A discounted stock price under the risk-neutral measure is a martingale. A trading strategy's P&L is a martingale if the underlying price is. An American option's value process — the upper envelope of expected payoffs — is a supermartingale. These facts power the entire edifice of continuous-time derivative pricing, yet they all reduce to a single line: the expected future value, conditioned on current information, equals the current value.
The introductory probability course introduced martingales via the binomial model. This note re-derives the definition from the σ-algebra machinery established in the previous lesson, extends it to sub- and supermartingales, and proves two fundamental structural results: the Doob decomposition (every submartingale is a martingale plus a predictable increasing process) and the martingale property of the discrete stochastic integral. Both are the discrete prototypes of the continuous-time results that drive Black-Scholes.
Learning objectives. By the end of this note you should be able to:
- Define a martingale, submartingale, and supermartingale with respect to a filtration, and reduce the definition to a one-step condition via the tower property.
- Verify or falsify the martingale property for the symmetric random walk, , the geometric martingale, and the exponential martingale.
- State and prove the Doob decomposition of a submartingale.
- Prove that a discrete stochastic integral with predictable integrand is a martingale.
- Explain why the doubling-bet strategy cannot produce bounded expected profit in finite time.
In this lesson
- Definition and filtration
- Core examples
- Doob decomposition
- Discrete stochastic integral as a martingale
Before you start
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