the fundamental theorems of asset pricing

The fundamental theorems of asset pricing are the two results that make modern derivative pricing logically sound — they say precisely WHY risk-neutral pricing works and WHEN it gives a unique answer. They translate two purely economic notions — 'there is no free lunch' and 'every payoff can be manufactured' — into two purely mathematical conditions about probability measures and martingales. They are the foundation on which Black-Scholes and all of quantitative finance rest.

First fundamental theorem: a market is arbitrage-free if and only if there EXISTS an equivalent martingale measure (a probability measure Q, equivalent to the real measure P, under which discounted asset prices are martingales). 'No arbitrage' means there is no self-financing trading strategy that starts from zero wealth, never goes negative, and ends with a positive payoff with positive probability — no riskless money machine. The equivalence is the punchline: the abstract economic axiom is EXACTLY captured by the existence of a martingale measure. In continuous time the correct statement needs a slightly stronger condition than naive no-arbitrage — No Free Lunch with Vanishing Risk (NFLVR) — and the precise theorem is due to Delbaen and Schachermayer; the discrete-time version (Dalang-Morton-Willinger) is cleaner and equivalent to plain no-arbitrage. Second fundamental theorem: an arbitrage-free market is COMPLETE (every contingent claim is replicable by a self-financing strategy) if and only if the equivalent martingale measure is UNIQUE. So existence of Q gives prices; uniqueness of Q gives a SINGLE price for every claim.

Together these theorems organise the entire pricing landscape: existence of Q = no-arbitrage = prices are well-defined; uniqueness of Q = completeness = every derivative has one unambiguous price obtained by replication and equal to the Q-expected discounted payoff. They are why the Black-Scholes model (which is complete, with a unique Q) gives a single number, while stochastic-volatility and jump models (incomplete, infinitely many Q) give only a no-arbitrage interval and need extra structure to pin a price. Honest caveats: the continuous-time first theorem genuinely requires NFLVR, not just the absence of obvious arbitrages — naive no-arbitrage is too weak in continuous time because of doubling-strategy pathologies, which is why admissibility (bounded-below wealth) and NFLVR are imposed. And completeness is a strong, special property: real markets are incomplete (you cannot perfectly hedge volatility or jumps), so the second theorem's clean uniqueness is the exception, not the rule.