Information Technology Reference
In-Depth Information
Assuming homogeneous Dirichlet boundary conditions on
∂G
R
, discretizing in time
with
θ
-scheme and denoting again by
N
:=
k
=
1
N
k
=|
G
|
the number of points in
the grid
, we obtain the fully discrete finite difference scheme for the truncated
pricing equation (
9.30
)
G
Find
w
m
+
1
N
∈ R
such that for
m
=
0
,...,M
−
1
θt
G
SV
)
w
m
+
1
θ)t
G
SV
)
w
m
,
(
I
+
=
(
I
−
(
1
−
w
0
=
w
0
.
Example 9.5.3
Consider the multi-scale SV model with
n
v
=
1 factor as described
0. By (
9.16
), the coefficients are
q
11
(x
1
)
1
,q
11
(x
2
)
in Example
9.1.1
with
ρ
=
=
=
ξ
2
(x
2
)
,
q
22
(x
1
)
1,
q
22
(x
2
)
β
2
,
μ
1
(x
1
)
1
,μ
1
(x
2
)
1
2
ξ
2
(x
2
)
,
μ
2
(x
1
)
=
=
=
=
r
−
=
1,
μ
2
(x
2
)
=
α(m
−
x
2
)
and
c
1
(x
1
)
=
1,
c
2
(x
2
)
=−
r
. The remaining coefficients are
zero. Hence, the finite difference matrix becomes
1
2
R
1
1
2
I
1
I
ξ
2
(x
2
)
R
β
2
G
MS
=
⊗
+
⊗
2
ξ
2
(x
2
)
C
1
I
r
−
I
1
C
α(m
−
x
2
)
I
1
I
−
r
.
−
⊗
−
⊗
−
⊗
1
,
2, one has
X
γ
1
w
1
+
γ
2
w
2
Since for any constants
γ
i
and any weights
w
i
,
i
=
=
γ
1
X
w
1
γ
2
X
w
2
, and since the Kronecker product is distributive, we simplify the
above expression to
+
1
2
R
1
I
ξ
2
(x
2
)
G
MS
C
1
I
1
=
⊗
−
⊗
Y
2
+
⊗
Y
1
,
where
1
2
β
2
R
1
1
2
I
ξ
2
(x
2
)
.
We now approximate the value of a call option in the model of Stein-Stein, i.e.
ξ(x
2
)
αm
C
1
α
C
x
2
I
1
,
r
I
1
Y
1
:=
−
+
+
Y
2
:=
−
=|
x
2
|
, whose exact price can be found in [145]. To t
h
is end, we set
K
=
100,
1
/
√
2,
ρ
T
0
for the model parameters. The computed option price is plotted in Fig.
9.1
.Wealso
give the rate of convergence
=
1
/
2 for the contract parameters and
α
=
1,
β
=
=
0,
m
=
0
.
2,
r
=
(N
−
1
)
of the
L
∞
-error with respect to the number of
O
grid points
N
=|
G
|
. The error is measured on the domain
(K/
2
,
3
/
2
K)
×
(
0
,
1
)
.
9.5.2 Finite Element Discretization
As for the finite difference discretization, we have to introduce weighted matrices.
For
w
, define the matrices
S
w(x
k
)
,
B
w(x
k
)
,
M
w(x
k
)
N
k
×
N
k
: R → R
∈ R
with entries
given by
Search WWH ::
Custom Search