In Finance, we use the counterintuitive, Physics-type “Randomness”
September 14, 2009
There are two kinds of randomness in this Universe. One, that comes from quantum mechanics and statistical mechanics in physics and is probabilistic in concept and the other that is known as Mathematical randomness and is tied to the concept of "program size complexity" in Computer Science. The former has completely monopolized the domain of quantitative finance.
It is the first kind of randomness, the probabilistic randomness, the one that comes from quantum mechanics, with which all of us in the area of Finance are preoccupied. This concept is tied to the idea of Brownian Motion, Random Walk, Normal Distribution, etc. and essentially defines a random event (random number) as something drawn from a certain probability distribution.
Using simple coin toss example (binomial distribution) we develop Monte Carlo simulation algorithm (Gaussian distribution) and predict stock price movements. The entire field of derivatives pricing is predicated upon and totally based on this probabilistic random walk model.
However, let's say you are shown a number (digital) series with the first 30 elements (data points) such as:
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
If now we ask you to predict the 31st value. You would immediately say 0, because the series is simply a recurring patter of 0s and 1s. Now if I tell you that the above series satisfies the definition of a random walk and that it is indeed a random series you may have difficulty in believing it. But such a series is indeed random, because it is one of the possibilities of a 30 toss experiment in a coin toss game. Toss a coin and if heads come up then put 1 and if tails come up then put 0. Do it 30 times and generate all possible series (there would be 230 possibilities, i.e. over a billion possibilities or paths). One of them is the above one, a perfect sequence of 0s and 1s.
So if 1 means the stock price goes up and 0 means the stock price goes down then you ask the computer to predict the 31st day's outcome it will simply say 0, i.e. the stock price goes down on the 31st day.
Your computer, as well as your computer programmer who writes the program does not understand Random Walk of physics (or finance). Therefore, they would look at the series - the entire set of data points - and write a simple algorithm, a computer program, if you will, to generate the entire series. If we ask the computer to generate the above series, all that we have to do is write a (pseudo code) program: "print 01 a million times" and the computer will generate a million data points of "0" followed by "1".
Mathematically speaking, and by that I mean, speaking the language of the Computer, the above series is neither random nor complex.
To the ordinary mind and to someone uninitiated in the world of quantum physics or statistical mechanics this will seem bizarre. How can a perfectly deterministic series - with a rule of "0" followed by "1" - exhibiting a clear pattern be called random? It is totally counterintuitive. Well, that is because the entire theory of probability is counterintuitive!
Of course, the computer does not know that the above coin toss experiment could have generated a series like this as well:
1 1 1 1 0 1 0 0 1 0 1 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 1 1 0 0
The above series definitely appears random and there is no discernable pattern in it. Even my grandmother would say this is a random series and the computer will agree. Even the smartest computer programmer would not be able to write a simple algorithm - a computer program - that would generate the entire set of data points of 1s and 0s in the above series. In fact, any computer program to generate the above series would be as big as the entire series. The size of the algorithm - in bits - would be as big as the data output.
Mathematically speaking, and in the language of Computer Science, this series is random and algorithmically complex. It displays "program size complexity".
In Computer Science the concept of "randomness" is tied to the "program size complexity", i.e. if the algorithm to generate a data series is as big as the series itself then the series is random. This is the mathematical definition of randomness. Algorithmic Information Theory (AIT) is built around this concept.
An example in nature of a mathematically random and a complex series is the series of Prime Numbers.
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