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I'm reading Pricing American Options using LU decomposition by Ikonen and Toivanen (IT).

They reference The valuation of American put options by Brennan and Schwartz, and cast it as method that uses LU decomposition to solve the linear complementarity problem (LCP) arising in the discretisation of the Black-Scholes "PDI" for American options.

On page 9, IT write:

"It is obvious that the use of the max-function in the backward substitution is possible because of the form of the solution of the option pricing problem."

Does this mean that the Brennan-Schwartz method of modified back substitution exactly the LCP exactly? If so, I don't think this is obvious. Is there a proof of it anywhere?

And on a related note, if Brennan-Schwartz solves the LCP exactly then why doesn't Wilmott use it in his "The Mathematics of Financial Derivatives"? He uses Projected Successive Over-relaxation (PSOR) exclusively, which is shown by IT to be a lot slower than using LU decomposition.

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Ikonen and Toivanen don't say that the LCP is solved exactly, they simply say that the modified back-substitution is a valid algorithm to solve the LCP.

A numerical error may arise around the location of optimal exercise, since it does not fall directly on the finite difference grid. I think that however, the error is of the same order as the discretization in space.

Incidentally, PSOR and the modified back-substitution are both presented in Elliot and Ockendon Weak and Variational Methods for Free Moving Boundary Problems, which is referenced by Ikonen and Toivanen.

The interest of PSOR is to be more general: there is some monotonicity condition that must be verified in order to apply the Brennan-Schwartz algorithm. So it is perfectly fine for a vanilla American put or call, but not for an American butterfly.

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