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

2.1. LEAST SQUARES CALIBRATION 65
A + ∫ ∞
−∞ x2 ν(dx) < C 3 for some constant C 3 , independent on the choice of the Lévy process Q.
Since for all α ∈ [0, 1], e x ∧ 1 ≤ e αx , we have:
S 0 − C Q (T, K) = Ke −rT E[e X T −m ∧ 1] ≤ Ke −rT E[e α(X T −m) ] = S α 0 (Ke −rT ) 1−α E[e αX T
],
where m = log(Ke −rT /S 0 ). The right-hand side can be computed using the Lévy-Khintchine
formula:
E[e αX T
] = exp T
{
− A 2 (α − α2 ) −
Choosing α = 1/2, the above reduces to
E[e X T /2 ] = exp T {− A 8 − 1 2
On the other hand, it is easy to check that
and therefore
A
8 + 1 2
∫ ∞
−∞
∫ ∞
−∞
∫ ∞
}
(αe αx − e αx − α + 1)ν(dx) .
−∞
(e x/2 − 1) 2 ν(dx)}.
(e x/2 − 1) 2 ν(dx) ≥ e−B
8 (A + ∫ ∞
−∞
x 2 ν(dx))
∫ √ √ ∞
A + x 2 ν(dx) ≤ 8eB
−∞
T log S0 Ke −rT
S 0 − C Q (T, K) ≤ 8eB
T
log S0 Ke −rT
S 0 − C ,
which finishes the proof of the lemma.
In the following lemma, L + B denotes the sets of all probabilities P ∈ L such that P [∆X t ≤
B ∀t : 0 ≤ t ≤ T ∞ ] = 1.
Lemma 2.4. Both M ∩ L B and M ∩ L + B
are weakly closed for every B > 0.
Proof. We give the proof for M ∩ L B ; the proof for M ∩ L + B
can be done in a similar fashion.
Let {Q n } ∞ n=1 ⊂ M ∩ L B with characteristic triplets (A n , ν n , γ h n) with respect to a continuous
bounded truncation function h, satisfying h(x) = x in a neighborhood of 0, and let Q be a Lévy
process with characteristic triplet (A, ν, γ h ) with respect to h, such that Q n ⇒ Q. Note that
a sequence of Lévy processes cannot converge to anything other than a Lévy process because
due to convergence of characteristic functions, the limiting process must have stationary and
independent increments. Define a function f by
⎧
0, |x| ≤ B,
⎪⎨
f(x) := 1, |x| ≥ 2B,
⎪⎩ |x|−B
B
B < |x| < 2B.