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Fig. 8.1
Basket (
top
)and
better-of-option price
(
bottom
)
(h
2
)
with respect to the mesh width
h
for both options. But if we plot the conver-
gence rate in terms of degrees of freedom
N
, we see the “curse of dimension” and
only obtain
O
(N
−
1
)
instead of the optimal convergence rate
(N
−
2
)
.
O
O
8.5 Further Reading
Analytic formulas for basket options on two underlying were derived by Zhang [165,
Chap. 27]. To avoid the 'curse of dimension', several authors (see Leentvaar and
Oosterlee [114] or Reisinger and Wittum [139] and the references therein) use finite
differences on the so-called sparse grids. For the finite element discretization, it is
possible to use sparse tensor product spaces for the discretization as in von Peters-
dorff and Schwab [159]. In both cases, we are able to reduce the complexity in the
number of degrees of freedom from
(h
−
d
)
to
(h
−
1
d
−
1
)
. Therefore, even
O
O
|
log
h
|
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