Fair in, fair out
The theorem needs a mild condition to rule out "cheating with infinity" — it holds if the stopping
time is bounded, or the martingale is bounded, or a suitable integrability
condition holds. Under any of these, stopping cannot change the expectation.
Watch it happen. Below, a fair coin drives a random walk that starts at
0 and stops the moment it first reaches +5 or
-5. Each individual run ends at +5 or
-5 — never at 0 — yet averaged over many
runs, the stopped value comes out at essentially 0. Press
Run:
// A symmetric ±1 random walk (a fair game / martingale) started at 0,
// stopped when it first hits +L or -L. The OST predicts E[stopped value] = 0.
const L = 5;
const runs = 20000;
let total = 0;
let hitsUp = 0;
for (let r = 0; r < runs; r++) {
let x = 0;
while (x > -L && x < L) {
x += Math.random() < 0.5 ? -1 : 1; // fair step
}
total += x; // x is now exactly +L or -L
if (x === L) hitsUp++;
}
console.log("average stopped value:", (total / runs).toFixed(3)); // ~ 0
console.log("fraction ending at +" + L + ":", (hitsUp / runs).toFixed(3)); // ~ 0.5
The average is a whisker from zero, and each barrier is hit about half the time — precisely
\mathbb{E}[M_\tau] = 0 = M_0. (More generally, starting at
0 between barriers +a and
-b, the theorem forces the hitting probabilities so the expectation stays
put.)
- For a martingale M and a stopping time
\tau (with a boundedness/integrability condition),
\mathbb{E}[M_\tau] = \mathbb{E}[M_0].
- No stopping strategy gives an edge in a fair game.
- The condition matters: without it, unbounded strategies can appear to beat the game.
Why it matters in finance
Under the risk-neutral measure, discounted asset prices are martingales — so the OST is the precise
statement that you cannot beat the market by timing alone when there's no
arbitrage. It also underpins the pricing of an
American
option, whose holder chooses an exercise (stopping) time: the theory of optimal
stopping decides the best rule and the fair price.
The infamous "martingale" betting system: bet £1 on a fair coin; if you lose, double to £2, then
£4, and so on, until you finally win — netting £1, guaranteed. It looks like free money, and it
seems to violate optional stopping. The catch is exactly the theorem's fine print: the stopping
time (the first win) is unbounded, and the strategy needs unbounded wealth to
survive a long losing streak. Cap either your bankroll or the number of rounds — as reality does —
and the OST reasserts itself: your expected profit snaps back to zero. The theorem's conditions
aren't pedantry; they are the whole reason the game stays fair.
-
The conditions are essential. "\mathbb{E}[M_\tau] = M_0"
can fail for an unbounded stopping time on an unbounded martingale — the doubling strategy is the
classic counterexample.
-
\tau must be a genuine stopping time: the decision to
stop can depend only on information seen so far. "Sell at the peak" is not a valid strategy —
it needs to see the future.