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Re: LMM and finite difference methods

Posted by Joseph Wang on Nov 15, 2005; 10:46am
URL: http://quantlib.414.s1.nabble.com/LMM-and-finite-difference-methods-tp4222p4224.html

Ferdinando Ametrano wrote:

>
>LMM are intrinsically high-dimension. If you model the 30-year yield
>curve as a strip of 6-month forward libor you have 60 factors. FD with
>more than 3 factors are less efficient than MC, and very hard (almost
>impossible?) to implement.
>
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The idea I had was to use Komogorov's equation to write a PDE for the
probability distribution functions for each of the factors.  You then
have a large number of interacting parallel PDE's to difference.  Messy,
but not impossible.  FDM's tend break down when you have systems that
are heavily path dependent, but I don't see the path dependence in the LMM.

Also you might be able to simplify even further.  Instead of trying to
calculate the whole PDF for each of the 60 PDE's you write down the
evolution equations of the moments of each PDE.  What should make this
tractable is that as time passes the higher order terms will get damped
down (unlike the CFD problem).

This gives you a large number of ODE's.  At that point things start
looking like a nuclear reaction matrix.

There are a large number of problems in the world that seem easy, but
when you brute force try to solve them the obvious way, you start
hitting a brick wall (turbulence and quantum gravity fall into this
category).  I was just wondering if someone can describe the brick wall
to me before I crash into it.  It may be some path dependence that I
don't see.  Or maybe the factors are coupled in such a way that one
can't disintangle them into more or less separate difference equations.

>The lack of literature on this topic suggests me that a) it must be
>undisclosed proprietary leading edge stuff, or that b) implementation
>details (i.e. boundary conditions) must be so complex to make the
>implementation unfeasible
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LMM also seems pretty recent as models go.