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For a (generalized) Brownian motion $Y = F(t,W)$, starting at $InitialValue$ and running for a total of $T$ time, if I want to "enclose" (in a visual way) "most" of the possible sample paths, I could consider, for instance,

$MinY = InitialValue - 3.5 * STD * sqrt(T)$ and
$MaxY = InitialValue +3.5 * STD * sqrt(T)$,

(where $STD$ is the standard deviation of the Normal increment). [A factor of 4 would also do, of course.]

Say now that I wish to find the similar bounds $MinY$ and $MaxY$ for a Geometric Brownian Motion.

Ho do I find reasonable bounds that I can use (as a function of: $InitialValue$, Total Time $T$, $Mean$ and $Variance$ of the underlying Normal distribution, given that the Y are Lognormal) ?

I am particularly interested in a formula for $MaxY$ (as I think that $MinY=0$ could be ok).

(For null drift, so far I am trying $MaxY= InitialValue * 20 ∗ STD∗sqrt(T)$ but not sure whether reasonable and justified.)

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  • $\begingroup$ The log of a GBM path is a BM path. So if you know how to put bounds around a BM, you can find the bounds around a GBM. $\endgroup$ – noob2 May 5 '20 at 21:31

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