[EDIT] Question is answered, just cleaned up some clerical errors in the formulas.

[EDIT] Based on the comment I got, let me clarify, I am not stuck on the relationship between the binomial model vs Black-Scholes, I'm trying to understand how the author 'chose' u, d and q. I am trying to understand it in his framework, based on his equations.

As described in the title, I'm reading through the mentioned paper / article ( which can be found here: https://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=2A91B9F3B554842EC77951BBB9691EFD?doi=

And I am at a loss on page 21 of the PDF:

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I have been trying to do this 'little bit of algebra' for hours now and I just don't get anywhere. Let me be more specific: If you simply plug in the equations for u,d and q which he provides, you indeed get the described results.

I tried to solve the two equations though, with the additional assumption that $u*d=1$. This reduces it to a system of two equations for two variables and I found what appear to be the solutions for $q$ and $d$ and by extension $u$:

$$ q=\frac{1}{2}\left(1+\frac{\mu}{\sigma}\sqrt{\frac{t}{n}}\right)\frac{1}{\sqrt{1+\frac{\mu^2t}{\sigma^2n}}} $$

$$ d=\exp\left({-\sqrt{\frac{t}{n}}\sigma}\sqrt{1+\frac{\mu^2t}{\sigma^2n}}\right)$$

With $c_n=\frac{\mu}{\sigma}\sqrt{\frac{t}{n}}$ we can write

$$ q=\frac{1}{2}(1+c_n)\frac{1}{\sqrt{1+c_n^2}} $$ $$ d=\exp\left(-\sqrt{\frac{t}{n}}\sigma\sqrt{(1+c_n^2)}\right) $$

At this point I felt pretty good about myself, these don't look too different from what's in the paper, except for some terms that should behave nicely as $n\rightarrow\infty$. But why are the solutions not the ones that they present in the article?


If you have any idea or hint or can spot where my brain left my skull, please let me know.

Thank you.

  • $\begingroup$ Try this excellent PDF from John Hull & let me know if you still need help after reading this: www-2.rotman.utoronto.ca/~hull/BSMBinomialProof/… $\endgroup$ Jan 5, 2021 at 10:32
  • $\begingroup$ @JanStuller Thank you for your reply! Unfortunately the paper simply states on the first page in an aside the quantities for u and d. I am still trying to understand the mathematical derivation of these quantities. I have seen the formula for p (=q) based on the risk free rate (also covered earlier in the article which I linked) but at the point where I am stuck it doesn't appear and isn't 'used' in the conjuring of u,d and q. $\endgroup$
    – et_85
    Jan 5, 2021 at 16:03

2 Answers 2


I can't believe how long it took me, but I finally found (almost) all the answers and I found my mistake.

This paper: Derivation of the Up and Down Parameters of the Binomial Option Pricing Model by R. Stafford Johnson and James E. Pawlukiewicz (1998) https://www.jstor.org/stable/41917712 does what I tried to do and from there I realized that setting $u*d=1$ was the mistake, I should have used the condition $q=0.5$ (zero skew) and voila, the values for $u$ and $d$ come out as expected in the limit for $n\rightarrow\infty$. Interestingly, $u*d=1$ in that case and that's where I got the wrong idea in the first place.

I still don't know where Cox et al. came up with their estimate for $q$ (since we are setting it to 0.5) but I will deal with that some other time.


I found their estimate for q, it's equation (4.8) at page 8 here: On Cox-Ross-Rubinstein Pricing Formula for Pricing Compound Option http://www.etamaths.com/index.php/ijaa/article/view/2026 I read your paper and, to me, the condition q=0.5 is too much arbitrary but even the assumption u=1/d of this other paper doesn't satisfy me either. Said that, in 4.8, being a quadratic equation, we get two values of p (it should be 1/2 +/- ecc... if i'm correct) but only p=1/2+ ecc.. is considered.

Let me know what you think, I can post my calculations if you want


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