- Cuchulainn
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I am looking for proofs (or otherwise) if convergence in NR method is monotone in step-size dt? Then there is a chance that Richardson extrapolation will work.RE seems to be erratic and unpredictable when applied to the untransformed lattice using CRR, JR etc. (unless a lot of juggling is done with the position of K and grids.)I suspect RE will not work with saw-tooth convergence. Mathematically it is obvious.

Last edited by Cuchulainn on April 28th, 2015, 10:00 pm, edited 1 time in total.

Don't waste your time with rubbish choices of parameters for binomial treesMark Joshi is the best person to look athttp://fbe.unimelb.edu.au/__data/assets/pdf_file/0007/806362/212.pdf

- Cuchulainn
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This is a nice paper.However, it does not seem to address an analysis of the NR method.

- Cuchulainn
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QuoteDon't waste your time with rubbish choices of parameters for binomial treesI love wasting time, especially if I can see light at the end of the tunnel.Seriously I am interested these issues on BM:1. low-order zig-zag (non-monotone) convergence 2. non-monotonicity of Richardson extrapolation (looks erratic).3. CRR and negative probability (well-known in PDE/fluid dynamics, called convection-dominance).4. How accurate are the Greeks in BM; I mean O(dt^??) in max norm?5. In general, I don't like RMS and such as a measure of accuracy; L infinity max norm is much stronger.6. The unsatisfactory tricks of using uneven NT, placing K on a node etc. etc. It looks almost like an ill-posed problem.So, back to NR; will it resolve 1-6? (BTW PDE/FDM does, but this is irrelevant here).

Last edited by Cuchulainn on May 2nd, 2015, 10:00 pm, edited 1 time in total.

My suggestion. You should be able to analyze the convergence in CRR directly since it is a binomial model and thediscrete transition density can be summed explicitly. This will likely be quite a tedious exercise with the asymptotics ofbinomial coefficients, but perhaps will give you what you seek.I also suspect it was probably already done by Mark Rubinstein or somebody in the early CRR literature, but don't have a cite.p.s. A little googling shows there is indeed a literature with explicit error terms for CRR.Google something like "explicit error", binomial model. I see a 2013 paper by Viitasaari that has explicit formulas and earlier cites.

Last edited by Alan on May 2nd, 2015, 10:00 pm, edited 1 time in total.

I can not remember the very paper(s) of Mark Joshi, but already coding more carefully can speed up a lot, http://axelvogt.de/axalom/CRR_optimized.zip

- Cuchulainn
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Quotep.s. A little googling shows there is indeed a literature with explicit error terms for CRR.Google something like "explicit error", binomial model. I see a 2013 paper by Viitasaari that has explicit formulas and earlier cites.Indeed. This is a good article and it addresses many issues. Further, the thesis by Stefanie Mueller hits the nail on the head. (a brilliant thesis IMO).In particular, smooth convergence and when Richardson extrapolation is allowed has been cleared up. P.S. The only place I've seen convection dominance discussed is Espen's book page 287sig < r * sqrt(dt)that give negative values for the quantities formerly known as probabilities.

Last edited by Cuchulainn on May 4th, 2015, 10:00 pm, edited 1 time in total.

- Cuchulainn
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QuoteOriginally posted by: AVtI can not remember the very paper(s) of Mark Joshi, but already coding more carefully can speed up a lot, http://axelvogt.de/axalom/CRR_optimized.zipI tried in C++ framework the convection-dominated:T = 1, r = 0.1, sig = 0.001, K = 65, S = 60, Put = 1.855678.NT = 5000method, value=========CRR, 0 (!!)Tian, 1.855ChangPalmer, 0Random Walk, 1.855 (not risk neutral)Jarrow-Rudd JR 1.8855 (not risk neutral)

Last edited by Cuchulainn on May 4th, 2015, 10:00 pm, edited 1 time in total.

- Cuchulainn
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The lack of smoothness is payoff influences accuracy.Take a cash-or-nothing futures put S = 100, K = 80, Q (G) = 10, T = 0.75, r = 0.06, q = r, s = 0.35; P = 2.6710.Stress NT = 500. Tian P = 6.85896. CRR P = 2.20454Tian with Thomee?Wahlbin (Heston/Zhao) with simple Trapezoid averaging P = 2.66888.

- cosmologist
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QuoteOriginally posted by: CuchulainnThe lack of smoothness is payoff influences accuracy.Take a cash-or-nothing futures put S = 100, K = 80, Q (G) = 10, T = 0.75, r = 0.06, q = r, s = 0.35; P = 2.6710.Stress NT = 500. Tian P = 6.85896. CRR P = 2.20454Tian with Thomee?Wahlbin (Heston/Zhao) with simple Trapezoid averaging P = 2.66888.Dear Daniel,Are you stating that for the given case, the Put values can either of 6.85896 or 2.20454 and hence both methods can not be trusted ?May be I am interpreting your results wrongly. Could you please explain? Thanks in advance.Sid

Last edited by cosmologist on May 5th, 2015, 10:00 pm, edited 1 time in total.

- Cuchulainn
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QuoteOriginally posted by: cosmologistQuoteOriginally posted by: CuchulainnThe lack of smoothness is payoff influences accuracy.Take a cash-or-nothing futures put S = 100, K = 80, Q (G) = 10, T = 0.75, r = 0.06, q = r, s = 0.35; P = 2.6710.Stress NT = 500. Tian P = 6.85896. CRR P = 2.20454Tian with Thomee?Wahlbin (Heston/Zhao) with simple Trapezoid averaging P = 2.66888.Dear Daniel,Are you stating that for the given case, the Put values can be either of 6.85896 or 2.20454?May be I am interpreting your results wrongly. Could you please explain? Thanks in advance.SidHi Cosmo,Nice to see you again.I tested 3 methods indeed, 2 of which are off the mark. A major issue is that discontinuous payoff has adverse effects on accuracy. This is well-known in PDE and Thomee/Wahlbin used averaging in 1974. Heston and Zhou do the same for lattices and have useful hints but their paper does not seem to be well-known.I tested a cash-or-nothing (~ scary Heaviside function). So, I tried maligned CRR which was better than 'straight' Tian (which never converges it seems). Now I took a simple (trapezoid) averaging and I got the improvements that you see.Stefanie Mueller uses Richardson extrapolation which may reduce computation time. I cannot give a definite conclusion (yet) on RE. Convergence is a bit erratic.I have not yet done averaging on plain options by averaging around the strike. hthDON THE RATE OF CONVERGENCE OF DISCRETE-TIME. CONTINGENT CLAIMS. STEVE HESTON. Goldman Sachs & Co., New York. GUOFU ZHOU.

Last edited by Cuchulainn on May 5th, 2015, 10:00 pm, edited 1 time in total.

QuoteT = 1, r = 0.1, sig = 0.001, K = 65, S = 60, Put = 1.855678 That sounds as if Call and Put are mixed up or Strike and Spot

Last edited by AVt on May 5th, 2015, 10:00 pm, edited 1 time in total.

- Cuchulainn
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QuoteOriginally posted by: AVtQuoteT = 1, r = 0.1, sig = 0.001, K = 65, S = 60, Put = 1.855678 That sounds as if Call and Put are mixed up or Strike and SpotYou've lost me on that one. Am I missing something?? CRR does not work for this case is the message, Tian does work.

Last edited by Cuchulainn on May 5th, 2015, 10:00 pm, edited 1 time in total.

- Cuchulainn
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QuoteOriginally posted by: AVtQuoteT = 1, r = 0.1, sig = 0.001, K = 65, S = 60, Put = 1.855678 That sounds as if Call and Put are mixed up or Strike and SpotOOPS, sorry, I made a typo; it should beQuoteT = 1, r = 0.1, sig = 0.001, K = 65, S = 60, Call = 1.1855678We can trigger convection-dominance in CRR by letting sig -> 0 and keeping other params fixed. When it breaks down, we increase NT to 'fix' it. For sig >= 0.1 things are more benign; CRR has a slight edge on Tian.

Last edited by Cuchulainn on May 6th, 2015, 10:00 pm, edited 1 time in total.

- Cuchulainn
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QuoteOriginally posted by: CuchulainnThe lack of smoothness is payoff influences accuracy.Take a cash-or-nothing futures put S = 100, K = 80, Q (G) = 10, T = 0.75, r = 0.06, q = r, s = 0.35; P = 2.6710.Stress NT = 500. Tian P = 6.85896. CRR P = 2.20454Tian with Thomee?Wahlbin (Heston/Zhao) with simple Trapezoid averaging P = 2.66888.The Tian method is a good all-rounder. For discontinuous payoffs we can use averaging a la above to improve the accuracy. For example, Simpson 3/8 is a good approach and more general than smoothing just before expiration.The jury is out with regard to Richard extrapolation; I do not seem to get the same conclusions as is so clear with PDE/FDM. Ex binary call S = K = 40, r = 0.05, sig = 0.3, Q = 1, T = 0.5; exact 0.46929NT = 500Tian (no averaging) 0.491427Tian (with Trapezoid) 0.476768Tian (3/8) 0.464044 (NT = 1000, 0.469795)Averaging deserves more attention?

Last edited by Cuchulainn on May 12th, 2015, 10:00 pm, edited 1 time in total.

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