# Distribution of the $90$th percentile of a geometric brownian motion simulation

I have run a simulation of a geometric brownian motion. The simulation runs from $t=0$ to $T=1000$. I generate $10000$ paths. For every moment for $t=1,2,3,\ldots, 100$ I calculate the $90$th percentile of the simulated paths. When I plot these points, depending on the used parameters, the curve that is produced is concave or convex. Please check the figure for an example.

When $\sigma$ is large relative to $\mu$, the curve of the $90$th percentile is concave (as above), and when $\sigma$ is small relative to $\mu$ the curve is convex. Is there any analytic formula for the distribution of percentiles of the GBM between two time instances? Also, is there an analytic or heuristic way to predict the shape of the 90th percentile (or any other percentile)? Thanks for the help.

• This curve obviously is neither concave nor convex in any standard sense, so what do you mean by these terms? – whuber Nov 24 '17 at 14:31

If I understand what you are doing right, the shape just shows which term is dominating. If $\mu$ (the drift) is large and the volatility is really close to zero, it should look like $e^{rt}$ since there is exponential growth. But, if $\mu$ is zero,and $\sigma$ is a realistic number, what you mwould see is the bounds growing as $\sqrt t$.
• The solution to the SDE for GBM is one that has a closed-form, exact solution. In other words, we can specify the distribution $P_t$ (usually this occurs in finance, and Price is the variable of focus) for all $t>t_0$. Since that gives you the mean and variance (and thus SD) at all times t, it should be easy to solve for the 90% bound. Think of it this way: the 95% bound is at just about the mean plus 2 x SD. – eSurfsnake Nov 25 '17 at 1:52