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

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

My message didn't show up in Nabble forum.  I am resending it just in
case it didn't go through.


On Wed, Sep 8, 2010 at 8:42 AM, Kakhkhor Abdijalilov
<kab...@gm...> wrote:
> 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.