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Fig. 10.2
Convergence rate
for the variance gamma
model
10.7 American Options Under Exponential Lévy Models
American options can be obtained similarly to the Black-Scholes case just by re-
placing the Black-Scholes operator
BS
J
.Thevalueofan
A
by the jump operator
A
American option in log-price
τ
∈
T
t,T
E
e
−
r(τ
−
t)
g(e
rτ
+
X
τ
−
)
x
,
v(t,x)
:=
sup
|
X
t
=
is, provided some smoothness assumption on
v
, the solution of a parabolic integro-
differential inequality
J
u
∂
t
u
−
A
≥
0
in
J
× R
,
≥
× R
u(t, x)
g(t,x)
in
J
,
J
u)(
(∂
t
u
−
A
g
−
u)
=
0
in
J
× R
,
g(e
x
)
u(
0
,x)
=
in
R
,
where we set
u(t, x)
=:
e
rt
v(T
−
t,x
−
(γ
g(t,x)
=
e
rt
g(e
x
−
(γ
+
r)t
)
.
As shown in Theorem
10.5.1
, we can localize the problem to a bounded domain
G
+
r)t)
and
=
(
−
R,R)
. For the variational formulation, we again consider
w
R
:=
u
R
−
g
|
G
,
the time value of the option, and obtain
L
2
(J
H
0
(G))
H
1
(J
L
2
(G))
such that
u
R
(t,
Find
u
R
∈
;
∩
;
·
)
∈
K
0
,R
and
a
J
(u
R
,v
a
J
(
(∂
t
u
R
,v
−
u
R
)
+
−
u
R
)
−
(∂
t
g,v
−
u
R
)
−
g,v
−
u
R
),
∀
v
∈
K
0
,R
,
u
R
(
0
)
=
0
.
Discretization using finite differences or finite elements as explained in Sect.
10.6
leads to the following sequence of linear complementary problems with
u
0
=
u
0
:
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