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

28 INTRODUCTION
7500
Taux de change DM/USD
7400
7300
7200
7100
7000
17/09/1992 2/10/1992
Figure 1: Jumps in the trajectory of DM/USD exchange rate, sampled at 5-minute intervals.
additional instruments, like in stochastic volatility models. Since in such a market every terminal
payoff can be exactly replicated, the very existence of traded options becomes a puzzle. The
mystery is easily solved by allowing for discontinuities: in real markets, due to the presence of
jumps in the prices, perfect hedging is impossible and options enable the market participants
to hedge risks that cannot be hedged by using the underlying only.
The third and the strongest argument for using discontinuous models is simply the presence
of jumps in observed prices. Figure 1 depicts the evolution of the DM/USD exchange rate over
a two-week period in 1992, and one can see at least three points where the rate moved by over
100 bp within a 5-minute period. Price moves like these ones clearly cannot be accounted for
in the framework of a diffusion model with continuous paths, but they must be dealt with if
the market risk is to be measured and managed correctly.
Although when this thesis was started, the field of financial modelling with Lévy processes
was already a well-developed one, with several Ph. D. theses [78, 79, 81, 86, 96] and a few hundred
research papers (see references in [27]) already written on the subject, two major issues, that
appear in the title of the present study, remained unresolved.
First, while the main concern in the literature has been to find efficient analytical and
numerical procedures for computing option prices in exponential Lévy models, an essential
step in using such models is to obtain the parameters — here the characteristic triplet of the