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

66 CHAPTER 2. THE CALIBRATION PROBLEM
By Proposition 1.7, ∫ ∞
−∞ f(x)ν(dx) = lim n→∞
∫ ∞
−∞ f(x)ν n(dx) = 0, which implies that the
jumps of Q are bounded by B.
Define a function g by
⎧
⎨ e x − 1 − h(x) − 1 2
g(x) :=
h2 (x), x ≤ B,
⎩
e B − 1 − h(B) − 1 2 h2 (B), x > B.
Then, by Proposition 1.7 and because Q n satisfies the martingale condition (1.2) for every n,
γ h + A 2 + ∫ ∞
−∞
(e x − 1 − h(x))ν(dx) = γ h + A + ∫ ∞
−∞ h2 (x)ν(dx)
2
= lim
n→∞
which shows that Q also satisfies the condition (1.2).
{γn h + A n + ∫ ∞
−∞ h2 (x)ν n (dx)
+
2
Proof of Theorem 2.1. Let {Q n } n≥1 ⊂ M ∩ L B be such that
Condition (2.9) implies that for every n,
∫ ∞
+
lim ‖C M − C Qn ‖ 2
n→∞
w = inf ‖C M − C Q ‖ 2 w
Q∈M∩L B
−∞
∫ ∞
−∞
and ‖C M − C Qn ‖ 2 w ≤ ‖C M − C Q 0
‖ 2 w for all n.
g(x)ν(dx)
g(x)ν n (dx)
}
= 0,
|S 0 − C Qn (T 0 , K 0 )| ≥ |S 0 − C M (T 0 , K 0 )| − |C M (T 0 , K 0 ) − C Qn (T 0 , K 0 )|
≥ |S 0 − C M (T 0 , K 0 )| − ‖C M − C Qn ‖ w
√
w0
≥ |S 0 − C M (T 0 , K 0 )| − ‖C M − C Q 0
‖ w
√
w0
> 0.
Therefore, by Lemmas 2.3 and 2.4, there exists a subsequence {Q nm } m≥1 of {Q n } n≥1 and
Q ∗ ∈ M ∩ L B such that Q nm ⇒ Q. By Lemma 2.2,
‖C M − C Q∗ ‖ 2 w = lim ‖C M − C Qnm ‖ 2
m→∞
w = inf ‖C M − C Q ‖ 2 w,
Q∈M∩L B
which shows that Q ∗ is a solution of the least squares calibration problem (2.4).
2.1.3 Continuity
Market option prices are typically defined up to a bid-ask spread and the close prices used for
calibration may therefore contain numerical errors. If the solution of the calibration problem