Can anyone here recommend one or more papers "based on PDE approach" which compare different solvers for pricing "multi-asset" European and American options, especially on computational time, complexity of the numerical system and preconditioning? Hopefully these papers tackle the BS PDE without transformation.Many thanks in advance.Leon

I don't think such a paper exists at this moment. I think it would need to be a multi-author.QuoteHopefully these papers tackle the BS PDE without transformation.Are you referring to specific kinds of transformations like log(S), etc.?

Dear Cuchulainn,Thank you for your reply first. Actually I asked this question because I hardly found such a paper either.And yes, I wish to find a paper comparing different solvers without using log transformation but dealing with BS PDE straight.Any paper focusing on either computational time, complexity of the numerical system or preconditioning will be interesting to me.Any suggestion will be welcome.Thanks in advance.

Hi LeonAtWilmott,I have recently compared several methods on the 3D Black Scholesequation discretized using finite differences. The methods compared were ADI, Crank-Nicolson, in-built matlab ode stiff solver ode15s with an alternative not used in finance but used in numerical PDEs. You can download the paper on ssrnhttp://papers.ssrn.com/sol3/papers.cfm?abstract_id=1799124Any comments or suggestions appreciated

Dear Mrbala,Thnak you very much for your paper. It's very impressive on the computational times.As I did not dig deeper in your paper, please pardon me if my question sounds silly.In your approach, when you solve your linear system at each time step, is the linear system 9-band in 2D and 27-band in 3D?Can the solver benefit from the banded structure somehow?Actually, I am solving the problem using FEM and would like to explot the band structure somehow to speed up the computation.Any idea of how the structure can be explouted from anyone?Once again, thank you for sharing your paper. You did save plenty of computational time in hight dimensional cases.

Mrbala,Nice paper. Some quick questions on the paper:1. Taking eq. (5)+(6) as a starting point, I suppose it is flexible enough in your framework to support other ODE solvers? At one extreme, Euler which is slow of course, but let's say ADE which is unconditionally stable and just requires my using a multiarray directly in memory? In that case nonlinear PDEs could be handled as well.(btw there's a few ADE threads here).2. I take it that the methods mentioned on page 10 are ODE solvers of the semi-discretisation of PDE (10)? 3. Which part of the scheme resolves possible errors with mixed derivatives (or is there a problem)? Some schemes take them explicitly at lower time level n. An issue that receives little attention is how to preserve ellipticity in the fdm scheme, which will ensure the scheme is monotone. 4. Let's say we do not wish x = log(S) and do domain transformation instead of trunction (which avoids some problems to some extent), would this be possible? just an idea.

QuoteActually, I am solving the problem using FEM and would like to explot the band structure somehow to speed up the computation.Any idea of how the structure can be explouted from anyone?LeonWilmott,What is possible is to do semi-discretisation of PDE using hat elements to get an ODEMdU/dt + KU = Fand then use ADE for the time discretisation. In 1d, for example it's a fd scheme at the end of the day and is faster than CN. In 2d,..

Dear LeonAtWilmott It is interesting that you have chosen the FEM given thedomain is so regular in financial applications. I would beinterested in how your results compare to finite differences.I agree with Cuchulainn after FEM you should have an ODEof the form MdU/dt + KU = F. You can modify most ode solversto add a mass matrix. In matlab the ode15s algorithm allowsyou to add a mass matrix as an argument. It should not be to difficult to modify ADI to include the mass matrix M.With regards to your point about taking advantage of the sparsitythat is less effective than taking advantage of the sparsity when doing ADI. In that case it is imperative to use pivoting when solving the linear systems.Dear CuchulainnSome quick answers1) Yes you can potentially use any ODE solver. I have planned to implement the ADE methods to see how they compare, I will let you know once I am done. With regards to your point on nonlinear PDEs that was my original application. In that case the method is part of a class of methods known as exponential integrators, maybe you are familiar with them.2) Yes all these methods are acting on the discetized PDE.3) The exponential methods treat the discretized linear operator, say A, as a large matrix, aiming to compute exp(A)b. This can be broken down into parts say A=A0+A1+A2, however we then need to use the BCH formula as the matrices do not compute. So we do not treat the mixed terms any differently as they do in ADI methods.4) I have tried the transformation you mentioned in the past when using sparse grid finite difference.

Dear Mrbala and Cuchulainn,I will look into the ADE method you mentioned here. And I use the FEM as it will provide me some elegant spatial discretization even in high dimenion.In fact, since the coefficient matrix after time and spatial discretization is so well-structured, I will be more interested in exploiting its banded structure if possible. (Any suggestion of this?)BTW, I know preconditioning is very popular in numerican computation, but why are there rare papers on preconditioning in mathematical finance?

Dear LeonYou might like to try the ODE solver RADAU5.It can solve the problem M dU/dt = KU + F.There are fortran, matlab and c++ implementationsavailable. See the websitehttp://www.unige.ch/~hairer/software.htmlin the Stiff Differential Equations and Differential-Algebraic Problems sections.

Dear Mrbala,Will have a try and thanks.

LeonWilmott, Mrbala,The mass matrix M can be reduced to a diagonal matrix by a 'row sum lumping mass matrix' method which is used in FEM by civil engineers in their notation.For example, with linear hat functions, M is tridiagonal with constant mesh h is M = H {1, 4, 1} --> h {0, 1, 0}. Then the form becomes dU/dt + KU = F.Theorem 3.2 here discusses stability of the ADE for the ODE http://www.math.ust.hk/~masyleung/Repri ... h05.pdfFor cubics, I remember that trapezoidal integration is used to achieve same ends. mrbala,Do you use Strang splitting?