A longer title would be “from random walk model to a stdev proportional to sqrt(t)”
Ignore the lognormal;
Ignore the rate of return;
Ignore stock prices. Just imagine a Weiner process. I find it more intuitive to consider the discrete time random walk. Assuming no drift, at each step the size and direction of the step is from a computer that generates a random number from a normal distribution like MSExcel normsinv(rand()), I’d like to explain/derive the important observation that t units of time into the Future, the UNKNOWN value of x has a Probability distribution that’s normal with mean 0 and stdev √t.
Now, time is customarily measured in years, but here we change the unit of time to picosecond, and assume that for such a short period, the future value of x has a ProbDist “b * ϵ(0,1)”, whose variance is b*b. I think we can also use the notatinon n(0,b*b).
Next, for 2 consecutive periods into the Future, x takes 2 random steps, so the sum (x_0to1 + x_1to2) also has a normal distribution with variance 2b*b. For 3 steps, variance is 3b*b…. All because the steps are independent — Markov property.
Now if we measure t in picosecond, then t means t picosecond, so the Future value after t random steps has a normal distribution with variance t b*b. So stdev is b*√t
For example, 12 days into the future vs 3 days into the future, the PD of the unknown value would have 2 normal distributions. stdev_12 = 2 * stdev_3.