A martingale is the precise mathematical statement of a fair game. Given a filtration (\mathcal{F}_n), an adapted, integrable process (M_n) is a martingale when the best forecast of tomorrow's value, using everything known today, is exactly today's value:

Read it as a conditional expectation: no information available at time n lets you predict an up-move or a down-move on average. A submartingale relaxes this to \mathbb{E}[M_{n+1} \mid \mathcal{F}_n] \ge M_n (the game drifts in your favour), and a supermartingale to \le M_n (it drifts against you).

The canonical example

Take the symmetric random walk: independent fair steps \xi_i = \pm 1 and partial sums S_n = \xi_1 + \xi_2 + \cdots + \xi_n. Because each increment has mean zero and is independent of the past, \mathbb{E}[S_{n+1} \mid \mathcal{F}_n] = S_n + \mathbb{E}[\xi_{n+1}] = S_n — a martingale. More generally, any running sum of independent mean-zero increments is a martingale.

Refresh the figure to draw a fresh path. It wanders, but it has no preferred direction: it is just as likely to drift above the dashed mean level M_0 = 0 as below it.

Constant mean, and why finance cares

Apply the tower property to the defining equation and the mean never moves:

A fair game is fair forever: your expected fortune at any future time equals your starting stake. This is exactly why martingales sit at the heart of pricing. Under the risk-neutral measure, a properly discounted, arbitrage-free asset price is a martingale — its expected discounted value tomorrow is its price today — and that single condition is what makes option prices well defined.