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I hope that I have understood the gist of your question, if not I may try to adjust this answer. Let the time step in a binomial tree be $\Delta t \equiv \frac{T}{N}$. For $N \to \infty$, the distribution of the stock price (of any specific point in time $t>t_0$ converges to the lognormal distribution with scale parameter $\log S_0 + (r-\frac{1}{2}\sigma^...

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