you misunderstood. Let me clarify a little bit

the stock price is not "50% higher, 50% lower at any given time"

if you know probability theory, the key is to find the most basic micro probability which is 50%, such as each lottery ticket. But you, having 1 ticket, are different from who have 10 ticket.

the change of stock price includes "drifting" and "shocking"

change = drift + shock

here drifting is not 50%-50%, while shocking is

let me use d for differential sign, change of stock is dS, then

dS = S (u dT + o e sqrt(dT))

here the first term is the drift and 2nd term is the shock, dT is the change of Time. This formula comes from the Brownian motion research.

when dT -->0, dT will be approching faster than sqrt(dT), therefore, the shocking effect is more important.

this is why the day trader pay more their attentions to the shocking effects.

when dT becomes larger, the drifting will more more dominant, this is whether "good" company or "bad" compay comes to play.

in one word, the stock price is not "50% higher, 50% lower at any given time". I said clearly before, only when dT ->0, which means at the trading moment, the stock price is 50% higher and 50% lower, definetely not at any given time!

please revisit my original post, even I gave an example using Monte Carlo whose results were not 50% higher and 50% lower, look it again, 10$ price after 10 days, 0% chance $11.

wrote in hurry, hope this clarify a little bit