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The variational formulation of the truncated problem reads then
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
BS
(u
R
,v
a
BS
(g, v
(∂
t
u
R
,v
−
u
R
)
+
−
u
R
)
≥−
−
u
R
),
∀
v
∈
K
0
,R
,
(5.7)
u
R
(
0
)
=
0
.
We calculate the right hand side in (
5.7
) for a put option, i.e.
g(s)
=
max
{
0
,K
−
s
}
.
H
0
(G)
. Then, by the definition of
a
BS
(
Let
ϕ
∈
·
,
·
)
in (4.9) and integration by parts,
2
σ
2
r
2
σ
2
ln
K
ln
K
1
1
a
BS
(g, ϕ)
e
x
)
ϕ
d
x
e
x
)
ϕ
d
x
−
=−
(K
−
+
−
(K
−
−
R
−
R
r
ln
K
e
x
)ϕ
d
x
−
(K
−
−
R
2
σ
2
e
x
ϕ
2
σ
2
r
2
σ
2
ln
K
ln
K
ln
K
1
1
1
e
x
ϕ
d
x
e
x
ϕ
d
x
=
R
−
−
−
−
R
−
R
−
r
ln
K
e
x
)ϕ
d
x
−
(K
−
−
R
rK
ln
K
1
2
Kσ
2
ϕ(
ln
K)
=
−
ϕ
d
x.
−
R
1
−
a
BS
(g, ϕ)
defines the linear functional
f
=
2
Kσ
2
δ
ln
K
−
Since this holds for all
ϕ
,
H
−
1
(G)
, where
χ
B
rKχ
{
x
≤
ln
K
}
∈
is the indicator function.
5.3 Discretization
As in the European case, we approximate the option price both by the finite dif-
ference and the finite element method. The finite difference method discretizes the
partial differential inequalities whereas the finite element method approximates the
solution of variational inequalities. In both cases, the discretization leads to a se-
quence of linear complementarity problems. These LCPs are then solved iteratively
by the PSOR algorithm.
5.3.1 Finite Difference Discretization
Discretizing (
5.6
) with finite differences and the backward Euler scheme, i.e. the
θ
-scheme with
θ
=
1, we obtain a sequence of
matrix linear complementarity prob-
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