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Stochastic Calculus / Discrete Martingales and the Gambling Blueprint

The Optional Stopping Theorem

American options give the holder the right to exercise at any time before expiry. The exercise rule is a stopping time — a random time τ\tau determined by information available up to time τ\tau. The central question is: is the option value at τ\tau still equal to the discounted expected payoff? In other words, does the martingale property of the discounted price process survive when we substitute a stopping time for a fixed future time?

The answer is yes — under precise conditions. The Optional Stopping Theorem (OST) makes this precise: if a stopping time is bounded (or if the martingale is "nicely bounded" up to that time), then E[Mτ]=E[M0]\mathbb{E}[M_\tau] = \mathbb{E}[M_0]. Without those conditions — as the doubling-bet counter-example showed — the equality can fail. The OST is the discrete-time foundation for the continuous-time theory of optimal stopping, American option pricing, and first-passage-time calculations in credit models.
Learning objectives. By the end of this note you should be able to:
  • Define a stopping time with respect to a filtration and verify the definition for concrete rules.
  • State and prove OST I (bounded stopping times) as a corollary of the discrete stochastic integral.
  • State OST II and OST III, identifying the conditions each imposes and why the doubling strategy violates them.
  • Apply the OST to solve the gambler's ruin problem and compute expected exit times.
  • Explain why American option pricing requires the OST, and what "uniform integrability" of the value process means financially.

In this lesson

  1. Stopping times
  2. The Optional Stopping Theorem
  3. Applications
  4. American option pricing and the OST

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