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I have a quick question: why does the Efficient Market Hypothesis (EMH) assume that stock prices follow a martingale process?

I understand that discounted prices under the risk-neutral probability measure are a martingale. This can be shown explicitly. But how is this related to the statement that (pre-discounted, and under the true distribution) prices are martingale under the EMH? Is there any mathematical framework that supports this statement?

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    $\begingroup$ "prices follow a martingale" is a very old formulation of EMH (1970's ?) , not what we would say today. (I assume you found this in an old book or paper). It is true only in the short term (a few days...) when we can assume the expected return is essentially zero. Of course the long term return to stocks is not zero. $\endgroup$ – noob2 Feb 2 at 14:24
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    $\begingroup$ I think this question ought to remain open, as I think there is a common confusion here (I myself might be wrong about the following). I think modern versions of EMH don't imply stock prices are a martingale process, but rather the more general Markov process. Please see this question. Maybe somebody is more familiar with the literature on this topic. $\endgroup$ – Daneel Olivaw Feb 2 at 14:53
  • $\begingroup$ I think @DaneelOlivaw and noob2 are right. The old notion (Fama's papers from the '60s and '70s) was that prices are martingales (something akin to a random walk). The modern formulation is that prices reflect all available information (the expectation for tomorrow's price, $P_{t+1}$, is the same whether you're given the entire price history $\sigma((P_s)_{s\leq t})$ or only the current price $\sigma(P_t)$), see Markov property. On the other hand, long-term returns are predictable, past volatility shocks matter etc. $\endgroup$ – Kevin Feb 2 at 15:15
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It only means that the directions of future movements of stock prices are impossible to forecast.

For a more mathematical explanation, consider this document.

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