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Stock market has been model as a random walk with a drift. Since it has a drift(bigger than zero) it is not a "Brownian Motion" but it still a Martingale? Is Stock market a Brownian Motion? Is it a Martingale?

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  • $\begingroup$ You could say it's an exponential martingale. $\endgroup$
    – ch-pub
    Jan 13 '15 at 20:02
  • $\begingroup$ A lot of people assume stock prices are a geometric Brownian motion, which is not a martingale. It should be noted that although some stock prices look like a geometric Brownian motion there is an enormous amount of evidence to suggest they are not. $\endgroup$
    – Wintermute
    Jan 14 '15 at 2:25
  • $\begingroup$ The question is invalid because the stock is NOT a martingale under the real-world measure. $\endgroup$
    – SmallChess
    Jan 14 '15 at 10:56
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The stock market is modeled as a brownian motian,with a real world drift usually larger than zero. This sort of model would be similiar to the CAPM or APT , VaR. The martingale is a mathematical condition that assure no arbitrage used in derivatives pricing, black scholes style. In that case the drift usually is the risk free rate of such economy.

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  • $\begingroup$ This is one of the simplest models, but yes, it makes the difference clear. $\endgroup$
    – vonjd
    Jan 14 '15 at 7:22
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In real world probabilities it is not a martingale as the expected value of the stock in the future will be different than its actual value, because of its non-zero drift.

In the risk-neutral probability world the stock price discounted by the risk-free rate can be considered as a martingale.

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