Information Technology Reference
In-Depth Information
14.7.2 Basket Options
Consider a basket option
u(t, x)
with payoff
g(x)
where the log price processes
of the underlyings are given by the pure jump process
X
(X
1
,...,X
d
)
and
correspondingly
u
1
(t, x)
,
u
2
(t, x)
for the processes
Y
1
,
Y
2
. We want to study the
error
=
u
i
(T , x)
1
,
2. Since we adjusted the drift to preserve
the martingale property, we additionally introduce the processes
Z
i,t
=
|
u(T , x)
−
|
for
ε
→
0,
i
=
(γ
i
X
t
+
−
γ)t,
i
=
1
,
2
,
whichhavethesamedriftas
Y
i
and the same Lévy measure as
X
.
Proposition 14.7.3
Assume g is Lipschitz continuous
.
Then
,
there are C
1
,C
2
>
0
such that
ε
d
E
Z
1
,T
))
≤
z
j
2
ν
j
(
d
z
j
),
d
,
(g(x
+
X
T
))
− E
(g(x
+
C
1
∀
x
∈ R
−
ε
j
=
1
(14.35)
ε
d
E
Z
2
,T
))
≤
z
j
3
ν
j
(
d
z
j
),
d
.
(g(x
+
X
T
))
− E
(g(x
+
C
2
∀
x
∈ R
−
ε
j
=
1
(14.36)
Proof
We have for
i
=
1
,
2,
Z
i,T
))
≤ E
g(x
γ)T)
E
(γ
i
(g(x
+
X
T
))
− E
(g(x
+
+
X
T
)
−
g(x
+
X
T
+
−
γ
j
d
γ
i,j
−
≤
T
.
j
=
1
Furthermore,
z
j
ν
ε
(
d
z)
≤
ε
|
γ
j
=
|
e
z
j
z
j
e
s
z
j
−
s
d
sν
j
(
d
z)
γ
1
,j
−
−
1
−
d
R
−
ε
0
ε
e
ε
2
z
j
2
ν
j
(
d
z),
≤
j
=
1
,...,d,
−
ε
z
j
ν
ε
(
d
z)
γ
j
=
Q
ε,jj
2
d
e
z
j
γ
2
,j
−
−
−
1
−
R
ε
|
|
z
j
1
2
e
s
(z
j
−
s)
2
d
sν
j
(
d
z)
≤
−
ε
0
ε
z
j
e
ε
6
3
ν
j
(
d
z),
≤
j
=
1
,...,d.
−
ε
The same error estimates are also obtained for the compound Poisson and Gaus-
sian approximation.
Search WWH ::
Custom Search