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Chapter 13
Multidimensional Diffusion Models
In the previous chapter, we introduced a spline wavelet basis
{
ψ
,k
}
for the dis-
cretization of infinitesimal generators
A
of Lévy processes
X
. The wavelet basis
served two purposes:
(i)
Preconditioning of linear systems
: We showed that the wavelet basis implies
in the context of
hp
-dG timestepping that the (generalized) condition number
of the linear systems to be solved in each time step is bounded independently
of jump intensity
α
, volatility
σ
and time step
k
. This robust preconditioning
was crucial for efficient iterative solution of the linear systems for the widely
varying time steps which occurred in connection with the
hp
-dG timestepping
and was crucial for the overall log-linear complexity of the proposed scheme,
and
(ii)
Compression of linear systems
: The compression of the stiffness matrix based
on a priori analysis reduces the number of non-zero matrix entries from
(N
2
)
O
to
O
(N)
and does not deteriorate the optimal convergence rate of the scheme.
In the present chapter, we develop efficient pricing algorithms for
multivariate prob-
lems
, such as the pricing of multi-asset options and the pricing of options in stochas-
tic volatility models, which exploit a third feature of the wavelet basis, namely that
wavelets constitute a
hierarchic basis
of the univariate finite element space
V
L
.This
allows constructing the so-called
sparse tensor product
subspaces for the numeri-
cal solution of
d
-dimensional pricing problems with complexity essentially equal to
that of one-dimensional problems. The tensor product structure of the infinitesimal
generators of the multidimensional BS operator and of stochastic volatility models
implies that the stiffness matrices corresponding to these operators in tensor product
wavelet bases can be built from tensor products of banded matrices corresponding
to univariate operators. Here, we will show that the same convergence rates can, in
fact, be achieved with
sparse tensor products of univariate matrices
. These matri-
ces are not sparse anymore and should therefore never be formed to maintain low
complexity of the pricing algorithm. To obtain linear complexity pricing algorithms,
sparse tensor product matrices are stored in factored form and are applied to a vector
in log-linear overall cost. Using wavelet preconditioning, the condition number of
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