Re: QuantLib enhancement proposals

Posted by Ferdinando M. Ametrano-2 on Oct 29, 2001; 4:07am
URL: http://quantlib.414.s1.nabble.com/QuantLib-enhancement-proposals-tp1779p1782.html

At 10:38 AM 10/26/01 +0100, Luigi Ballabio wrote:
>what do you mean by free boundary conditions? If you mean "no particular
>boundary condition given", it would be exactly that---one wouldn't give
>any particular boundary condition, either by passing a null Handle<BC> or
>implementing a NullBC class whose methods do nothing. Of course this
>wouldn't exactly mean "no boundary condition", since as long as you have a
>first row and a last row in your operator you have two corresponding
>boundary conditions... but I'm probably missing something here. Could you
>give more details, please?

Luigi already got my answer since we're in the same office, anyway for the
sake of this list discussion, here is a sketch. Please feel free to add,
correct and/or give more detailed discussion.

An example of free boundary condition is the American early exercise.
With trees you're used to roll back and then simply apply the non-arbitrage
(call) condition:
V=max(V, S-K),
that is you enforce the option value V to be greater or equal to the option
intrinsic value.

To use the same approach with finite difference is OK with explicit
methods, not with implicit methods, because each option value on the
fixed-time grid depends on all the others on the same fixed-time grid.
That is you cannot solve the equation system and then apply the
non-arbitrage condition (if you do that you lower the accuracy order of
your solution), instead you have to solve the system enforcing the
non-arbitrage condition as early as you can, at the same time as the values
are found.

This is usually accomplished solving the system with the "Successive
Over-relaxation" (SOR) technique instead of LU decomposition.

Has anyone other example of free boundary problems? I can only think of
passport option.

ciao -- Nando