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Cuchulainn
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ADE Method

September 20th, 2012, 3:22 pm

Here is my v1 of the SABR PDE and its domain. This is pluggable with the ADE in FD2.feedback welcome. @chocolatemonkey ... the BCs are just Dirichlet.@mtsm ... this might be a starting point for you.
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mtsm
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ADE Method

September 21st, 2012, 12:05 pm

oh thank you, I had not seen this. very handy. will check it out. I am actually working on your thing right now. I am going to ask you a few questions, but since it is on 2f BS, I will stick to the old thread.
 
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Cuchulainn
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September 21st, 2012, 12:16 pm

No problem, fire away when you are ready.
 
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chocolatemoney
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September 25th, 2012, 8:18 am

Quote@chocolatemonkey ... the BCs are just Dirichlet. Hi, I have been playing with BCs.I would like to understand the meaning and the purpose of Zvan 1998 BCs (Finite Difference methods in FE dy Duffy on page 243), in particular the far-field BC for vol:According to my numerical experiments, it is much less robust that a simpler external BC, for example Heston 1993.It basically forces the user to make use of mollifiers or other tools (smoothing conditions, ..) to reduce the oscillations around the strike.Thanks!
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Cuchulainn
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September 26th, 2012, 6:00 am

QuoteOriginally posted by: chocolatemoneyQuote@chocolatemonkey ... the BCs are just Dirichlet. Hi, I have been playing with BCs.I would like to understand the meaning and the purpose of Zvan 1998 BCs (Finite Difference methods in FE dy Duffy on page 243), in particular the far-field BC for vol:According to my numerical experiments, it is much less robust that a simpler external BC, for example Heston 1993.It basically forces the user to make use of mollifiers or other tools (smoothing conditions, ..) to reduce the oscillations around the strike.Thanks!Most(?) articles do not motivate how they got these boundaries. In fact, formulating these PDEs on 1/2 infinite regions is mathematically incorrect IMO. Even more. no BC should be given as we should really be talking about NUMERICAL BC, of which there are many. You prescribe BC a posteriori, not a priori imo. Which BC has Roelof Sheppard in his thesis?Example: take FTCS scheme for u_t + au_x = 0 on 0 < x < infinity (a > 0) by approximation on (0,1). What numerical BC on x = 1?I do not understand the Zvan BC. Did you try domain transformation z = v/(v+1) and then examine Fichera BC at z = 1? As far as I can remember it becomes a characteristic boundary.In your post you seem to pose two issues:1. BC 2. Smoothing of payoff at the strikeAre these related here?//Who understands BC?
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Alan
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September 26th, 2012, 1:37 pm

QuoteOriginally posted by: chocolatemoneyQuote@chocolatemonkey ... the BCs are just Dirichlet. Hi, I have been playing with BCs.I would like to understand the meaning and the purpose of Zvan 1998 BCs (Finite Difference methods in FE dy Duffy on page 243), in particular the far-field BC for vol:According to my numerical experiments, it is much less robust that a simpler external BC, for example Heston 1993.It basically forces the user to make use of mollifiers or other tools (smoothing conditions, ..) to reduce the oscillations around the strike.Thanks!For t < T, large v drives the call price to S and the put price to K e^(-r tau), where tau = T-t. (See Ch.10 in my book for exceptions to this general rule for stoch. vol. poblems)This bc is probably ok and will likely accomplish that but it seems a kludge. For euro-style, personallyI would just factor out the discount factor, and use put=K at v=infinity if I had to use something.
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Cuchulainn
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September 26th, 2012, 6:43 pm

Let's not forget this thread Neumann BC 2.7 For ADE and Neumann BC you use a mini_ADE at the first nodes.
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chocolatemoney
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October 11th, 2012, 9:22 am

If I am not in error, the far-field BC on vol is external. What is a "characteristic boundary" in this context?Unit-transformation is sexy but I get oscillations in the solution due to the (1-sigma) as denominator, when sigma -> 1 (I tested various implementations in the domain range of the products in am working with) I think this is a numerical issue that does not affect only vol=1, so a (1-sigma-epsilon) adjustment at vol=1 is not fixing this on my side.The oscillations I get are quite an expensive price to pay for the luxury of avoiding domain truncation, in my view.Furthermore, in my trading activity, the "interesting box" is in the neighborhood of the strike price: Unit-transformation forces me to spend time to calculate grid points for very high vols and prices. In order to get a higher resolution in the "interesting box", a non-uniform-mesh could be used, but that increases the complexity and the computational efforts.The results I am most happy with are:-> ADE (40% faster than ADI)-> Standard SABR, which is numerically more solid-> Heston BCs where I've found a nice trade-off between the point of truncation and how much of "a good approximation" of the solution the far-field BCs are... but there's lot of research to be done on this topic.[Edited: .. I've tried to improve my english]
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Cuchulainn
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October 11th, 2012, 2:36 pm

V = 1 is a nasty one as the coefficient blows up there.QuoteIn order to get a higher resolution in the "interesting box", a non-uniform-mesh could be used, but that increases the complexity and the computational efforts.Maybe a finer mesh in order to damp oscillations? I have documented it already in Appendix 3! Quote-> ADE (40% faster than ADI)This is Barakat-Clark,If you use the original Saul'yev the averaging is not done and this savesNT*N*NY additions and NT*N*NY multiplications. It will be even faster.
Last edited by Cuchulainn on October 11th, 2012, 10:00 pm, edited 1 time in total.
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