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I dont understand the introduction and/or idea of the variable $X$ on page $80$ in the following handout.

http://www.maths.lth.se/matstat/kurser/fmsn25masm24/ht17/Ch3.pdf

Does someone know whats the deal with it, why do we change and what are the implications of the change?

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  • $\begingroup$ Stock prices follow a random trajectory, described by the stock prices $S_0, \cdots,S_T$, but it can equally well be described by the successive daily "logarithmic returns" $X_t = \ln(S_t/S_{t-1})$. From the sequence of S's you can find the X's and vice versa. It is more convenient to describe the random process in terms of the X's. $\endgroup$
    – Alex C
    Sep 7 '19 at 16:57
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As Alex said, the $X_i$ correspond to i.i.d. log-returns. The benefit is that $$\ln(S_T)=\ln(S_0)+\sum_{i=1}^N X_i.$$ And for sums of i.i.d. random variables, we know the limiting distribution due to the central limit theorem. Then, you can easily show that the stock price in the binomal model converges to a geometric Brownian motion (and hence, you end up in the Black-Scholes model).

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