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

136 CHAPTER 4. DEPENDENCE OF LEVY PROCESSES
shocks do we need to describe sufficiently rich dependence structures? The answer depends
on the particular problem, but to describe all possible dependence structures, such that the
d-dimensional process remains a Lévy process of compound Poisson type, one needs a total of
2 M − 1 shocks (M shocks that affect only one stock, M(M−1)
2
shocks that affect two stocks etc.,
adding up to 2 M − 1). It is clear that as the dimension of the problem grows, this kind of
modelling quickly becomes infeasible. Not only the number of parameters grows exponentially,
but also, when the number of shocks is greater than one, one cannot specify directly the laws
of components because the laws of jumps must be given separately for each shock.
Comparison of advantages and drawbacks of these two methods leads to an understanding
of the required properties of a multidimensional modelling approach for Lévy processes.
general, such an approach must satisfy the following conditions:
• One should be able to choose any one-dimensional Lévy process for each of the components.
In particular, it should be possible to couple a compound Poisson process with
a process which has infinite jump intensity. This is particularly important for financial
applications because information about margins and about dependence may come from
different sources. For instance, for pricing basket options the market practice is to estimate
the correlations (dependence) from historical data while using implied volatilities,
computed from the prices of traded options.
In
• The range of possible dependence structures should include complete dependence and
complete independence with a “smooth” transition between these two extremes.
• It should be possible to model dependence in a parametric fashion (e.g., not through
the entire multidimensional Lévy measure). A parsimonious description of dependence
is especially important because one typically does not have enough information about
the dependence structure to estimate many parameters or proceed with a nonparametric
approach.
To implement this program, we suggest to model separately the dependence structure of
a Lévy process and the behavior of its components (margins). It has long been known that
the dependence structure of a random vector on R d can be disentangled from its margins
via the notion of copula.
Since the law of a Lévy process is completely determined by its