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Does anyone here actually use wavelets in their work? I am very interested in wavelets.

QuoteOriginally posted by: CactusManDoes anyone here actually use wavelets in their work? I am very interested in wavelets.Hi,For the numerical analysis of derivatives the following are used:MCFDMFEMLatticesMeshlessWaveletsThe last two methods are less well known. I looked at wavelets some time and I confess that I did not try it for PDE, only having seen them for simple PDEs. Do you know if they can be used for 'nasty' problems (discontinuities and jumps) and where the solution is not continuous. That's the challenge. here is an article on another thread, (suggested by Jan Dash), written by Gil Strang. I think that it might be useful Heatlet If one could apply it to the BS one-factor equation that would be nice.

For meshless methods, see http://math.iit.edu/~fass/ for a couple of papers applied to finance. This references some other names interested in this. Fasshauer Greg, Abdul Khaliq, David A. Voss 'A Parallel Time Stepping Approach Using Meshfree Approximations For Pricing Options With Non-Smooth Payoffs' Bachelier Conference 2004

Cuchulainn, do you have some financial data you cold send me? I would like to study this problem and applying wavelets with the actual financial data. Yes, I've seen this application to PDEs. I have wondered about this work (and he mentions the "...little to do with physical waves..."). (Basic) wavelets are good for some things and not good for others.QuoteDo you know if they can be used for 'nasty' problems (discontinuities and jumps) and where the solution is not continuous. That's the Wavelets are supposed to be ideal for 'nasty' problems like you described, because they are highly localized. That is, behavior in one area does not affect behavior at other areas. I want to see actual examples, though.

QuoteCuchulainn, do you have some financial data you cold send me? I would like to study this problem and applying wavelets with the actual financial data. Yes, I've seen this application to PDEs.Take any standard problem from JCF or the books by Paul Wilmott.That' a good start.If you like I can send you a test set and we can compare with FDM, for example. You might like to look at the famous CFD thread. Some examplesBS

That sounds great. I would love to see this test set.Oh, what I meant was that wavelets are great for some things, but I've seen some strange papers where people are trying to apply them to problems where I wouldn't expect at least the basic ones to give a good solution.But, where their special properties are useful, I am VERY INTERESTED.

Dempster has done alot of wavelet stuff. He gave a nice presentation in france last month. It was nothing new as he's been working on it for years but he is definitely trying to apply wavelets to option pricing.

QuoteOriginally posted by: desucofDempster has done alot of wavelet stuff. He gave a nice presentation in france last month. It was nothing new as he's been working on it for years but he is definitely trying to apply wavelets to option pricing.Any articles? Do you have a link?To be honest, FDM, FEM and FVM are not bad.

http://mahd-pc.jims.cam.ac.uk/publish/finance.htmlgo down to the 2001 section and there it is ..... rumour has it a new paper is due any time now with 1 of his phd students, think it was meant to be out a week or 2 ago

QuoteOriginally posted by: CactusManThat sounds great. I would love to see this test set.Oh, what I meant was that wavelets are great for some things, but I've seen some strange papers where people are trying to apply them to problems where I wouldn't expect at least the basic ones to give a good solution.But, where their special properties are useful, I am VERY INTERESTED.here is test set from BladeQuoteJust thought I'd share the benefits of some C++ coding I did with the forum. Before I worked in finance, I worked for a few years in a CFD (computational fluid dynamics) company. I took one of the basic finite difference discretisation schemes (MacCormack scheme) used for around 15 years in CFD and a implemented a numerical solution to Black-Scholes. This scheme is fully explicit so lends itself easily to american style exercise, but is second order accurate in time and space, just like Crank-Nicholson. I used two forms, the non-conservative (NC) form which most people are used to seeing, and the conservative formulation (C), which should help to reduce some of the error due to discretisation in S. For a call option with K = 80.0, sigma = 0.3, r ( risk-free rate ) = 0.07, S = 100 and T = 0.5, I get the following analytic resultAnalyticValue = 23.758, Delta = 0.907077, Gamma = 0.0078387, theta = -8.21389and the following result for the non-conservative formulation. Numerical ( NC ) - dS = 0.1, solution time 103.819sValue = 23.758, Delta = 0.907076, Gamma = 0.00783868, theta = -8.21388Numerical ( NC ) - dS = 1.0, solution time 0.032349sValue = 23.7583, Delta = 0.907054, Gamma = 0.00783643, theta = -8.21291Numerical ( NC ) - dS = 5.0, solution time 0.000827sValue = 23.7357, Delta = 0.906996, Gamma = 0.00777608, theta = -8.19161What can be seen is that the result is pretty much spot on for dS = 0.1, but is also pretty accurate for large time steps such as 1.0, with a solution time sub 1 second, and even dS = 5.0 with a solution time of sub 1-millisecond. The conservative results are shown below:Numerical ( C ) - dS = 0.1, solution time 195.699sValue = 23.758, Delta = 0.907077, Gamma = 0.00783868, theta = -8.21388Numerical ( C ) - dS = 1.0, solution time 0.090287sValue = 23.7582, Delta = 0.907057, Gamma = 0.00783633, theta = -8.2128Numerical ( C ) - dS = 5.0, solution time 0.001376sValue = 23.7343, Delta = 0.907071, Gamma = 0.00777471, theta = -8.18929The results are a little better in most cases, especially in the delta, but it seems that the extra accuracy doesn't warrant the extra solution time. This formulation should however fare alot better over the non-conservative form when there is variable volatility, sigma = f(S). Finally these can be compared to the results I got from my Crank-Nicholson solver:Numerical ( CN ) - dS = 0.1, solution time 14.0876sValue = 23.7546, Delta = 0.907130, Gamma = 0.007839, theta = -8.21444Numerical ( CN ) - dS = 1.0, solution time 1.18434sValue = 24.6647, Delta = 0.914685, Gamma = 0.007283, theta = -8.08372Numerical ( CN ) - dS = 5.0, solution time 0.235688sValue = 28.3642, Delta = 0.939682, Gamma = 0.005295, theta = -7.549203I have attached the C++ code for both schemes and they should compile on either windows or linux. I develop on linux/unix but have attempted to keep the code portable. Simple change the parameters in the sections " Common " and " finite difference " to adjust the option parameters and space discretisation. The time step is automatically calculated. You can also change the option type ( call, put ,etc ) in a section further down. Then simply compile and get the results. The code was pretty much done over two days, so bear in mind it is a bit rough around the edges, but I would interested to see what you think of the scheme and how it fares against some the FD solvers you guys are using. Thanks, Blade

To answer question about Dempster papers:Also see General Forum:Wavelet for more discussion I can't believe I forgot to mention Michael Dempsters presentation.http://www.msri.org/publications/ln/msr ... ster/1/and papersThere are two papers vrs. presentations at http://mahd-pc.jims.cam.ac.uk/publish/finance.html. One of I think is the same as below.Dempster, M.A.H., Eswaran, A. and Richards, D. (2000) "Wavelet methods in PDE valuation of financial derivatives." In Leung, K.S., Chan, L.-W. and Meng, H. (eds.) Data mining, financial engineering, and intelligent agents: Intelligent Data Engineering and Automated Learning - IDEAL 2000 International Conference (2nd), 13-15 December 2000, Shatin, N.T., Hong Kong, China. Berlin: Springer Verlag, pp.215-238Dempster, M.A.H. and Eswaran, A. (2001) "Wavelet based PDE valuation of derivatives." In Casacuberta, C., Miro-Roig, R.M., Verdera, J. and Xambo-Descamps, S. (eds.) Shaping the 21st century: European Congress of Mathematics, Vol. 2 (3rd), 10-14 July 2000, Barcelona, Spain. Basel: Birkhauser, pp.347-365The other is a 104 page paper on Wavelets that is more general on the topic.