Fundamental theorem of asset pricing

A family of results linking no-arbitrage conditions in a financial market to the existence of an equivalent martingale measure. Its precise form depends on the market model and the definition of arbitrage.

Equivalent martingale measure: A probability measure equivalent to the real-world measure under which discounted asset prices are martingales. Its existence is the theorem’s central mathematical counterpart to no-arbitrage.
No-arbitrage principle: The condition that no admissible trading strategy can produce a riskless gain without investment. It is the economic condition paired with existence of a pricing measure.
No Free Lunch with Vanishing Risk: A strengthened no-arbitrage condition ruling out limits of strategies whose losses vanish while gains persist. In many continuous-time models it supports existence of an equivalent martingale measure.
Risk-neutral valuation: A pricing method that discounts expected payoffs computed under a risk-neutral probability measure. The theorem supplies the measure that makes this valuation consistent with no-arbitrage.
Risk-neutral measure: A probability measure under which discounted asset prices are martingales, enabling valuation by discounted expected payoffs. It expresses the pricing implications of the theorem in a familiar valuation framework.
Equivalent probability measures: Probability measures that assign probability zero to exactly the same events. Equivalence preserves which outcomes are possible while allowing prices to be evaluated under another measure.
No Free Lunch with Bounded Risk: A no-arbitrage condition that excludes approximate free gains among strategies with uniformly bounded downside risk. It appears in discrete-time formulations of the theorem under suitable assumptions.
Option pricing: The determination of fair values for contracts whose payoffs depend on future asset prices. Equivalent martingale measures turn option values into discounted expectations.
Stochastic discount factor: A random variable that converts future payoffs into current prices through expectations under the real-world probability measure. It provides an alternative representation of the pricing measure and asset prices.
Numeraire: A reference asset or unit in which other asset prices and payoffs are measured. Choosing a numeraire determines which prices the theorem requires to be martingales.

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