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

96 CHAPTER 3. NUMERICAL IMPLEMENTATION
3.1 Discretizing the calibration problem
A convenient way to discretize the calibration problem is to take a prior Lévy process P with
Lévy measure supported by a finite number of points:
ν P =
M−1
∑
k=0
p k δ {xk }(dx). (3.1)
In this case, by Proposition 1.5, the Lévy measure of the solution necessarily satisfies ν Q ≪ ν P ,
therefore
ν Q =
M−1
∑
k=0
q k δ {xk }(dx), (3.2)
that is, the solution belongs to a finite-dimensional space and can be computed using a numerical
optimization algorithm. The advantage of this discretization approach is that we are solving the
same problem (2.27), only with a different prior measure, so all results of Section 2.5 (existence
of solution, continuity etc.) hold in the finite-dimensional case.
Taking Lévy measures of the form (3.1) we implicitly restrict the class of possible solutions
to Lévy processes with bounded jumps and finite jump intensity. However, in this section we
will see that this restriction is not as important as it seems: the solution of a calibration problem
with any prior can be approximated (in the weak sense) by a sequence of solutions of calibration
problems with priors having Lévy measures of the form (3.1). Moreover, in Section 3.6.1 we
will observe empirically that smiles produced by infinite intensity models can be calibrated with
arbitrary precision by such jump-diffusion models.
We start with a lemma showing that every Lévy process can be approximated by Lévy
processes with atomic Lévy measures.
Lemma 3.1. Let P be a Lévy process with characteristic triplet (A, ν, γ) with respect to a
continuous bounded truncation function h, satisfying h(x) = x in a neighborhood of 0, and for
every n, let P n be a Lévy process with characteristic triplet (A, ν n , γ) (with respect to the same
truncation function) where
ν n :=
2n∑
k=1
δ {xk }(dx) µ([x k − 1/ √ n, x k + 1/ √ n))
1 ∧ x 2 ,
k
x k := (2(k − n) − 1)/ √ n and µ is a finite measure on R, defined by µ(B) := ∫ B (1 ∧ x2 )ν(dx)
for all B ∈ B(R). Then P n ⇒ P .