Geometric Brownian Motion Assumes Independent Returns

Summary

The document explains a basic implication of geometric Brownian motion: returns over separate periods are modeled as independent, so a sequence of rising past returns does not, within that model, make the next return more likely to be positive. The model’s drift parameter describes the assumed average return, while its randomness comes from Brownian increments that are independent across time.

The answer contrasts this assumption with real market behavior, where returns may show dependence, and notes that a Brownian model can still serve as a useful approximation at very short horizons, such as in some market microstructure settings. The discussion is conceptual and brief; it provides no derivation, data, or test of when the approximation is appropriate. The conclusion follows from the model’s assumptions, not from a claim that observed stock returns are always independent.

Key ideas

• Geometric Brownian motion models returns across periods as independent.
• Past positive returns do not imply a positive next return within this model.
• The drift parameter represents the model's average return, not a running average of realized past returns.
• Brownian assumptions may be useful approximations at very short horizons, though actual returns can exhibit dependence.

Tags

# If the price of a stock follows a Geometric Brownian motion, then does stock return depends on past stock returns?


# If the price of a stock follows a Geometric Brownian motion, then does stock return depends on past stock returns?












Got this question from my homework. I think if past returns are keep raising then current return should also be positive, but the answer is it's not related to past returns, why? I tried to ask chatgpt and it gave a function of $r = μ + σ^2/2$ but it couldn't explain why μ is not related to past returns. I think $μ$ can be the mean value of past returns and would connect current stock return to past stock returns.

## Answer by quantinho (score 2, accepted)

https://quant.stackexchange.com/a/75366

If the returns follow Geometric Brownian Motion model then by definition the returns are independent of past returns. You are right in that returns are affected by past returns but there are models that assume Brownian motion for returns. For example, if you are doing some high-frequency or working in market microstructure models assuming Brownian motion will still be useful since looking at price moves in milliseconds looks very much random.

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