Stock Price Drift, Brownian Motion, and Martingale Measures

Summary

The discussion separates a stock-price model from the martingale condition. A Brownian-motion-based model can include drift under real-world probabilities; having nonzero drift does not mean the process ceases to be Brownian motion. The first answer connects such modeling to broad asset-pricing and risk frameworks, though it does not explain those connections in detail.

The second answer clarifies that a stock with nonzero expected return is generally not a martingale under the real-world probability measure. In derivative pricing, the relevant condition is typically that the appropriately discounted price is a martingale under a risk-neutral measure, with the risk-free rate setting the drift in a basic model. This is a concise conceptual exchange rather than a full treatment: it does not specify assumptions about dividends, discounting, or the underlying price process, all of which affect the precise martingale statement.

Key ideas

• A Brownian-motion model may include a nonzero drift under real-world probabilities.
• A stock with nonzero expected return is generally not a martingale under the real-world measure.
• Derivative pricing commonly uses a risk-neutral measure under which the discounted stock price is a martingale.
• The risk-free rate supplies the drift in the basic risk-neutral model, subject to model assumptions.

Tags

# Martiglale and Brownian Motion


# Martiglale and Brownian Motion












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?

## Answer by Tulio Carnelossi (score 1)

https://quant.stackexchange.com/a/16197

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.

## Answer by Maxime (score 1)

https://quant.stackexchange.com/a/16204

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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