Optional Stopping Theorem

Imagine a perfectly fair game — the kind where your expected wealth never drifts up or down, so at every moment your best guess for tomorrow is exactly today's balance. That's a martingale. Now suppose you're allowed to quit whenever you like, using any rule you want, as long as the rule doesn't peek into the future. Can a clever quitting rule leave you ahead on average? The optional stopping theorem says no: however you time your exit, your expected value at the stop equals your starting value.

$$E[M_\tau] = E[M_0].$$

Here $M_\tau$ is the martingale's value at the random time $\tau$ you chose to stop, and $M_0$ is where you began. The theorem is both a deep impossibility result — the exact reason no betting system beats a fair game — and a surprisingly practical calculator, because it lets you read off the value of a random process at a random time without tracking every path.

What counts as a legal stopping rule

The catch is the phrase "doesn't peek into the future." A stopping time $\tau$ is a quitting rule you can execute in real time: to know whether you've stopped by now, you only need the information you have now. "Stop the first time my bankroll hits $100" is legal. "Stop the day before the peak" is not — you'd need tomorrow's data to obey it. This non-anticipating requirement is precisely why you can't use the theorem to manufacture profit: the profitable rules are exactly the illegal ones.

The fine print that makes or breaks it

The theorem is not free — it needs one of a few regularity conditions to rule out pathological, infinitely-patient strategies. Any one of these is enough:

Condition	What it says	Typical use
Bounded time	$\tau \le K$ for some fixed $K$	you must stop within $K$ rounds no matter what
Bounded steps + finite mean time	jumps are capped and $E[\tau] < \infty$	gambler's ruin: $\pm 1$ steps, finite expected duration
Dominated	the stopped process stays below some integrable $Y$	prices bounded by an integrable payoff

Skip the check and the theorem can fail loudly — which is the whole lesson of the doubling scheme below.

Worked example: gambler's ruin, two lines

Take a symmetric The Random Walk $S_n$ that starts at $i$ and stops at $\tau$, the first time it hits $0$ or $N$. Since $S_n$ is a martingale and $\tau$ satisfies the bounded-steps-plus-finite-mean condition, optional stopping gives $E[S_\tau] = S_0 = i$. But $S_\tau$ can only be $0$ or $N$, so with $p = P(\text{hit } N)$,

$$E[S_\tau] = 0\cdot(1-p) + N\cdot p = Np = i \;\Longrightarrow\; p = \frac{i}{N}.$$

The ruin probability drops out with no recursion. Want the expected duration? Use a second martingale, $M_n = S_n^2 - n$. Optional stopping gives $E[S_\tau^2 - \tau] = i^2$, and since $E[S_\tau^2] = N^2 p = Ni$,

$$E[\tau] = E[S_\tau^2] - i^2 = Ni - i^2 = i(N - i).$$

The art is choosing the right martingale: $S_n$ delivers the probability, $S_n^2 - n$ delivers the time. Build it, stop it, read off the answer.

Where it fails: the doubling system

The famous "martingale" betting system: on a fair coin, bet $1; if you lose, double to $2, then $4, and so on until your first win, then walk away up $1. Let $\tau$ be that first win. At the stop you're always ahead by exactly $1, so $E[M_\tau] = 1$, yet you started at $M_0 = 0$. The theorem appears violated — but it isn't, because none of the conditions hold: $\tau$ is unbounded (you might lose arbitrarily many times), the bets grow without limit, and the strategy needs infinite capital to survive the losing streaks. With a real, finite bankroll you almost surely hit a streak that wipes you out, and the expectation snaps back to zero. This is the canonical demonstration that the regularity conditions are the theorem, not decoration.

The pricing connection

Optional stopping is the machinery under Risk-Neutral Pricing. Under the risk-neutral measure, a discounted asset price is a martingale, so the value today equals the expected discounted value at any legal future stopping time — maturity for a European claim, or the exercise time for an American one. Pricing an option is, at bottom, applying this theorem to a carefully chosen martingale and reading off its value at time zero. That through-line — from "you can't beat a fair game" to "the price is the discounted expected payoff" — is the same theorem wearing two hats.