i have read on the meshless method on these days.in a post dating back to year 2006 (http://www.wilmott.com/messageview.cfm? ... SGDBTABLE=) , Cuchulainnasked :======What's the best Numerical method for the Black Scholes? The candidates are:M1: Binomial/trinomialM2: Monte CarloM3: FDM (many flavours)M4: FEM M5: Collocation methodsM6: MeshlessM7: Numerical Integration methods (QUAD)M8: 'approximate' exactM9: Transform methods (Laplace, Fourier, Lie)M10: 'Real' exact======i am wondering :- how would you rank the meshless method, TODAY ?- what are its latest developments ?- is it actually used by praticians ?

- Cuchulainn
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I suppose I better reply Doing a google, there seems to be healthy interest in meshless. 7-8 years ago about 1 article.It could be a good solution for PIDE (a pain in FDM/FEM) and n-factor PDEs. But every method has its own idiosyncracies. It is a Method of Lines in which space is using RBF. The Crank Nicolson is used a lot to discretise the ODE. What about the ADE sweeps?meshless for PIDELooks doable since you get an ODE system when using RBF. How stable are those matrices? No magic bullet.2 cents

Last edited by Cuchulainn on January 1st, 2011, 11:00 pm, edited 1 time in total.

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There is a recent wilmott article about RBF with cubic spline. Option Pricing with Radial Basis Functions: A TutorialFor Black Scholes for me the best method is ADE. Quick (fully explicit), unconditionnaly stable (for your chosen step size it will NEVER crash) and second order accuracy. Unconditionally Stable and Second-Order Accurate Explicit Finite Difference Schemes Using Domain Transformation: Part I One-Factor Equity Problems And as you may know we have an ADE expert here (the previous poster).

trugarez, and merci i am going to study the papers you suggested above

Last edited by tagoma on January 6th, 2011, 11:00 pm, edited 1 time in total.

i wanted to get the Sheppard's working paper "Pricing Equity Derivatives under Stochastic Volatility: A Partial Differential Equation Approach" from www.datasimfinancial.com as pointed out in "Unconditionally Stable and Second-Order Accurate Explicit Finite Difference Schemes Using Domain Transformation: Part I One-Factor Equity Problems". i found the url, but it seems that the whole website doesn't work properly. ! que lástima eso !can someone confirm that www.datasimfinancial.com is currently broken out ?

Last edited by tagoma on January 6th, 2011, 11:00 pm, edited 1 time in total.

Here the paper Edouard! It's a MsC thesis so it's quite heavy.Sheppard MsC thesisDatasim should be broken or under maintenance (as the forum this afternoon ).

Hi Edouard,The meshless method is quite cute as an idea, but it doesn't really work in practice. For one thing, nobody uses it in the industry. But one can say the same about FEM (or about wavelets for that matter). The main weakness of the meshless method is that the radial basis function method of interpolation doesn't really work. Sure, you'll find some body of research that claims it actually works, but one thing is to show it works in some cases, quite another to build a robust method that works for a wide class of useful functions. What's the problem with RBF ? Several: 1. in order to interpolate some function you need to solve a large system of equations2. that system becomes more unstable if you use more basis functions3. that means that in practice nobody uses the RBF with more than about 1000 basis functions. In Daniel's book he mentions one result where the author uses 121 basis functions and gets results that are favorable compared to FD. Who would use FD with only 121 points? If I use FD I'm free to use 10000 points; but RBF doesn't work in that case anymore, so the comparison result wouldn't support the author's conclusion ... 4. there is a claim that RBF's are dimension blind. And herein lies their big promise. You would seem to get the best of FD and Monte Carlo in the same package. Unfortunately things are different on paper and in reality. Any decent Monte Carlo scheme needs at least 10000 paths to be worth anything. But this is way beyond the number of points where RBF's are stable. 5. besides, where do you pick the collocation points? you would need to run a MC, which is perfectly fine, except that there is a slight problem: you would get different points at each point in time. Which is again fine, except everything that was written in the literature so far doesn't cover this case. It's still ok, it's not a biggie to implement it yourself, only that now you don't have the Method of Lines anymore; since you change the collocation points, you need to redo the interpolation at each point in time, which means you have to solve that large linear system 6. one of the biggest problems with RBF is that they are not adaptive. Or rather they are, but not in a computationally efficient way. For example, say you realize you need to get more samples in a certain region. You need to resolve that large system again. There is currently no locally adaptive RBF methodology. If you can come up with one, you hit the jackpot; the applications would go far beyond math finance. 7. back to RBF's being dimension blind. They aren't that blind. In particular if you change the unit in one dimension, you change the radial basis functions (because you change the "circles" of the geometry). That means they are not scale invariant. FD are not scale invariant either, but then again, they don't promise to break the curse of dimensionality. Monte Carlo on the other hand is completely scale agnostic. Now this is a fundamental question that none of the proponents of RBF's have bothered to address: how do you choose the units in each dimension? I'll stop here. This is my 2 cents, if anybody had a better experience with RBF's, please share. Best,V.

By the way, if you are interested in RBF's there are several books; the best one in my opinion is "Meshfree Approximations Methods with Matlab" by G. E. Fasshauer. Best,V.

@Costeanu. thank you for your comments above.i guess that each method has some kind of drawbacks.it's interesting to see that practicioners don't use the RBF method.and what's your answer to the following question ? (assuming the BS is worth )QuoteWhat's the best Numerical method for the Black Scholes?The candidates are: M1: Binomial/trinomialM2: Monte Carlo M3: FDM (many flavours) M4: FEM M5: Collocation methods M6: Meshless M7: Numerical Integration methods (QUAD)M8: 'approximate' exactM9: Transform methods (Laplace, Fourier, Lie)M10: 'Real' exact

Last edited by tagoma on January 7th, 2011, 11:00 pm, edited 1 time in total.

Many people (some quite famous, e.g. Pablo Triana) think that BS is used for pricing and risk-management. BS is only used for quoting prices, and as such it cannot be good or bad. As such, the only useful way to price BS is the analytical formula. No sane person uses a numerical method for BS. What they do is use numerical methods for more complicated models, and options. Typically you have to price some more or less exotic options, like a TARN. You have to assume a model, it's up to you if you use stoch vol, local vol, or a combination, or some Levy process (although nobody uses Levy processes, that's only academic stuff). Once you figure out what model is appropriate, you need to calibrate it, i.e. to find the parameters in the model that make it recover the vanilla prices traded in the market. After that you price your exotic option, and calculate its greeks. So the question should be what numerical methods are used in general, not for BS. The answer, in my opinion is: - trinomial (I'm not sure anybody uses binomial)- FD (usually CN with ADI)- SALI tree- Monte Carlo- American Monte Carlo- transformsIn addition to that, I would also add adjusters. Best,V.

- Cuchulainn
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The electricity generator broke down on friday , that's why you cannot access roelof's thesis. fxthesis is good on analysis.

Last edited by Cuchulainn on January 7th, 2011, 11:00 pm, edited 1 time in total.

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- Cuchulainn
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QuoteOriginally posted by: CosteanuHi Edouard,The meshless method is quite cute as an idea, but it doesn't really work in practice. For one thing, nobody uses it in the industry. But one can say the same about FEM (or about wavelets for that matter). V.Hardly anyone does FEM at school. Hence it is less used. It does not necessarily mean it is less useful.One issue:FEM is great for boundary conditions and adaptive meshes. FDM is a pain in this regard.

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Well, that's my point. FEM are useful. So the fact that nobody uses them in the industry (for math fin applications) is not by itself sufficient reason to discard a method. By the way, I threw out there the wavelets, to see if I get any reaction. Meshless methods can be used with wavelets, and wavelets are the ultimate adaptive method. So I'm going to contradict myself here: I think there might be a future for the meshless methods, only not by using RBF's, but wavelets. The problem with wavelets however is that they were studied extensively only for 2-dim (maybe 3-dim, but not higher than that). Best,V.

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On the topic of the curse (of dimensionality), what do you think is the root cause? I think it is 70% the time discretisation (mulltiplicative operator schemes like ADI and Soviet Splittiing) and 30% space (boundary conditions).If we take the extreme (an unrealistic) case of explicit Euler for an n-factor PDE, the only bottleneck is lack of virtual memory because multiarray can handle the data structure and no matrix inverison is needed. The problem is no longer memory (the reason why ADI was born?) but we do need a 64-bit OS.On the other hand, n factor FEM is too much? How can one create mass and stiffness matrices in 4d?

Last edited by Cuchulainn on January 9th, 2011, 11:00 pm, edited 1 time in total.

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