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Fama’s Fair Game Condition and Martingale Price Changes

Article Quant Q&A · Author: s5s
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Summary

The document clarifies how Fama’s fair game condition relates to the martingale property. The question arises because the paper states that the expected change in a security’s price over the next period is zero, while a martingale is commonly described as having a conditional expected future value equal to its current value.

The answer explains that the variable in Fama’s equation is the price change from one period to the next. If the conditional expectation of that change is zero, then the conditional expected next-period price equals the current price, matching the martingale condition. The excerpt is brief and focuses on interpreting the notation; it does not assess the paper’s broader assumptions, empirical evidence, or whether actual market prices satisfy the condition.

Key ideas

  • Fama’s fair game statement concerns the conditional expectation of a price change.
  • A zero expected price change implies that the expected next-period price equals the current price.
  • This interpretation aligns the fair game condition with the martingale property.
  • The discussion explains notation but does not test whether market prices empirically behave as martingales.

Tags

StatisticsEquitiesMarket microstructure
Full text
# Fama: Efficient Capital Markets: A Review of Theory and Empirical Work - are martingales incorrect?


# Fama: Efficient Capital Markets: A Review of Theory and Empirical Work - are martingales incorrect?












In his paper, Eugene Fama gives the definition of a "fair game" as given below. I disagree. AFAIK, a martingale has the following property: $E[X_{t+\tau} | X_t] = X_t$. What am I missing?

Footnote 9 says:

And again, in a previous statement:

## Answer by Bob Jansen (score 6)

https://quant.stackexchange.com/a/55698

The way I understand it is:

In equation 2 $x_{j, t + 1}$ is defined as the change in of $p_j$ over the period $t$ to $t + 1$. The formula says that the expectation of the change is zero which is the same as saying that the expectation of the original variable at $t+1$ is equal to its current value.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.