= Solution
We prove the <positive state-price density alternative> using finite-dimensional <convex cones>. Put $w=(1+\|A_1\|)^{-1}$ and define
$$
D=\{\mathbb E[w fA_1]:f\text{ bounded, measurable and nonnegative}\},\qquad K=\overline D,\qquad L=\operatorname{span}K.
$$
All displayed expectations are finite, since $w\|A_1\|\leq1$. The set $D$ is a <convex cone>, so $K$ is a <closed convex cone>. A vector $x$ is nonnegative on $K$ exactly when $x^\top A_1\geq0$ almost surely. Indeed, one direction follows by taking every nonnegative bounded test weight $f$; for the other direction, if $x^\top A_1<0$ on a set of positive probability, take its indicator as $f$ to obtain a strictly negative expectation. This also identifies $K$ as the closed cone generated by the <support of a probability distribution> of $A_1$.
The three cases $A_0\notin K$, $A_0\in K\setminus\operatorname{ri}K$, and $A_0\in\operatorname{ri}K$ are treated below. They give respectively a negative-cost separating <portfolio>, a supporting zero-cost <portfolio> with a strictly positive payoff on an event of positive probability, or the required positive pricing weight. Thus the alternatives are exhaustive.
They are mutually exclusive. If $A_0=\mathbb E[\nu A_1]$ with $\nu>0$ and a <portfolio> has nonnegative payoff, then $x^\top A_0=\mathbb E[\nu x^\top A_1]\geq0$. A negative initial cost is impossible. If the payoff is strictly positive with positive probability, this expectation is strictly positive, so nonpositive initial cost is impossible too. The given weighted integrability permits taking the expectation of this scalar product.
In a one-period market, $x^\top A_0$ is the <portfolio>'s initial cost and $x^\top A_1$ its terminal value. The first two cases are <investment-consumption arbitrage>: nonnegative wealth can be taken out at both dates, with a strict benefit at one of them. If a strictly positive bank account is traded, any negative initial cost can be invested in it, converting the opportunity into a zero-cost <arbitrage> with nonnegative terminal payoff and positive payoff with positive probability.
In the third case, $\nu$ is a strictly positive <state-price density>. If a traded <numéraire> has price $B_0>0$ and terminal value $B_1>0$, its pricing identity is $\mathbb E[\nu B_1]=B_0$. Consequently
$$
\boxed{\frac{dQ}{dP}=\frac{\nu B_1}{B_0},\qquad\frac{A_0}{B_0}=\mathbb E_Q\left[\frac{A_1}{B_1}\right].}
$$
The density has mean one and is strictly positive, so $Q$ is equivalent to the physical measure. Thus discounted asset prices are <martingales> in the one-period sense. This is the <fundamental theorem of asset pricing> formulation of absence of arbitrage; normalization to a probability measure uses the positive traded numéraire, not an unsupported assumption that $\mathbb E\nu=1$.
Back to article page
