I have here a series of three random walks. Which one would you rather call a stock rate. The following?
And would you say, it is biased? Most likely, you would. In fact, it is not. It is a random walk with 0-1-normal distributed steps.
How about the next one?
Does it look more realistic? I like the fact, that is has these jumps. The steps here have a distribution with density 1/(1+x^3), which is Talib’s power law with exponent 2 (the Black Swan Talib from my previous post). Then the probability of taking a step larger than a tends towards 0 like 1/a^2. This distribution has a definite expected value, but the variance is infinite.
If we make this more extreme (power law with exponent 1), we take the density 1/(1+x^2), we get a distribution with undefined mean value. Here is a simulated random walk.
We get normal behavior, and suddenly an outbreak of immense size. It is close to the turkey story in the book. The turkeys are fed well for a long time. They come to the conclusion that things are stable, and then all of a sudden …
The graphs and simulations were done with my Euler Math Toolbox. See the link on the link bar.