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

30 INTRODUCTION
Lévy models by Fourier transform. This section is the only one of the first chapter to contain
new results: we propose original improvements to this method and obtain estimates for the
truncation and discretization errors, that are not given in the original reference [23].
The second chapter is dedicated to the theoretical treatment of the calibration problem.
We start by discussing the least squares calibration method, commonly used by academics
and practitioners. We show that in the context of non-parametric calibration of exponential
Lévy models, least squares calibration does not always allow to find a solution and even if it
does, the solution is typically very sensitive to small perturbations of input data. To solve the
calibration problem in a stable manner, we first reformulate it as the problem of finding an
exponential Lévy model that has the smallest relative entropy with respect to the prior among
all solutions of the least squares calibration problem. This problem, called minimal entropy
least squares calibration, is still ill-posed, so we regularize it using the technique of minimal
entropy regularization, from the theory of ill-posed inverse problems.
The third chapter discusses the numerical implementation of our calibration algorithm. To
solve the calibration problem numerically, it is expressed in terms of the characteristic triplets
of the prior and the solution and the Lévy measure of the prior is discretized on a uniform grid
so that the calibration problem becomes finite-dimensional.
In the fourth chapter we first review the two available methods to model dependence between
components of Lévy processes. Understanding the drawbacks of these methods allows to
formulate the desirable properties of a multidimensional modelling approach. These properties
lead us to defining the notion of Lévy copula, first in the case of Lévy processes with only
positive jumps in every component and then in the general case. After proving a representation
theorem, which shows that Lévy copulas completely characterize dependence structures of Lévy
processes, we compute Lévy copulas that correspond to various special types of dependence.
The fifth and the last chapter provides the tools necessary to apply Lévy copulas to finance.
We first give methods to construct parametric families of Lévy copulas and then develop an efficient
algorithm to simulate multidimensional Lévy processes with dependence structures given
by Lévy copulas. The last section of this chapter contains an example of a multidimensional
exponential Lévy model for option pricing, constructed using Lévy copulas.