 Fundamental theorem of asset pricing

In a finite-state, finite-horizon financial market with frictionless trading, a strictly positive bank account $B$, and unrestricted predictable self-financing strategies, absence of arbitrage is equivalent to existence of an equivalent martingale measure $Q$. Discounted asset prices $S_t/B_t$ must be martingales under $Q$. Equivalence means that $Q$ and the physical probability measure have the same null events; it does not mean their probabilities agree.^[ref]

One direction follows immediately: a zero-cost self-financing gain has zero discounted expectation under $Q$, so it cannot be nonnegative everywhere and positive with positive probability. Conversely, Hahn-Banach separation theorem produces strictly positive Arrow state prices annihilating the attainable zero-cost gains; normalizing by the bank account gives $Q$. The more general infinite-state and continuous-time statements require attention to closure and admissibility.

With trivial initial information and terminal information equal to the chosen final sigma-algebra, a finite arbitrage-free market is a complete market exactly when $Q$ is unique. Every terminal contingent claim $H$ is then replicable, with value $V_t=B_t{\mathbb{E}}_Q[H/B_T\mid {\mathcal{F}}_t]$.^[ref]

For example, take $B_0=B_1=1$, $S_0=1$, and terminal stock prices $0.8$ and $1.2$. The martingale equation forces both $Q$-probabilities to be $1/2$. A strike-one call option pays zero or $0.2$, so its price is $0.1$. Holding half a share and borrowing $0.4$ replicates those two payoffs. The physical up probability could instead be $3/4$: the pricing weights express attainable-payoff consistency, rather than a forecast of frequencies.

The Fundamental Theorem of Asset Pricing is a key concept in financial mathematics and economics that establishes a connection between the pricing of financial assets and the existence of arbitrage opportunities in a market. It essentially provides the theoretical foundation for understanding how assets should be priced in a no-arbitrage market. The theorem can be summarized in a few main points: 1. **No Arbitrage Condition**: The first part of the theorem states that if there are no arbitrage opportunities in a market (i.e.