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

58 CHAPTER 2. THE CALIBRATION PROBLEM
Calibration problem for data, consistent with exp-Lévy model.
of call options {C M (T i , K i )} i∈I , find Q ∗ ∈ M ∩ L, such that
Given market prices
∀i ∈ I, C Q∗ (T i , K i ) = C M (T i , K i ). (2.1)
In most cases, however, Equations (2.1) cannot be solved exactly, either because the observed
option prices contain errors or because the market in question cannot be described by an
exponential Lévy model. In this case a common practice is to replace the exact constraints
(2.1) by a nonlinear least squares calibration problem. Section 2.1 takes a critical look at
this approach and establishes some limits of its applicability to non-parametric calibration of
exponential Lévy models.
An important difficulty of least squares calibration is its lack of identification, meaning
that a finite number of observed option prices does not allow to reconstruct the law of a Lévy
process in a unique fashion. To address this issue, we suggest, in Section 2.2 to reformulate
the calibration problem as that of finding the risk-neutral exponential Lévy model that has the
smallest relative entropy with respect to a given prior probability measure among all solutions
of the least squares calibration problem. Section 2.3 reviews the literature on the use of relative
entropy for pricing and calibration and places the present study into the context of previous
work on this subject.
The use of relative entropy for selection of solutions removes to some extent the identification
problem but the resulting calibration problem is still ill-posed: small errors in market data may
lead to large changes of its solution. The last section of this chapter uses the method of
regularization to approximate the solutions of this problem in a stable manner in presence of
data errors.
2.1 Least squares calibration
When the market data is not consistent with the class of exponential Lévy models, the exact
calibration problem may not have a solution. In this case one may consider an approximate
solution: instead of reproducing the market option prices exactly, one may look for a Lévy triplet
which reproduces them in the best possible way in the least squares sense. Let w be a probability