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JohanSollie
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Multidimensional binomial trees

March 19th, 2014, 12:32 pm

hey guys,I am having problems with multidimensional binomial trees. I simply cannot get them to give any reasonable answer, and I think that there is something fundamentally wrong with my approach. I have attached a spreadsheet (had to zip it to get it uploaded) with two versions of Hull and White 1990 decoupling. You can find the example in 26.7 Hull 7 ed. pp 609 - 610.Can any of you see what goes wrong? I have implemented a spread option for reference and included the Margrabe '78 model.{S1,S2} ~gbmzero interest rateLet me know if the sheet does not work and I will explain in detail what I do.Any help is greatly appreciated!
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Cuchulainn
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Multidimensional binomial trees

March 19th, 2014, 6:56 pm

I have Hull 6th ed. but was unable to find it.. which chapter is it?Espen Haug's book on Options does describe Rubinstein's model (VBA) as well as Clewlow-Strickland (also for spread options). (pseudo-code).QuoteProgram testing can be a very effective way to show the presence of bugs, but it is hopelessly inadequate for showing their absence. Edsger Dijkstra
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JohanSollie
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Multidimensional binomial trees

March 20th, 2014, 6:52 am

I dont know which chapter it is in the 6 ed, but in 7 ed it is 26.7.Rubinstein did have an implementation in "return to oz", but it is not what I am looking for. The reason I want the Hull White '90 is because it can easily be generalized to m dimensions. Have a look at http://www.risk.net/digital_assets/4239/v12n3a1.pdfsec 3.1.2 (chol tree)This is what I am implementing, and the HW model is a spesific case treated in the paper. But since I cannot get the HW model working, I cannot get the cholesky tree working either. It is for sure something that I am overlooking, and I bet it is something simple. I just can't find it!
 
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JohanSollie
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Multidimensional binomial trees

March 20th, 2014, 7:09 am

Does Haug have the Clewlow-Strickalnd spectral decomposition? Can t find it in my copy, which chapter is that? Here is a link to the Hull White 90 paper, the decoupling is done in section V.http://efinance.org.cn/cn/FEshuo/250105 ... 87-100.pdf
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Cuchulainn
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Multidimensional binomial trees

March 20th, 2014, 6:34 pm

QuoteOriginally posted by: JohanSollieDoes Haug have the Clewlow-Strickalnd spectral decomposition? Can t find it in my copy, which chapter is that? Here is a link to the Hull White 90 paper, the decoupling is done in section V.http://efinance.org.cn/cn/FEshuo/250105 ... 7-100.pdfI don'ty know what spectral decomposition means in this context.The FD method in that paper are kind of outdated.
 
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Cuchulainn
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Multidimensional binomial trees

March 20th, 2014, 6:41 pm

QuoteOriginally posted by: JohanSollieI dont know which chapter it is in the 6 ed, but in 7 ed it is 26.7.Rubinstein did have an implementation in "return to oz", but it is not what I am looking for. The reason I want the Hull White '90 is because it can easily be generalized to m dimensions. Have a look at articlesec 3.1.2 (chol tree)This is what I am implementing, and the HW model is a spesific case treated in the paper. But since I cannot get the HW model working, I cannot get the cholesky tree working either. It is for sure something that I am overlooking, and I bet it is something simple. I just can't find it!Had a _very_ quick look at 3.1.2. Seems they want diffusion term == I. Reminds me Nelson/Ramaswamey
 
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JohanSollie
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Multidimensional binomial trees

March 21st, 2014, 7:29 am

Yes, the choleksy tree is normalized with the inverse cholesky so all diffusion terms have unit variance. I may have taken the thread in the wrong direction by introducing the korn and muller paper. What I am struggling with is the implementation of _any_ decoupled trees, and I used the Hull and White as an example that people might know and it should be well tested. I am sure that someone on this forum has done multivariate trees ( or any other method) using decoupling of the underlying processes prior to tree construction.I will optimize my trees and solution methods as soon as I can get the simple excel trees working! Thanks for the Nelson/Ramaswamey link.I really appreciate any thoughts on this as I have been struggling with it surprisingly long. I thought this would be straight forward and 2 hours of work - as it turns out, that was not the case...(btw, korn and muller had an article on their trees in Wilmott journal, I attached it. It is the simplest way of treating decoupling I have seen, but there is nothing in there that is not in the original paper. The variance - covariance matrix is decomposed and used for decoupling of the underlying processes. There are many ways of doing it, the spectral decomposition had been used before they landed on the obvious cholesky solution.)
 
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Cuchulainn
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Multidimensional binomial trees

March 21st, 2014, 7:44 am

Someone has looked and solved it seems but maybe not by a unit diffusion.http://www.wilmott.com/messageview.cfm? ... id=95863If I look at the PDE case transforming PDEs to a unit diffusion in nd is not trivial IMO. Closeley related to Choslesky is Sylvester's law of inertia.http://en.wikipedia.org/wiki/Sylvester's_law_of_inertia
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GiuseppeAlesii
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Multidimensional binomial trees

March 21st, 2014, 8:47 am

Probably you may find convenient reading two publications of mine:1) Alesii, Giuseppe. Esercizi e complementi di finanza aziendale Roma Aracne, 2008, find summary at http://bit.ly/1awdE0FAs you may notice, in the appendix there are the GAUSS codes of (Boyle et al., 1989), (Kamrad and Ritchken, 1991), (Ekvall, 1996) and (Gamba and Trigeorgis, 2005).The last two models do use a decomposition approach. Although I am not sure this is what you are actually looking for.Within the text, a plain expanation of the decomposition approach by the last two models is reported with worked out examples.2) Alesii, Giuseppe, Going Parallel Over the Rainbow (December 8, 2013). Available at SSRN, find a copy of the paper at http://bit.ly/1hILEMhThis paper reports a parallel version --pseudo codes only-- of the previous multivariate lattice models.I hope it helps.To be more supportive, it would be good to know the Hull and White reference as you may download it from Google Scholar, else poring on Hull "The Textbook" one may miss a model for another.
 
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JohanSollie
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Multidimensional binomial trees

March 21st, 2014, 12:55 pm

Ok, solved it. stupid mistake.http://d-nb.info/999701673/34pp 136 - 137.Thanks to Cuchulainn, GuiseppeAlesii and all others who has had a look at it.
 
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Cuchulainn
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Multidimensional binomial trees

March 21st, 2014, 4:37 pm

Hope we were of some help at least.Out of curiosity, was it a software bug or a mahs bug?
 
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JohanSollie
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Multidimensional binomial trees

March 21st, 2014, 6:33 pm

Well it is somewhat embarrassing to be honest. As I said it must be something that I am overlooking and that it must be something fundamentally wrong with what I was doing. The short story is that of two dimensions I managed to lose one.. The cholesky three is now up and running, and it looks pretty promising. I will play around with it and see how well it converges at higher dimensions. It is built on the RB discretization meaning that all prob. are 1/2 which is nice. With optimizing it should also run pretty fast, but that is work for the week to come. You did help, sometimes it is good to ask the question regardless of the answer you get. That being said, your answers were good