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

86 CHAPTER 2. THE CALIBRATION PROBLEM
are within an error δ of C M , and want to construct an approximation to MELSS(C M ), the
solution of problem (2.11) with the true data, it is not a good idea to solve problem (2.11) with
the noisy data C δ M
because MELSS(Cδ M ) may be very far from MELSS(C M). We therefore
need to regularize the problem (2.11), that is, construct a family of continuous “regularization
operators” {R α } α>0 , where α is the parameter which determines the intensity of regularization,
such that R α (CM δ ) converges to MELSS of the calibration problem as the noise level δ tends to
zero if, for each δ, the regularization parameter α is chosen appropriately. The approximation
to MELSS(C M ) using the noisy data CM δ is then given by R α(CM δ ) with an appropriate choice
of α.
Following classical results on regularization of ill-posed problems (see [40]), we suggest to
construct a regularized version of (2.11) by using the relative entropy for penalization rather
than for selection, that is, to define
J α (Q) = ‖C δ M − C Q ‖ 2 w + αI(Q|P ), (2.26)
where α is the regularization parameter, and solve the following regularized calibration problem:
Regularized calibration problem
Given prices C M of call options, a prior Lévy process
P and a regularization parameter α > 0, find Q ∗ ∈ M ∩ L, such that
J α (Q ∗ ) =
inf J α(Q). (2.27)
Q∈M∩L
Problem (2.27) can be thought of in two ways:
• If the minimum entropy least squares solution with the true data C M exists, (2.27) allows
to construct a stable approximation of this solution using the noisy data.
• If the MELSS with the true data does not exist, either because the set of least squares
solutions is empty or because the least squares solutions are incompatible with the prior,
the regularized problem (2.27) allows to find a “compromise solution”, achieving a tradeoff
between the pricing constraints and the prior information.
In the rest of this section we study the regularized calibration problem. Under our standing
hypothesis that the prior Lévy process has jumps bounded from above and corresponds to