Hi everyone, I was wondering if the finite element method is used by quants? Almost no course about numerical finance mention it !

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The reason FEM isn't used that much is that it is primarily used for (a) complicated domains (domains in fin math are relatively straightforward (square,rectangle, triangle etc)(b) provides nicer, more powerful math to prove certain mathematical properties and used for more complex equations/problemsthe downside of FEM is that it it is(1) overkill for quant finance(2) not as easy to apply to higher dimensional cases(3) Theoretically it is very nice but few quants care too much about the beauty of the method, if it cant price quick, be used in generality for range of options and most importantly be calibrated .... then its of no use to themalso, there has been stuff done using FEM for fin math .... Topper has a book, Pironeau done some stuff on it(he developed FreeFEM), others also have done stuff .... there is a lot out there on FEM for quant finance actually but for me its a dead end. FEM will never take off in fin math but if its pure math knowledge and things you are after and dont care if its ever used in industry then go for it as its a very enjoyable field of studyanyway just my $0.02

yep, Pironneau was one of my teachers (and of course made a lot of advertising for the use of FEM), and because I knew he wrote some book about it, I wanted to know what was the situation in real..

yes, I have his book. I like the look of it and have briefly skimmed it but am too busy to look through it in detail as have other things to do. He will for sure boast about FEM as that is his bread and butter but I like his research from a academic/mathematical point of view ..... but do not see it being used regularly in industry (which I do not claim is the holy grail, you can be plenty happy and do nice math that will not be used by banks/hedge funds)

I am with CompPDE on this. The PDE problems in finance can be solved more easily using FDM. For difficult geometries indeed FEM is king.And FEM demands more mathematical backgound than FDM. It takes longer to set up the discrete system of equations.Paris VI has a major reputation in FEM. FDM is not so popular. //One area where FEM has a possible advantage is modelling n-factor American options using variational techniques.

QuoteOriginally posted by: Alekkyep, Pironneau was one of my teachers (and of course made a lot of advertising for the use of FEM), and because I knew he wrote some book about it, I wanted to know what was the situation in real..His book is very good in my opinion with lots of practical examples (in C++). The theory is explained well.

QuoteThe PDE problems in finance can be solved more easily using FDM. For difficult geometries indeed FEM is king.Anyone tried FVM yet? Rarest, but might be interesting.

QuoteOriginally posted by: quartzQuoteThe PDE problems in finance can be solved more easily using FDM. For difficult geometries indeed FEM is king.Anyone tried FVM yet? Rarest, but might be interesting.See Mr Blade here CFD threadPeter Forsyth and colleagues in Waterloo.

Yes, I use it because I'm a quant with too much spare time on my hands so I try to solve the black scholes equation in different ways just to compare the results I also think it's a bit of an over kill when FD CNS is very fast and accurate and much more straight forward to impliment.I do have a problem with the collocation method using finite elements. My source of reference is pg93 to 99, pg 114-115 of Topper's book on cubic-hermite elements. It involves collocating at the gauss points as such the problem is not self adjoint. I am having stability issues when I step the solution over time using the usual backward Euler method. I solve the matrix for each time step using LU decomposition.Has anyone built a collocation model that has local support (as opposed to the psudo spectral method)? If so any suggestions on what basis functions to use?and how to overcome the stability issue for non self adjoint problems? (I don't have problems for self adjoint problems using the Galekin approach for linear and cubic basis functions).

QuoteIt involves collocating at the gauss points as such the problem is not self adjoint. I am having stability issues when I step the solution over time using the usual backward Euler method. I solve the matrix for each time step using LU decomposition.I don't have the book on my desk at the moment but will have a look soon.Just one question: When you solve MU(n+1) = F(n) at each time level 1) what does the matrix M look like; diagonally dominant, M_matrix?? (diagonal positive, off-diagonals negative)2) what kinds of stability problems do you get? like oscillations or negative solution, whatever?3) high drift/low vol?I don't think it is Euler in time that is the problem, it is more stable than CN , albeit lower accuracy (In theory). Is it useful to do the 'perpetual' case. i.e. just an elliptic problem just to see if that is not stable as well?

Two things to watch out for:1) Topper uses the forward PDE (thus, initial condition u(x,0) instead of terminal condition). This can be very confusing (e.g. Hull uses FDM with backward PDE)2) LU decomposition can fail if the matrix has zero pivots. It's nice when the matrix is strictly diagonally dominant, but this may not hold in the FEM case. Failing that, it might be necessary to use an iterative method (like Jacobi, also discussed in Topper)

I'm finding that the jacobi, Gauss-Sediel and SOR methods don't perform well when the matrix is not srtrictly diagonal. These were the solvers I first tried when implimenting the Galekin method using cubic polynomials. That's when I moved to LU.Maybe there are "fixes" to improve the performance - I dunno.

QuoteOriginally posted by: wildmanI'm finding that the jacobi, Gauss-Sediel and SOR methods don't perform well when the matrix is not srtrictly diagonal. These were the solvers I first tried when implimenting the Galekin method using cubic polynomials. That's when I moved to LU.Maybe there are "fixes" to improve the performance - I dunno.It's not much of a consolation to you at the moment, but these problems do not occur when we use FDM for non self-adjoint parabolic equations. In fact, using expoentially fitting schemes we can get an M-matrix and this is ideal because then the scheme satisfies the discrete maximum principle. And it takes a few minutes to write down the matrix elements (no inner products needed).Having said that, LU decomposition assumes that diagonal elements are non-zero; failing that I think Plan B is to use pivoting using row interchanges (even when the diagonal elements is nearly zero). Even though the matrix can be invertible, the diagonal elements can be zero even though pivotal elements can be(come) zero.Hope this helps a bit.D // You are right about Jacobi. must be diag dom otherwise no fixed point and hence no solution.