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=
Example 15.4.9
We approximate the price of a European call with strike
K
1 and
maturity
T
1
/
2 within the Bates model (see (
15.13
) for its infinitesimal genera-
tor). For the one-factor model (
n
v
=
=
1), we choose the model parameters
(α,β,λ
0
,λ
1
,ρ,m,μ,δ,r)
=
(
2
.
5
,
0
.
5
,
0
.
5
,
0
,
−
0
.
5
,
0
.
025
,
0
,
0
.
2
,
0
),
and for the two-factor model (
n
v
=
2) we let
(α
1
,α
2
,β
1
,β
2
,λ
0
,λ
1
,λ
2
,ρ
1
,ρ
2
,m
1
,m
2
,μ,δ,r)
=
0
.
08
,
0
.
1
,
0
).
To obtain rates of convergence on the sparse tensor product space
V
L
,wetake
L
(
1
.
6
,
0
.
9
,
0
.
5
,
0
.
2
,
0
.
1
,
5
,
0
.
3
,
−
0
.
1
,
−
0
.
3
,
0
.
039
,
0
.
011
,
−
=
4
,...,
9. Furthermore, in the
hp
-dG time stepping, we take the partition
M
M,γ
of
(
0
,T)
with
M
0
.
4 in the polynomial degree
vector (see Definition 12.3.1). We measure both the error
e
=
L
,
γ
=
0
.
3 and let the slope
μ
=
U
dG
(T )
in
:=
u(T )
−
the
L
2
- and
L
∞
-norm on the domain
G
0
=
(
−
0
.
25
,
0
.
75
)
×
(
0
.
01
,
1
.
21
)
for the
one-factor model and on
G
0
=
(
0
.
25
,
0
.
64
)
for the
two-factor model. To illustrate once more the superiority of
V
L
, we also approximate
the option value for the two-factor model using the full tensor product space
(
−
0
.
25
,
0
.
75
)
×
(
0
.
25
,
0
.
64
)
×
V
L
with
L
=
3
,...,
6. We find for the one-factor model
e
L
2
(G
0
)
=
O
N
−
2
(
log
2
N
L
)
2
.
75
,
e
L
∞
(G
0
)
=
O
N
−
2
(
log
2
N
L
)
3
.
9
,
L
L
whereas for the two-factor model we have
L
∞
(G
0
)
=
O
N
−
L
(
log
2
N
L
)
5
.
7
.
The experimental
L
2
-rates are in very good agreement with the approximation prop-
erty of the projector
P
L
, compare with Theorem 13.1.2 and Remark
15.4.8
, while
therateinthe
L
∞
-norm is slightly smaller. Note that the curse of dimension is
clearly visible in the rate
L
2
(G
0
)
=
O
N
−
2
(
log
2
N
L
)
5
.
1
,
e
e
L
e
L
2
(G
0
)
=
e
L
∞
(G
0
)
=
O
(N
−
2
/
3
)
of the full tensor
L
product space, compare with Fig.
15.2
.
Example 15.4.10
We consider the BNS model and its corresponding (transformed)
pricing equation (
15.18
). We compute the price and its sensitivity with respect to
ρ
of a European call with strike
K
0
.
5. We choose for
the background driving subordinator
L
λt
an IG(
a,b
)-OU process, for which (with
w
=
=
1 and maturity
T
=
1
2
b
2
z
,
a
κ
(ρ)
=
aρ(b
2
2
ρ)
−
1
/
2
,
k(z)
=
2
√
2
π
z
−
3
/
2
(
1
+
b
2
z)e
−
1) there holds
−
see, e.g. [128] and (
15.9
) for the definition of
κ
(ρ)
. We take the parameters
(ρ,λ,a,b)
=
(
0
,
2
.
5
,
0
.
09
,
12
)
. Rates of convergence (for the error measured in the
L
2
-norm) are calculated on the domain
G
0
=
−
×
(
2
,
0
.
5
)
(
0
.
22
,
1
.
1
)
using sparse
tensor product spaces
V
L
,
L
10
−
4
in the back-
ward Euler time stepping. A closed form solution using the characteristic function
of the log-price process can be found in [9]. The resulting rates of convergence are
showninFig.
15.3
. We observe that for both subordinators the price and sensitivity
converge with same rate, which is
=
5
,...,
9, and time step
t
=
5
·
e
L
2
(G
0
)
=
O
(N
−
2
(
log
2
N
L
)
7
.
5
)
.
L
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