Martingales, Market Efficiency, and Evidence of Price Randomness

# Evidence that supports the assumption that prices are random processes


# Evidence that supports the assumption that prices are random processes












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.

## Answer by Richi Wa (score 3)

https://quant.stackexchange.com/a/38219

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.

## Answer by vonjd (score 3)

https://quant.stackexchange.com/a/38222

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!