# Scaling Property of Randomness

15 Jan 2015I really enjoyed the book Fooled by Randomness by Nassim Taleb, and I came upon many interesting statements about randomness in markets, the most interesting one being the scaling property of randomness.

In the stock market, the general direction that stocks move in are for a reason, but the second or millisecond deviations that stocks have can be mostly attributed to noise or randomness. As Taleb did in his book, let’s take a stock that has a 15% return with 10% volatility per year. This translates into a 93% chance that the stock will go up in that year, assuming that the noise can be approximated by a normal distribution.

At a second time interval, there is only a 50.02% chance that the stock will go up. At a day granularity, there is a 54% chance that the stock will go up. At a month granularity, there is a 67% chance that the stock will go up, and at a year granularity, there is a 93% chance that the stock will go up. When I first read this, I was shocked. Randomness does not scale linearly, and therefore when analyzing the markets at a small time scale, stock price variations are often meaningless. This means that at the smallest of time scales, markets are inefficient, a very interesting conclusion.