Optional stopping theorem
Quantitative Finance glossary · Stochastic Calculus & Continuous Time Finance
Definition
The result that a martingale evaluated at a suitable random stopping time has expected value equal to its starting value.
Optional stopping theorem in plain English
A martingale is a running total in a perfectly fair game, where at every moment your expected next value is just wherever you are right now (a fair coin-flip bet is the classic example). The optional stopping theorem says that even if you use a smart rule to decide WHEN to quit, as long as that rule is reasonable (you can't peek into the future and your game can't drag on forever unbounded), your expected ending value equals your starting value. In plain terms: no clever stopping strategy lets you beat a fair game on average. This is why it's a favorite tool for proving that certain betting or trading tricks can't secretly generate free money.
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