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Fig. 13.2
Convergence rate
of the wavelet discretization
in terms of the mesh width
h
(
top
)andintermsofdegrees
of freedom (
bottom
)
13.5.2 Low-Rank d-Dimensional Black-Scholes
We consider the geometric call option as in Sect.
13.5.1
written on
d
underlyings
under the Black-Scholes model and we are now interested in the case of larger di-
mensions, i.e.
d>
8. This occurs when pricing contingent claims on stock indices
considering all
d
price processes in comparison to handling the index as one sin-
gle process. Straightforward computations in such high dimensions would currently
require too high discretization levels as previously noted. Instead, we rely on the di-
mensionality reduction by
ε
-aggregation to identify a rank
dε
-aggregated process
driving a
d
-dimensional market. In particular, we focus on the Dow Jones indus-
trial index where
d
=
30. We compute the principal components of the volatility
d
×
d
of their realized daily log-returns over 252
periods resulting in the spectrum
(s
1
,...,s
d
)
, normalized by
s
1
U
DU
covariance matrix
Q
:=
∈ R
as in Sect.
13.4.1
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