# Binomial Option pricing, paper by John C. Cox, I don't understand the calculation / choice of u.d.q

[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=10.1.1.379.7582&rep=rep1&type=pdf)

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

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?

[SNIP]

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

Thank you.

• 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/… Jan 5, 2021 at 10:32
• @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. Jan 5, 2021 at 16:03

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.