Talk:Binomial options pricing model

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Model accuracy

It is possible to use various finite difference methods for both American and Bermudan options (It's an assignment to write such a program in my numerical methods class.). While it is also true that the binomial tree model can accommodate discrete dividends, so can these finite difference methods. Why do you think that binary trees are "more accuratte"? Eweinber (talk) 14:53, 27 October 2010 (UTC)

The increase in accuracy for the BOPM really only pertains in comparison to models that do not allow the placement of discrete dividends, such as the closed-form Black-Scholes model. I don't think there is a claim that the BOPM is inherently more accurate than finite differencing. In practice, I have found BOPM slightly easier to wrassle to the ground and cough up the correct answer in a reasonable time than other techniques. BTW, a "reasonable time" is measured in microseconds, if not nano-seconds, these days. Ronnotel (talk) 15:23, 27 October 2010 (UTC)

Model practicality

I reverted a change by 59.162.216.53 that claimed the binomial model is impractical. This is not true. By in large, the binomial model is used by practioners for a variety of reasons, mostly having to do with accuracy. As a developer of quantitative option pricing models, I have a fairly broad exposure to this issue and have yet to come across a serious option trader relying on the Black-Scholes model for actual trading. Ronnotel 13:25, 27 September 2006 (UTC)

Agreed. Plus what else do you use for Bermudan options (other than simulations and Finite Difference methods of course... ) Fintor 15:31, 28 September 2006 (UTC)

Discrete Dividends

I believe most practical binomial models use discrete dividends rather than a continuous dividend yield. Using a continuous dividend yield can result in significant mis-pricing at or near dividend dates. Any objections to changing the formula to reflect this? Ronnotel 16:55, 28 November 2006 (UTC)

Why not show both formulae - with commentary as to differences? Fintor 17:49, 28 November 2006 (UTC)

Issues with the formula

I was using this article as a guideline for constructing a simple binomial model for valuing a stock option and came across an interesting problem/issue/quirkiness/something. I would get all my values do up the formula and I realized that when I changed the value for the volatility of the stock the end result did not change at all. Not even by the lowliest of decimal points. After reducing the formulae for the binomial value and the probability, p, I found out why.

The formula for the binomial value given in the ariticle is equivalent to ${\displaystyle Se^{-qt}}$

where:

S is the current price of the underlying

q is the dividend yield

t is the time step

(I hope I've given the correct key, all the variables above correspond to those in the article).

Why then, would the formula and the model require so much information if you only really need these three things? Is there a mistake somewhere in there maybe?

Or am I just making things up?

Douglas Robinson 18:28, 29 November 2006 (UTC)

Doug, I'll take a look at the formula and see what can be done. In the mean time, check and see if you have access to a copy of Black-Scholes and Beyond by Neil Chriss or Option, Futures and Other Derivatives by John Hull. Both provide excellent, implementable algorithms. Ronnotel 18:50, 29 November 2006 (UTC)
Um, the probability of an up (and down) transition incorporates the volatility. Go back and check how you are calculating your transition probabilities. But you should still try to get ahold of one of both of these references. Ronnotel 18:54, 29 November 2006 (UTC)

Binomial Formula

=pSu+(1-p)Sd)/(e^{rt})

=S(p({u-d}/{e^{rt}})+d/e^{rt})

=S({e^{(r-q)t+sigma*sqrt(t)}-1}/{e^{2sigma*sqrt(t)}-1}(e^{sigma*sqrt(t)-rt}-e^{-sigma*sqrt(t) -rt})=d/{e^{rt}})

=S({e^{-qt+2*sigma*sqrt(t)}-e^{-qt}}/{e^{2*sigma*sqrt(t)}-1}

=Se^{-qt}({e^{2*sigma*sqrt{t}}-1}/{e^{2*sigma*sqrt{t}}-1})

=Se^{-qt}

(Sorry for the confusing look of it, but I couldn't quite get the syntax of the math parser)

Douglas Robinson 21:43, 29 November 2006 (UTC)

I think your approach is a little off - you are using stock price (S) where you should be using option prices (C). Start with two trees, one filled with stock prices (S), the other with option prices (C). The stock prices are easy, base node is just S(0). Next step has Su and Sd. Third step has Suu, S(0) and Sdd, etc. Now, start filling in the option tree from back to front. The last step is also easy, just calculate expiration value of the option for each stock price from the stock tree. Next, go back one time step in the option tree using the binomial formula. However, use the values from the option tree, i.e.

${\displaystyle C=e^{-rt}(pC_{u}+(1-p)C_{d})\,}$

Ronnotel 21:53, 29 November 2006 (UTC)

Okay,

I definitely have been approaching the model in the wrong way. Thanks for clearing that up for me. But the formula still has that weirdness to it.

If you start with ${\displaystyle C=e^{-rt}(pC_{u}+(1-p)C_{d})\,}$

You can continue with:

(going to try out this math parser again... had a nice image of the equation and working out but I don't see how to upload it)

${\displaystyle ={pCu+(1-p)Cd}/(e^{rt})\,}$

${\displaystyle =C(p({u-d}/{e^{rt}})+d/e^{rt})\,}$

${\displaystyle =C({e^{(r-q)t+sigma*sqrt(t)}-1}/{e^{2*sigma*sqrt(t)}-1}(e^{sigma*sqrt(t)-rt}-e^{-sigma*sqrt(t)-rt})=d/{e^{rt}})\,}$

${\displaystyle =C({e^{-qt+2*sigma*sqrt(t)}-e^{-qt}}/{e^{2*sigma*sqrt(t)}-1}\,}$

${\displaystyle =Ce^{-qt}({e^{2*sigma*sqrt(t)}-1}/{e^{2*sigma*sqrt(t)}-1})\,}$

${\displaystyle =Ce^{-qt}\,}$

Douglas Robinson 15:09, 30 November 2006 (UTC)

I think I see a problem, please note that ${\displaystyle C_{u}}$ is not the same as ${\displaystyle Cu}$. ${\displaystyle C_{u}}$ can only be calculated through the backward induction step described above. Again, I would strongly encourage you to get ahold of either of the references I mentioned - going off this article alone is going to be a tough journey. Ronnotel 15:16, 30 November 2006 (UTC)

User:Jesse 20/09/2013 Hey, I've just implemented this algorithm in Matlab, and it's apparent that the exercise price at time t is NEVER more than the expected price from the two possible branches at time t+dt, which means the time zero American call is always valued identically to the European call. — Preceding unsigned comment added by 130.130.37.85 (talk) 05:15, 20 September 2013 (UTC)

User:Jesse 21/09/2013

I've since learned that without any dividends, the above mentioned call values are in fact the same. — Preceding unsigned comment added by 101.161.174.12 (talk) 01:18, 21 September 2013 (UTC)

Yep - only two cases when the model will indicate an early exercise for call: i) just prior to ex dividend date, and ii) if the short stock rebate rate suddenly goes negative (i.e. the stock becomes hard-to-borrow). In the absence of dividends, the binomial model collapses to the Black-Scholes model. Ronnotel (talk) 13:39, 23 September 2013 (UTC)

1st: page doesn't work (some PHP Zend Optimizer problem).
2nd and 3rd: xls files don't work in MS Office 2007, because macros need that the cells are unlocked to work, but IDK the password to unlock them. Perhaps it works in older Excel versions - IDK this and i can't check.

Maybe this section should be revised?
--Loonatic (talk) 21:48, 18 December 2010 (UTC)

closed-form solution

I have no particular issue with discussing the recent paper proving no closed-form solution (CFS) in the article. However, in order to feature this result (should it hold up to scrutiny, no particular doubt that it won't) in very first paragraph would require some evidence that the world of quantitative finance has been searching for a closed-form solution to BOPM. I'm a practitioner in this exact field with too many years of experience to mention and I can't recall reading about anyone ever searching for such a thing. In my experience, it's always been assumed that one didn't exist although I suppose it's nice to know that no-one has wasted their time looking, I guess. Is there reliably sourced evidence of a search for a CFS to BOPM? Ronnotel (talk) 17:00, 2 March 2011 (UTC)

However, in order to feature this result (should it hold up to scrutiny, no particular doubt that it won't) in very first paragraph would require some evidence that the world of quantitative finance has been searching for a closed-form solution to BOPM.

Aha! I agree but such evidence will almost surely have failed to surface given failed attempts of obtaining a closed form solution unless they would come in form of a conclusive proof; such is the result of this paper. (Conceivably, failed attempts would have not been worthy of publications). Additionally, this result cements in an elegant manner your belief that none exists. It should be worthy to be mentioned in the very first paragraph as it highlights BOPMs limitations. — Preceding unsigned comment added by Algotheorist (talkcontribs) 18:27, 2 March 2011 (UTC)

I agree - how about the current text which reads: In general, binomial options pricing models do not have closed-form solutions., which is the essential result you describe and can now be asserted based on the proof, but without the clutter. Support for this statement can be given further down in the article with appropriate citations. Agreed? Ronnotel (talk) 18:48, 2 March 2011 (UTC)

Almost there! just as we enlist the authors of BOPM by name, it would be only be honorable to give the author what he deserves in that context. In general, Georgiadis in xxrefxx proves that binomial options pricing models do not have closed-form solutions. — Preceding unsigned comment added by Algotheorist (talkcontribs) 19:43, 2 March 2011 (UTC)

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