What is a martingale?

Question

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

Answer 1

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).

Answer 2

A martingale is a random process $X(t)$ which has the following properties:

$ E[X(T)|\mathcal{F}_t] = X(t) $

for $T > t$ and

$ E[|X(T)|] < \infty $

where $\mathcal{F}_t$ is the filtration at time $t$.

A martingale is a random walk, but not every random walk is a martingale. A Brownian random walk is a martingale if it does not have drift.

Also, a martingale does not have to be a Markov process.

EMH is not directly related to martingales.

Answer 3

Often one will find the argument that a random walk of price changes would be a proof of the efficient market hypothesis, but this is (IMO) a logical fallacy: Only because the EMH does imply random walks in the price changes, the finding of random walks does not imply automagically that the EMH is true.

Answer 4

A martingale can be viewed as a fair game (a game in which there is no arbitrage strategy)

A (centered) random walk is a martingale (think of it as the total Gain of the fair game)

If EFH is in order, then you can think that all information is in the current price, I think this more comparable to Markov Property than to Martingale property.

Hope that this helps a bit

Regards