mathematics - What is a martingale?

What is a martingale and how it compares with a random walk in the context of the Efficient Market Hypothesis?

Answer

Samuelson suggested in 1965 that the stock prices follow a martingale (see P. Samuelson “Proof That Properly Anticipated Prices Fluctuate Randomly”).

Assume there is a security with a random payoff $X_T$ at date $T$. Let $..., P_{t–1}, P_t, P_{t+1},...$ be the time series of prices of a security with this payoff. Finally, define the price change $\Delta P_{t+1}=P_{t+1} – P_{t}$ for any pair of successive dates $t$ and $t + 1$. Samuelson begins by defining “properly anticipated prices” as prices that are equal to the expected value of $X_T$ at every date $t \leq T$, based on the information $\Phi_t$ available at date t (which, in particular, includes the present and all past price realizations for that security, $...,P_{t–2}, P_{t–1}, P_t$). That is, for all $t \leq T$: $$P_t = \mathbb E(X_T|\Phi_t).$$

In particular, $P_T = X_T$. He then proves that the “prices fluctuate randomly” since it follows that for all $t \leq T$, $P_t = \mathbb E(P_{t+1}|\Phi_t)$ or alternatively that $\mathbb E(\Delta P_{t+1}|\Phi_t) = 0$, and $$\mathbb E(\Delta P_{t+1}\Delta P_{t+2}...\Delta P_T|\Phi_t) = \mathbb E(\Delta P_{t+1}|\Phi_t) \mathbb E(\Delta P_{t+2}|\Phi_t)...\mathbb E(\Delta P_T|\Phi_t)=0.$$ In words, prices follow a martingale, and successive price changes are mutually uncorrelated.

This implies that if “prices are properly anticipated,” all the information in the past price series that is useful for forecasting next period’s expected price is contained in the current price. Note that this is a much weaker statement than to say that all information in the past price series that is useful for forecasting the probability distribution of next period’s price is contained in the current price (which is the random walk hypothesis suggested by Fama in his thesis).