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

1.4. PRICING EUROPEAN OPTIONS 55
Since the bound ˜g n (v) is increasing on (−∞, 0) and decreasing on (0, ∞),
∆ 3 N−2
∑
sup
∂ 2
∫ L/2
24π ∣∂v 2 e−ivk ζ T (v)
∣ ≤ ∆3
12π ˜g 2(0) + ∆2
24π
m=0
−L/2
⎧
2∑
≤ ∆2 C
⎨
3−l
l∑
12π (3 − l)! ⎩ 2∆ |k − rT | j ∑l−1
|k − rT | j
+
j!
j!
≤ ∆2
6π
l=0
2∑
l=0
j=0
j=0
{
C (
3−l
∆ + π )
e |k−rT | + log
(3 − l)! 2
˜g 2 (v)dv
∫ L/2
−L/2
(
L
2 + √
L 2
4 + 1 )
dv |k − rT |l
+
1 + v2 l!
}
|k − rT | l
l!
.
∫ L/2
−L/2
⎫
dv
⎬
√
1 + v 2 ⎭
To complete the proof of (1.32) it remains to substitute this bound and a similar bound for
∆
∑
∣ ∣ 3 N−2 ∣∣
24π m=0 sup ∂ 2
e −ivk ζ Σ ∣∣
∂v 2 T (v) into (1.34).
The Simpson rule (cf. [32]) is defined by:
∫ x+2h
x
f(ξ)dξ = 1 h(f(x) + 4f(x + h) + f(x + 2h)) + R,
3
where R = 1
90 h4 f (4) (x ∗ )
for some x ∗ ∈ [x, x + 2h]. The proof of (1.33) can be carried out in the same way as for the
trapezoidal rule.
Example 1.1. Let us compute the truncation and discretization errors in the Merton model
(1.16) with parameters σ = 0.1, λ = 2, δ = 0.1, µ = 0 for options with maturity T = 0.25.
Taking Σ = 0.1, we obtain the following values of the coefficients C k and C Σ k :
k C k Ck
Σ
1 0.0871 0.05
2 0.0076 0.0025
3 0.0024 2.85 · 10 −4
4 3.29 · 10 −4 1.88 · 10 −5
5 1.82 · 10 −4 2.99 · 10 −6
With 4048 points and the log strike step d equal to 0.01, the truncation error is extremely small:
ε T = 2 · 10 −59 , and the discretization error for at the money options is given by ε D = 0.0013
for the trapezoidal rule and by ε D = 3.8 · 10 −5 for the Simpson rule. Since the call option price
in this setting (S = K = 1, r = q = 0) is C = 0.0313, we conclude that the Simpson rule gives
an acceptable pricing error for this choice of parameters.