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I have heard that the price of stock or future changing over time is a random process, namely, a martingale, and no one can have an edge. Is there any evidence supporting this assumption?

Why do so many quantitive traders profit from trading?

If it is not a martingale, then it is not fair play.

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    $\begingroup$ You can test the randomness of a market using different techniques. Your question is very general and for this reason, difficult to ask. I recommend you to study non-parametric test (Bachelier test) or Random Walk based test (Variance ratio test) to test if a market is a random process or not, at a specific market frequency. $\endgroup$ – NunodeSousa Feb 13 '18 at 14:35
  • $\begingroup$ @NunodeSousa Has anyone done this test? reference, thanks. $\endgroup$ – XL _at_China Feb 13 '18 at 15:39
  • $\begingroup$ Yes, I did it as an exercise some years ago. My implementation was based on chap. 7 of the book "High Frequency Trading" of Irene Aldridge. $\endgroup$ – NunodeSousa Feb 13 '18 at 15:52
  • $\begingroup$ @NunodeSousa Then what is your definition of randomness? The one based on Kolmogorov complexity? $\endgroup$ – XL _at_China Feb 13 '18 at 16:07
  • $\begingroup$ If one of the answers were helpful it would be great if you could accept one - Thank you! $\endgroup$ – vonjd Jun 24 '18 at 10:03
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Actually there are many different approaches to prove randomness (academic) or disprove randomness (fund managers to persuade their clients or their bosses ;-) in financial markets.

One approach I find especially interesting is based on algorithmic information theory. Basically what that does is to find an algorithm to compress financial data. The fewer regularities (= randomness) the more complex the algorithm will be. While e.g. $01010101$ will just be "repeat $01$ four times", $11010010$ seems to be "more random" so that the resulting algorithm will be more complex.

The paper
Brandouy, Olivier and Delahaye, J. P. and Ma, L., Estimating the Algorithmic Complexity of Stock Markets (May 1, 2011). International Conference of the French Finance Association (AFFI), May 11-13, 2011; Algorithmic Finance 2015, 4:3-4, 159-178. Available at SSRN: https://ssrn.com/abstract=1836886 or http://dx.doi.org/10.2139/ssrn.1836886

Abstract
Randomness and regularities in finance are usually treated in probabilistic terms. In this paper, we develop a different approach in using a non-probabilistic framework based on the algorithmic information theory initially developed by Kolmogorov (1965). We develop a generic method to estimate the Kolmogorov complexity of numeric series. This approach is based on an iterative “regularity erasing procedure” (REP) implemented to use lossless compression algorithms on financial data. The REP is found to be necessary to detect hidden structures, as one should “wash out” well-established financial patterns (i.e. stylized facts) to prevent algorithmic tools from concentrating on these non-profitable patterns. The main contribution of this article is methodological: we show that some structural regularities, invisible with classical statistical tests, can be detected by this algorithmic method. Our final illustration on the daily Dow-Jones Index reveals a weak compression rate, once well- known regularities are removed from the raw data. This result could be associated to a high efficiency level of the New York Stock Exchange, although more effective algorithmic tools could improve this compression rate on detecting new structures in the future.

Translation
Markets seem to be quite efficient!

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  • $\begingroup$ Thank you for your answer, Kolmogorov complexity is first developed by Chaitin, Solomonoff, and Kolmogorov, The randomness is defined as incompressable, loosely speaking, It is non-computable, and algorithm fail to capture any random process. $\endgroup$ – XL _at_China Feb 14 '18 at 1:38
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    $\begingroup$ Interesting reference. 1 up! $\endgroup$ – Richard Feb 14 '18 at 7:25
  • $\begingroup$ @Richard: Thank you, i upvoted your answer too :-) $\endgroup$ – vonjd Feb 14 '18 at 8:04
  • $\begingroup$ Actually, there is a definition of randomness based on finite data/sequence, Martin-Löf rondomness can not be verified, we can only determine it is not random if the sequence is not random. If it is, we can verify by any means $\endgroup$ – XL _at_China Feb 14 '18 at 10:33
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You have to distinguish (at least) two approaches:

1) derivatives pricing: Here you assume that there is a probability measure other than (but somehow tied to ) the real world measure - say $Q$. Under $Q$ your underlying is a martingale. Then pricing derivatives is calculating expectations. This measure $Q$ is linked to the principle of no-arbitrage. If you take other prices than the no-arbitrage price then (in a liquid market) other participants can form riskless portfolios with the asset that you price wrongly and gain a profit.

The assumptions of no-arbitage are not always true but they often serve as a valid benchmark for prices that take into account frictions and other restrictions that happen in reality.

2) Risk management/trading: Here you don't assume that the stocks/assets are martingales. You usually use whatever turns out to be useful and backed by statistics and evidence.

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  • $\begingroup$ Thank you for your answer, rather, I think your answer explains why some traders profit from market. And you assume that the underlying is random without proof or disproof. $\endgroup$ – XL _at_China Feb 14 '18 at 1:34
  • $\begingroup$ Furthermore we can think of random modelling as: the truth might be a dynamic system but it is that complex that we turn to probabilistic modelling. $\endgroup$ – Richard Feb 14 '18 at 7:37
  • $\begingroup$ You mean NP-randomness or p-randomness? Possibly, but it is not a fair play, and NP-randomness or p-randomness can be verified or falsified, And then some traders are doomed to profit while others are doomed to loss their money. $\endgroup$ – XL _at_China Feb 14 '18 at 10:59
  • $\begingroup$ I don't know about the difference between the two. But you mentioned martingale theory. The setting where you use martingale theory and probabilistic modelling without martingale theory - this is what my answer is about. If your question is about something very different then you could clarify it there. If it is a bout algorithmic trading then my answer is of limited use. $\endgroup$ – Richard Feb 14 '18 at 11:12

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