[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.
