I'm having some trouble following the development of a discrete model in Baxter & Rennie from Page 41-42 (2002 printing). Any help would be greatly appreciated.Model with constant growth and noise-------------------------------------------------(let & = delta)The model is parameterized by the intertick time &t. As the quantity gets smaller, the model should ever more closely approximate a continuous-time model. There are also three fixed constant parameters: volatility, sigma; growth, mu; and risk-free rate, r.The cash bond B_t has the simple form B_t=exp(rt), which does not depend on interval size.The stock process follows the nodes of a recombinant tree, which moves from value s at some particular node along the next up/down branch to the new value:s exp(mu*&t + sigma*Sqrt(&t)) if up,s exp(mu*&t - sigma*Sqrt(&t)) if downThe jumps are all equally likely to be up as down, that is p=0.5 everywhere.For a fixed time t, if we set n to be the number of ticks until time t, then n=t/&t andS_t = S_0 exp(mu*t + sigma*Sqrt(t)*(2X_n - n)/Sqrt(n))where X_n is the total number of the n separate jumps which were up jumps. The random variable X_n has the binomial distribution with mean n/2 and variance n/4, so that (2X_n -n)/Sqrt(n) has zero mean and variance one. By the central llimit theorem, this distribution converges to that of a normal variable with zero mean and unit variance. So as &t gets smaller and n gets larger, the distribution of S_t becomes log-normal, as log S_t is normally distributed with mean logS_0+mu*t and variance sigma^2*t.I understand how they arrived with the factor (2X_n - n)/Sqrt(n). But how did they shoehorn that factor into the equation for S_t? The increments are +/- sigma*Sqrt(&t) and &t = t/n. How does this term get transformed to the normalized value?Thanks so much. I wish these authors would not skip so many steps in their derivations. I'd happily pay for the extra paper--I don't see what they're trying to achieve by being terse. On page 7 of this book, there is a transformation that involves a couple of tricky substitutions. Although I figured them out, I don't see why the authors did not include them in the text.

Isn't that just S = S0 * u^X * d^(n-X),where u = exp(mu*dt + sig *sqrt(dt)), d = exp(mu*dt - sig *sqrt(dt)), so you getS = S0 * exp(mu * t + X *sig *sqrt(dt) - (n-X) *sig *sqrt(dt)), which yields the formula ?

And yes, they are increments, but not for S but for log(S), does that help ?

You are correct. I stared at the problem for some time and could not see that step--but it's clear now. Thanks so much.