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Fig. 13.5
Convergence rates
of the approximation of a 30
dimensional option price on
the Dow Jones by
d
=
2
,...,
5 dimensional
options in the Black-Scholes
model
We numerically solve (
13.34
)for
d
2
−
L
=
2
,...,
5 and various mesh widths
h
=
with the Dow Jones realized volatility covariance matrix
Q
whose principal compo-
nents are plotted in Fig.
13.4
(top), weights
α
i
=
0
.
3,
i
=
1
,...,d
, maturity
T
=
1,
strike
K
=
1 and interest rate
r
=
0
.
045, and plot the convergence rate of the relative
L
2
-error
e
L
:=
v(T,
L
2
(G
1
)
v(T,
·
)
L
2
(G
1
)
·
)
−
v
L
(T ,
·
)
,G
1
=
(
3
/
4
K,
5
/
4
K)
×{
K
}×···×{
K
}
,
at maturity
T
1inFig.
13.5
. The flattening behavior of the convergence rates is
explained by further expanding the error into
(a)
an
ε
-aggregation error made by
artificially setting volatilities to zero (Theorem
13.4.3
) and
(b)
a discretization error
[159]as
=
v(t,y)
−
v
L
(t,
y)
ˆ
=
v(t,y)
−
v(t,
y)
ˆ
+
v(t,
y)
ˆ
−
v
L
(t,
y)
ˆ
≤
v(t,y)
−
v(t,
y)
ˆ
+
v(t,
y)
ˆ
−
v
L
(t,
y)
ˆ
.
(a)
(b)
It follows that the lower bounds observed in Fig.
13.5
therefore stem from the
ε
-
aggregation error which diminishes as
d
is increased.
13.6 Further Reading
A rigorous error analysis for the numerical solution of high dimensional parabolic
equations is given in von Petersdorff and Schwab [159], where a wavelet based
sparse grid discretization is used in conjunction with a discontinuous Galerkin time-
discretization. Reisinger and Wittum [139] use a finite difference based discretiza-
tion on sparse grids employing the combination technique. Leentvaar and Oost-
erlee [113] use high order finite difference discretizations on sparse grids for the
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