Martingale
Quantitative Finance glossary · Stochastic Calculus & Continuous Time Finance
Definition
Stochastic process M_t adapted to filtration mathcalF_t with mathbbE[M_t]<∞ and mathbbE[M_t,|,mathcalF_s] = M_s for all s ≤ t — a fair game. The fundamental theorem of asset pricing says no-arbitrage is equivalent to existence of an equivalent measure under which discounted prices are martingales (FTAP-I); market completeness is uniqueness of that measure (FTAP-II). Itô integrals of L^2-integrands against Brownian motion are martingales by construction.
Martingale in plain English
A martingale is the math definition of a perfectly fair game: whatever your fortune is right now, your best guess for its future value is exactly what you have today — no built-in edge up or down. Flipping a fair coin for even money is the classic example; your expected pile of chips after the next flip equals your current pile. This idea is the backbone of modern pricing theory: the deep result (the Fundamental Theorem of Asset Pricing) says a market has no free-lunch arbitrage exactly when you can view discounted prices as a fair game under some suitably reweighted set of probabilities. That's not just abstract elegance — it's the engine that lets us assign one consistent, arbitrage-free price to complicated derivatives.
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