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13.5 Numerical Examples
We give numerical examples using the wavelet finite element discretization as de-
scribed in Sect.
13.3
.
13.5.1 Full-Rank d-Dimensional Black-Scholes Model
We consider the geometric call option with payoff
max
(
0
,e
i
=
1
α
i
x
i
d
,
=
−
∈ R
g(x)
K),
x
for strike
K
=
1. The antiderivatives of
g
for
α
i
>
0,
i
=
1
,...,d
are given by
e
i
=
1
α
i
x
i
d
log
K
k
1
d
2
d
1
k
g
(
−
2
)
(x)
α
−
2
i
=
−
α
i
x
i
−
{
i
=
1
α
i
x
i
≥
.
log
K
}
!
i
=
1
k
=
0
i
=
1
2
−
L
.Us-
We first set
d
=
2 and solve problem (2.24) for various mesh widths
h
=
ing interest rate
r
=
0
.
01, covariance
Q
=
(σ
i
σ
j
ρ
ij
)
1
≤
i,j
≤
d
,
σ
1
=
0
.
4,
σ
2
=
0
.
1,
ρ
12
=
0
.
2 and weights
α
i
=
1
/d
,
i
=
1
,...,d
, we plot the convergence rate of the
L
2
-error
(K/
2
,
3
/
2
K)
2
,
e
L
:=
u(T ,
·
)
−
u
L
(T ,
·
)
L
2
(G
0
)
,G
0
=
at maturity
T
1inFig.
13.2
. To compare the rates we also solved the problem on
full grid. In the top picture, it can be seen that sparse grid has (up to log terms) the
same rate as full grid. To better show the advantage of a sparse grid, we additionally
plot the convergence rate in terms of degrees of freedom. For the full grid, we have
N
L
=
O
=
(
2
2
L
)
and for the sparse grid
N
L
=
O
(L
2
L
)
. The convergence rate in the
full grid shows the “curse of dimension”, whereas for the sparse grid we still obtain
the optimal rate (up to log terms).
For
d>
2weset
σ
i
=
0
.
3,
i
=
1
,.
..,d
and
ρ
i,j
=
0,
i
=
1
,...,d
−
2,
j
=
i
+
ρ
1
ρ
2
,
i
2
,...,d
,
ρ
12
=
ρ
,
ρ
i,i
+
1
=
−
=
1
,...,d
−
1, with
ρ
=
0
.
1 and weights
α
1
=
2
,...,d
. We plot the convergence rate of the absolute error at
the point
S
0
=
(K,...,K)
in Fig.
13.3
.
For low dimensions
d
∈{
1,
α
i
=
0,
i
=
, the second order convergence rate on the sparse
grid can be well observed over all levels. For higher dimensions
d
∈{
2
,
3
,
4
}
,thelog
terms prevail at low levels, hence the flattened behavior of the convergence curves
which then exhibit the expected second order rates at finer discretizations. Despite
the smoothness of the solution, we report a steeply increasing constant in the rates
as
d
is raised, which forces us to already set
L
6
,
8
}
8 in order to have a
relative
L
2
-error on the order of 10
−
2
which currently prevents us from reasonably
increasing
d
beyond 8. The size of the constant can be traced back to the initial
condition
u
h
,the
H
-projection of the payoff
g
onto
V
h
(Sect.
13.3
), showing similar
relative
L
2
-errors.
=
11 for
d
=
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