4 contains new material: we improve the method due
to Carr and Madan [23] in a number of ways and provide estimates of the truncation and
discretization error, not found in the original paper.
1.1 Lévy processes
Proofs of the the results of this section can be found, unless otherwise mentioned in [87]. For
additional details on Lévy processes the reader may consult [17] or [54]. The latter book treats
a far more general class of semimartingales but properties of Lévy processes are often discussed
as examples or corollaries of the general results.
Definition 1.1 (Lévy process). A stochastic process {X t } t≥0 on (Ω, F, P ) such that X 0 = 0
is called a Lévy process if it possesses the following properties:
1. Independent increments: for every increasing sequence of times t 0 . . . t n , the random variables
X t0 , X t1 − X t0 , . . . , X tn − X tn−1
are independent.
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