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where the matrices
Y
1
,
Y
2
are given by
1
2
βρ
B
x
2
1
8
(
1
4
βκρ)
M
x
2
r
M
1
,
Y
1
:= −
+
−
−
1
2
β
2
S
1
1
2
(α
−
1
2
(
4
αm
−
β
2
)
B
x
−
1
2
β
2
κ)
B
x
2
Y
2
:=
+
−
2
1
2
κ(α
−
β
2
κ)
M
x
2
2
ακm)
M
1
.
+
+
(r
−
Applying the
θ
-scheme to discretize in time, we finally obtain the fully discrete
scheme to approximate the solution
w
of the (truncated) weak formulation (
9.29
)
Find
w
m
+
1
N
∈ R
such that for
m
=
0
,...,M
−
1
M
θt
A
SV
w
m
+
1
=
M
θ)t
A
SV
w
m
,
+
−
(
1
−
(9.33)
w
0
N
=
w
0
.
Note that (
9.33
) gives approximations
w
N
(t
m
,x,y
1
,...,y
p
)
=
w
N
(t
m
,z)
to the
function
w(t
m
,z)
of (
9.18
). To obtain approximations
v
N
(t
m
,z)
to
v(t
m
,z)
of (
9.11
),
we have, by (
9.17
), to set
v
N
(t
m
,z)
w
N
(t
m
,z)e
η(z)
.
=
Example 9.5.6
We consider a European call with strike
K
=
100 and maturity
T
=
0
.
5 within the Heston model, for which we set the parameters
α
=
2
.
5,
β
=
0
.
5,
ρ
0. As shown in Fig.
9.2
, the option price at maturity con-
verges in the
L
∞
-norm at the rate
=−
0
.
5,
m
=
0
.
025,
r
=
O
(N
−
1
)
.
9.6 American Options
American options in stochastic volatility models can be obtained similar to the
Black-Scholes case by replacing the Black-Scholes operator
BS
by the corre-
sponding stochastic volatility operator. The value of an American option is given
as
A
∈
T
t,T
E
e
−
r(T
−
t)
g(e
X
T
)
z
,
V(t,z)
:=
sup
|
Z
t
=
τ
T
t,T
denotes the set of all stopping times for
Z
. A similar result to Theo-
rem 5.1.1 is not available for general stochastic volatility models due to the possible
degeneracy of the coefficients in the SDE. We therefore make the following assump-
tion.
where
Assumption 9.6.1
Let
v(t,z)
be a sufficiently smooth solution of the following
system of inequalities
in J
× R ×
G
Y
,
∂
t
v
−
A
v
+
rv
≥
0
g(e
x
)
G
Y
,
v(t,x)
≥
in J
× R ×
(9.34)
G
Y
,
(∂
t
v
−
A
v
+
rv)(g
−
v)
=
0
in J
× R ×
g(e
x
)
G
Y
,
=
R ×
v(
0
,z)
in
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