SERVING THE QUANTITATIVE FINANCE COMMUNITY

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Alan
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### American options -- Reference prices

Apparently we don't have a thread for this, so here is a good place to start.I am investigating convergence of some numerical methods and would appreciate it ifsomebody could confirm a 7-digit reference value under the Black-Scholes model.It is an American-style put option with $S_0 = K = 50$, $T=1$, $\sigma=0.40$, and $r = 0.08$ (no dividends).The put value is 6.299XXX; I need confirmation of the value I have for the 3 remaining digits, XXX.So as to not prejudice the answers, I'll give my value later in the thread.Thanks!
Last edited by Alan on January 1st, 2016, 11:00 pm, edited 1 time in total.

Cuchulainn
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### American options -- Reference prices

Actually, we had quite a discussion here
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Alan
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### American options -- Reference prices

Thanks, looks like a good thread. Browsing through it quickly, I didn't see any numerical examples good to 7 digits.I don't want to start a new thread on numerical methods, just need confirmation of this 7-digit value.

DavidJN
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### American options -- Reference prices

Your number of tree steps please?Do you use a control variate technique?

Cuchulainn
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### American options -- Reference prices

I have tested the Binomial method (NT = 13000) because the early exercise is 1st order and you get an extra digit accuracy by doubling NT.I rounded up the usual suspects (no names at the moment:)) The results I get are:A. 6.2995361B. 6.2995339C. 6.2995361D. 6.2995660E. 6.2995339Which? (D looks suspicious, might be a coding issue, anyway D is anonymous for the moment). I need to reconfigure my FDM framework but is waiting in the aisles if you need more backup.
Last edited by Cuchulainn on January 1st, 2016, 11:00 pm, edited 1 time in total.
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Alan
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### American options -- Reference prices

QuoteOriginally posted by: DavidJNYour number of tree steps please?Do you use a control variate technique?I am checking a value from Mathematica, using their built-in FinancialDerivative function. Unfortunately, they don't really document what they do. They have an option, "GridSize" -> {xsteps, tsteps}, which I have taken diagonally up to xsteps = tsteps = 640,000 if that helps.
Last edited by Alan on January 1st, 2016, 11:00 pm, edited 1 time in total.

Alan
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### American options -- Reference prices

QuoteOriginally posted by: CuchulainnI have tested the Binomial method (NT = 13000) because the early exercise is 1st order and you get an extra digit accuracy by doubling NT.I rounded up the usual suspects (no names at the moment:)) The results I get are:A. 6.2995361B. 6.2995339C. 6.2995361D. 6.2995660E. 6.2995339Which? (D looks suspicious, might be a coding issue, anyway D is anonymous for the moment). I need to reconfigure my FDM framework but is waiting in the aisles if you need more backup.My value agrees to 6.2995 but does not agree with any of the remaining digits. So, at his point we have 6.2995XX and I would like to get consensus on the XX.
Last edited by Alan on January 1st, 2016, 11:00 pm, edited 1 time in total.

Cuchulainn
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### American options -- Reference prices

I have run the FDM (ADE) with large NS and NT (~20000 and 300000, respectively)I get a range [6.299588, 6.299590] depending in NS and NT. (BTW I use Excel to display values). // Increasing NS, NT too much might lead to round-off. For example, keep NS fixed, and increase NT until the value stabilizes or that convergence is no longer monotone.
Last edited by Cuchulainn on January 2nd, 2016, 11:00 pm, edited 1 time in total.
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frolloos
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### American options -- Reference prices

Using Trigeorgis binomial tree with 10,000 time steps in Matlab I get 6.299518. Seems to be off compared to Cuch's results (I'd trust Cuch's results more! )

spursfan
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### American options -- Reference prices

6.299597
Last edited by spursfan on January 2nd, 2016, 11:00 pm, edited 1 time in total.

Alan
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### American options -- Reference prices

QuoteOriginally posted by: spursfan6.299597Excellent! This is the closest to my result, which is 6.299596How many time steps?So, can we nail it down: 96 or 97 ? Or, if others think this is far off -- make your case.Remember I mentioned mine was based on Mathematica's built-in. It was stable at these 7 digits with both 320,000 and 640,000 time steps.So, this suggests one needs 500,000+ time steps for the end game here, or some really convincing extrapolation.
Last edited by Alan on January 2nd, 2016, 11:00 pm, edited 1 time in total.

spursfan
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### American options -- Reference prices

I used a truncated tree with 2,097,152 steps extrapolated with another tree with 4,194,304 steps - the answer was 6.2995968 - will keep it running to see what the next answer isWith 524,288 and 1,048,576 steps my answer was 6.2995961
Last edited by spursfan on January 2nd, 2016, 11:00 pm, edited 1 time in total.

Alan
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### American options -- Reference prices

Thanks, Mike, I appreciate it. Also thanks, Daniel and frolloos

Cuchulainn
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### American options -- Reference prices

QuoteOriginally posted by: frolloosUsing Trigeorgis binomial tree with 10,000 time steps in Matlab I get 6.299518. Seems to be off compared to Cuch's results (I'd trust Cuch's results more! )Actually, I get similar results for CRR, JR and Chang-Palmer when NT = 10000 (BTW what do you get for Trigeorgis when NT = 13000?)(NT = 13000)A. 6.2995361 Random walkB. 6.2995339 CRRC. 6.2995361 JRD. 6.2995660 TianE. 6.2995339 Chang PalmerI create a 2d lattice in memory so NT = 13000 is my max for 32-bit version.
Last edited by Cuchulainn on January 3rd, 2016, 11:00 pm, edited 1 time in total.
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Jean Piaget

frolloos
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### American options -- Reference prices

QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: frolloosUsing Trigeorgis binomial tree with 10,000 time steps in Matlab I get 6.299518. Seems to be off compared to Cuch's results (I'd trust Cuch's results more! )Actually, I get similar results for CRR, JR and Chang-Palmer when NT = 10000 (BTW what do you get for Trigeorgis when NT = 13000?)(NT = 13000)A. 6.2995361 Random walkB. 6.2995339 CRRC. 6.2995361 JRD. 6.2995660 TianE. 6.2995339 Chang PalmerI create a 2d lattice in memory so NT = 13000 is my max for 32-bit version.For 13,000 I get 6.299536, which is practically the same as your A and C above.