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

1.4. PRICING EUROPEAN OPTIONS 53
By the first part of Lemma 1.10,
∫
1 ∞
|Φ T (v − i)|
2π L/2 v 2 dv ≤ 1 ∫ ∞
dv
2π L/2 v 2 e−T Av2 /2 = 1 ∫ ∞
4π L 2 /4
≤ 1
4π
dz Az/2
e−T
z3/2 ∫
8 ∞
L 3 dze −T Az/2 =
L 2 /4
4
πT AL 3 e− L2 AT
8 ,
and since a similar bound can be obtained for Φ Σ T
, this proves the first part of the proposition.
To prove the second statement, observe that from the second part of Lemma 1.10,
∫
1 ∞
2π L/2
|Φ T (v − i)|
v 2 dv ≤ |Φ ∫
T (L/2 − i)| ∞
dv
2π L/2
v 2 = |Φ T (L/2 − i)|
.
πL
To compute a bound for the discretization (sampling) error ε D in Equation (1.28), we define
constants C k , k ≥ 1 by
⎧
⎪⎨
C k =
⎪⎩
|Φ (k)
T
(−i)|, k is even,
|Φ (k+1)
T
(−i)| k
k+1 , k is odd,
(1.31)
and start by proving a technical lemma.
Lemma 1.12. Let {X t } t≥0 be a real-valued Lévy process with triplet (A, ν, γ) and characteristic
function Φ T , such that the condition (1.23) is satisfied. Let Γ T (v) := Φ T (v−i)−1
v
. Then for all
n ≥ 0,
|Γ (n)
T (v)| ≤ C n+1
n + 1
∀ v ∈ R
Proof. Under the condition (1.23), for all n ≥ 1, E[|X T | n e X T
] < ∞. Therefore, by Lebesgue’s
dominated convergence theorem, for all n ≥ 1,
Φ (n)
T
∫ ∞
(v − i) = (ix) n e i(v−i)x µ T (dx),
where µ T is the probability distribution of X T . For even n > 1 this implies:
−∞
∫ ∞
|Φ (n)
T
(v − i)| ≤ x n e x µ T (dx) = C n ,
−∞
and for odd n ≥ 1, using Jensen’s inequality for the probability measure ˜µ T = e x µ T yields:
∫ ∞
(∫ ∞
) n
|Φ (n)
T
(v − i)| ≤ |x| n e x µ T (dx) ≤ |x| n+1 e x n+1
µ T (dx) = Cn ,
−∞
−∞