Choosing Parameters in the Cox Binomial Option Model

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


# 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.

## Answer by et_85 (score 2)

https://quant.stackexchange.com/a/60363

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.

## Answer by decentralized (score 1)

https://quant.stackexchange.com/a/74956

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