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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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.
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.