Risk-neutral measure

A probability measure under which discounted asset prices are martingales. It represents derivative prices as discounted expected payoffs without implying that investors are actually indifferent to risk.

Fundamental theorem of asset pricing: A result connecting no-arbitrage in financial markets to the existence of equivalent martingale measures. It states when a risk-neutral measure exists and when it is unique.
Probability measure: A mathematical assignment of probabilities to events in a sample space. A risk-neutral measure is a particular probability measure used for valuation.
Risk-neutral valuation: A pricing method that discounts expected future payoffs computed under a risk-neutral measure. It is the main practical use of the measure.
Physical probability measure: The probability measure intended to describe the actual statistical behavior of market outcomes. Unlike the pricing measure, it represents real-world probabilities rather than risk-adjusted valuation weights.
Complete market: A market in which every contingent claim can be replicated by trading available assets. Completeness is the condition that makes the equivalent martingale measure unique.
Equivalent martingale measure: A probability measure equivalent to the physical measure under which discounted asset prices are martingales. This is the standard formal type of risk-neutral measure in no-arbitrage pricing.
Martingale: A stochastic process whose conditional expected future value equals its current value. Discounted asset prices must have this property under the risk-neutral measure.
Black–Scholes model: A model for pricing European options using geometric Brownian motion and continuous hedging assumptions. Its option prices can be derived as discounted expectations under a risk-neutral measure.
Risk-neutral probability: A state probability used in a discrete pricing model to make discounted asset prices martingales. It is the discrete-state counterpart of the broader risk-neutral measure.
Incomplete market: A market in which some contingent claims cannot be replicated using available traded assets. Such markets generally admit multiple risk-neutral measures and do not determine a unique price.

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