From the point of view of cryptanalysis, the algorithmic view based on frequency analysis may be taken as a hacker approach to the financial market. While the goal is clearly to find a sort of password unveiling the rules governing the price changes, what we claim is that the password may not be immune to a frequency analysis attack, because it is not the result of a true random process but rather the consequence of the application of a set of (mostly simple) rules. Yet that doesn’t mean one can crack the market once and for all, since for our system to find the said password it would have to outperform the unfolding processes affecting the market – which, as Wolfram’s PCE suggests, would require at least the same computational sophistication as the market itself, with at least one variable modelling the information being assimilated into prices by the market at any given moment. In other words, the market password is partially safe not because of the complexity of the password itself but because it reacts to the cracking method.
Whichever kind of financial instrument one looks at, the sequences of prices at successive times show some overall trends and varying amounts of apparent randomness. However, despite the fact that there is no contingent necessity of true randomness behind the market, it can certainly look that way to anyone ignoring the generative processes, anyone unable to see what other, non-random signals may be driving market movements.
Von Mises’ approach to the definition of a random sequence, which seemed at the time of its formulation to be quite problematic, contained some of the basics of the modern approach adopted by Per Martin-Löf. It is during this time that the Keynesian kind of induction may have been resorted to as a starting point for Solomonoff’s seminal work (1 and 2) on algorithmic probability.
Per Martin-Löf gave the first suitable definition of a random sequence. Intuitively, an algorithmically random sequence (or random sequence) is an infinite sequence of binary digits that appears random to any algorithm. This contrasts with the idea of randomness in probability. In that theory, no particular element of a sample space can be said to be random. Martin-Löf randomness has since been shown to admit several equivalent characterisations in terms of compression, statistical tests, and gambling strategies.
The predictive aim of economics is actually profoundly related to the concept of predicting and betting. Imagine a random walk that goes up, down, left or right by one, with each step having the same probability. If the expected time at which the walk ends is finite, predicting that the expected stop position is equal to the initial position, it is called a martingale. This is because the chances of going up, down, left or right, are the same, so that one ends up close to one’s starting position,if not exactly at that position. In economics, this can be translated into a trader’s experience. The conditional expected assets of a trader are equal to his present assets if a sequence of events is truly random.
If market price differences accumulated in a normal distribution, a rounding would produce sequences of 0 differences only. The mean and the standard deviation of the market distribution are used to create a normal distribution, which is then subtracted from the market distribution.
Schnorr provided another equivalent definition in terms of martingales. The martingale characterisation of randomness says that no betting strategy implementable by any computer (even in the weak sense of constructive strategies, which are not necessarily computable) can make money betting on a random sequence. In a true random memoryless market, no betting strategy can improve the expected winnings, nor can any option cover the risks in the long term.
Over the last few decades, several systems have shifted towards ever greater levels of complexity and information density. The result has been a shift towards Paretian outcomes, particularly within any event that contains a high percentage of informational content, i.e. when one plots the frequency rank of words contained in a large corpus of text data versus the number of occurrences or actual frequencies, Zipf showed that one obtains a power-law distribution
Departures from normality could be accounted for by the algorithmic component acting in the market, as is consonant with some empirical observations and common assumptions in economics, such as rule-based markets and agents. The paper.
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