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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.

Josef Teichmann authored lectures on the fundamental theorem of asset pricing. The notes connect no-arbitrage, equivalent local martingale measures and replication, including a finite-state convex-geometric argument. This explains why a collection of quoted prices can be consistent precisely when an appropriate positive pricing measure exists under the theorem's hypotheses.^[ref]

Consider an arbitrage-free complete finite-state market with $n$ traded assets, strictly positive physical probabilities on all successor states, and conditional second-moment matrixpositive definite. Completeness permits at most $n$ successors from each parent atom, while positive definiteness forces at least $n$ independent successor price vectors. Thus there are exactly $n$.

The fundamental theorem of asset pricing supplies strictly positive one-step state prices. If their vector is $q$, their physical probabilities are $p_s$, and the independent successor price vectors are the columns of $C$, then $P_{t-1}=Cq$ and $V_t=C\mathrm{diag}(p)C^{\top }$. ConsequentlyThese unique ratios satisfy $\mathbb{E}[Z_t{P}_t\mid {\mathcal{F}}_{t-1}]=P_{t-1}$. Their successive product is a martingale deflator. The no-arbitrage hypothesis is essential: completeness and an invertible second-moment matrix alone do not make the regression expression positive.^[ref]