[Quantlib-dev] Longstaff-Schwartz method, SVD and OLS

[Kakhkhor A. <kab...@gm...>]

It seems that LongstaffSchwartzPathPricer implementation may not deal
with basis collinearity properly.

SVD can deal with collinearity, but the cutoff threshold for small
singular values in LinearLeastSquaresRegression is n*QL_EPSILON (n is
matrix size).

Btw, the cutoff should be applied to the ratio of singular values, not
to the singular values directly. That is something we need to fix as
well.

The above cutoff  threshold  is OK for OLS purposes, but not for MC
(at least how it is implemented in QL). For OLS purposes we compute
coefficients by performing SVD of the design matrix and then use those
coefficients with the same design matrix to compute the projection of
the dependent variable. This way all singular values cancel out as
long as the cutoff was applied.

But LongstaffSchwartzPathPricer does compute projection (option
continuation values) from a new sample. It is like using the old
coefficients with a new design matrix. There is chance that singular
values of this new design matrix won't cancels out inverse singular
values of the old design matrix. Both design matrices have the same
statistical properties, so that singular values should be more or less
comparable, except very small ones. With typical sample sizes, SVD
cutoff threshold can be as small as 1.0E-012, which is much smaller
than statistical uncertainty of MC (inverse square root of sample
size). Small singular values will show up whenever there is a
collinearity in the basis system.

Fortunately, the problem is fixed if SVD cutoff (n*QL_EPSILON) is
replaced  with something commensurable with MC tolerance. I want to
code and submit new LongstaffSchwartzPathPricer and
LinearLeastSquaresRegression classes, but would like to know others
think.

With best regards,
Kakhkhor Abdijalilov.