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

68 CHAPTER 2. THE CALIBRATION PROBLEM
Proposition 2.5. Suppose that there exists a couple (T 0 , K 0 ) with w 0 := w({T 0 , K 0 }) > 0 and
let C M be such that the condition (2.9) is satisfied for some Q 0 ∈ M ∩ L B . Then the solutions
of the least squares calibration problem (2.4) on M ∩ L B depend continuously on the market
data at the point C M .
Proof. Let {CM n } n≥0 be a sequence of data such that ‖CM n − C M‖ w −−−→
n→∞
n let Q n be a solution of the calibration problem (2.4) on M ∩ L B with data CM
n
0, and for every
(we can
suppose without loss of generality that a solution exists for all n because it exists starting with
a sufficiently large n by Theorem 2.1). Then, using the triangle inequality several times, we
obtain:
|S 0 − C Qn (T 0 , K 0 )|
≥ |S 0 − C M (T 0 , K 0 )| − |C n M(T 0 , K 0 ) − C Qn (T 0 , K 0 )| − |C n M(T 0 , K 0 ) − C M (T 0 , K 0 )|
≥ |S 0 − C M (T 0 , K 0 )| − ‖Cn M − CQ 0
‖ w
√
w0
− |C n M(T 0 , K 0 ) − C M (T 0 , K 0 )|
≥ |S 0 − C M (T 0 , K 0 )| − ‖C M − C Q 0
‖ w
√
w0
− 2 ‖Cn M − C M‖ w
√
w0
> C ′ > 0
for some C ′ , starting from a sufficiently large n. Therefore, by Lemmas 2.3 and 2.4, {Q n } has
a subsequence that converges weakly towards some Q ∗ ∈ M ∩ L B .
Let {Q nm } ⊆ {Q n } with Q nm ⇒ Q ∗ ∈ M ∩ L B and let Q ∈ M ∩ L B . Using Lemma 2.2 and
the triangle inequality, we obtain:
‖C Q∗ − C M ‖ w = lim ‖C Qnm − C M ‖ w ≤ lim inf
m m
{‖CQnm − C nm
M ‖ w + ‖C nm
M − C M‖ w }
≤ lim inf
m
‖CQnm − C nm
M ‖ w ≤ lim inf
m
‖CQ − C nm
M ‖ w ≤ ‖C Q − C M ‖ w ,
which shows that Q ∗ is indeed a solution of the calibration problem (2.4) with data C M .
2.1.4 Numerical difficulties of least squares calibration
A major obstacle for the numerical implementation of the least squares calibration is the nonconvexity
of the optimization problem (2.4), which is due to the non-convexity of the domain
(M ∩ L), where the pricing error functional Q ↦→ ‖C Q − C M ‖ 2 is to be optimized. Due to this
difficulty, the pricing error functional may have several local minima, and the gradient descent