# Minimal bounds to enclose most sample paths of a GBM (Geometric Brownian Motion)

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.)

• 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. – noob2 May 5 '20 at 21:31