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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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