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

64 CHAPTER 2. THE CALIBRATION PROBLEM
Then the problem (2.4) has a solution in M ∩ L B : there exists Q ∗ ∈ M ∩ L B , such that
‖C M − C Q∗ ‖ 2 w =
The proof is based on the following lemmas.
inf ‖C M − C Q ‖ 2 w.
Q∈M∩L B
Lemma 2.2. The pricing error functional Q ↦→ ‖C M − C Q ‖ 2 w, defined by (2.2), is uniformly
bounded and weakly continuous on M ∩ L.
Proof. From Equation (1.14), C Q (T, K) ≤ S 0 . Absence of arbitrage in the market implies that
the market option prices satisfy the same condition. Therefore, (C M (T, K) − C Q (T, K)) 2 ≤ S0
2
and since w is a probability measure, ‖C M − C Q ‖ 2 w ≤ S0 2.
Let {Q n } n≥1 ⊂ M ∩ L and Q ∈ M ∩ L be such that Q n ⇒ Q. For all T, K,
lim
n
C Qn (T, K) = e −rT lim
n
E Qn [(S 0 e rT +X T
− K) + ]
= e −rT lim
n
E Qn [S 0 e rT +X T
− K] + e −rT lim
n
E Qn [(K − S 0 e rT +X T
) + ]
= S 0 − Ke −rT + e −rT E Q [(K − S 0 e rT +X T
) + ] = C Q (T, K).
Therefore, by the dominated convergence theorem, ‖C M − C Qn ‖ 2 w → ‖C M − C Q ‖ 2 w.
Lemma 2.3. For all B > 0, T > 0, K > 0 and all C with 0 < C < S 0 , the set of Lévy processes
Q ∈ M ∩ L B satisfying C Q (T, K) ≤ C is relatively weakly compact.
Proof. By Prohorov’s theorem, weak relative compactness is implied by the tightness of this
family of probability measures. Since the jumps are bounded, to prove the tightness, by Proposition
1.6 it is enough to show that there exist constants C 1 and C 2 such that for every Lévy
process Q ∈ M∩L B such that C Q (T, K) ≤ C, its characteristic triplet (A, ν, γ) satisfies |γ| ≤ C 1
and A + ∫ 1
−1 x2 ν(dx) ≤ C 2 .
By the risk-neutrality condition (1.2),
γ = − A ∫ B
2 − (e x − 1 − x1 |x|≤1 )ν(dx)
−B
It is easy to see that |e x − 1 − x1 |x|≤1 | ≤ e B∧1 x 2
e B∧1 (A + ∫ ∞
−∞ x2 ν(dx)).
for x ∈ (−∞, B] and therefore |γ| ≤
This means that to prove the lemma, it is sufficient to show that