One period replication and state prices

Definition A one-period model has two dates, $0$ and $1$, and a finite set of states $\Omega = \{\omega_1, \dots, \omega_K\}$, exactly one of which is revealed at date $1$. There are $N$ traded assets. Asset $n$ costs $c_n$ at date $0$ and pays $A_{jn}$ at date $1$ in state $\omega_j$, both in currency units. The market is frictionless, which means that any real holdings, including short and fractional ones, can be traded at these prices with no transaction costs, taxes or limits on borrowing. A physical probability measure $\mathbb{P}$ gives every state a probability $p_j = \mathbb{P}(\{\omega_j\}) > 0$, so every listed state can occur. In this lesson one traded asset is a cash account with price $B_0 = 1$ that pays $B_1 = R$ in every state, where $R > 0$ is the gross one-period return. A claim is a payoff vector $X \in \mathbb{R}^K$, with $X_j$ paid in state $\omega_j$.

Definition The payoff matrix of a one-period model is the $K \times N$ matrix $A$ whose row $j$ lists the payoffs of all assets in state $\omega_j$ and whose column $n$ is the payoff vector of asset $n$. A portfolio is a vector $\theta \in \mathbb{R}^N$ of holdings, in units of each asset. With the price vector $c \in \mathbb{R}^N$, its payoff and its cost are

A claim $X$ is attainable when $A\theta = X$ has a solution, which means that $X$ lies in the column space $\operatorname{Col} A$. Every claim is attainable exactly when $\operatorname{rank} A = K$, and an attainable claim has only one replicating portfolio exactly when $\operatorname{rank} A = N$, that is, when the columns are linearly independent.

Definition A replicating portfolio for a claim $X$ is a portfolio $\theta$ with $A\theta = X$, so that it pays exactly $X_j$ in every state $\omega_j$. Its replication cost is $V_0 = c^{\top}\theta$. Suppose two portfolios $\theta$ and $\theta'$ both replicate $X$ but have different costs. Buying the cheaper one, selling the dearer one and placing the difference in the cash account costs nothing today, and since the two payoffs cancel, it pays $R$ times that difference in every state, which is an arbitrage. So in a market without arbitrage that trades the cash account, every attainable claim has one replication cost, whichever replicating portfolio is used. This is the law of one price for attainable claims. It is weaker than the absence of arbitrage, because a market can give equal payoffs equal prices and still offer an arbitrage.

Definition A state-price vector is a vector $\pi \in \mathbb{R}^K$ that prices every traded asset as the sum of its state payoffs weighted by $\pi$,

which says $c_n = \sum_{j} \pi_j A_{jn}$ for each asset $n$. The number $\pi_j$ is the state price of $\omega_j$, in date-$0$ currency per unit of currency paid in state $\omega_j$. When the Arrow security that pays $1$ in state $\omega_j$ and $0$ in every other state is attainable, every state-price vector gives it the value $\pi_j$, which is its replication cost. The cash account forces $R \sum_j \pi_j = 1$, so the state prices add up to the discount factor $1/R$, not to $1$. A result proved in The fundamental theorem of asset pricing says that this finite market, which trades the cash account, has no arbitrage if and only if it has a state-price vector with $\pi_j > 0$ for every $j$.

Definition The replication system $A\theta = X$ and the state-price system $A^{\top}\pi = c$ form a dual pricing system. The first combines the columns of $A$ to build a payoff, and the second weights the rows of $A$ to reproduce the prices. For any solutions of the two systems,

So every state-price vector values every attainable claim at its replication cost. When $A$ is square and invertible, both systems have unique solutions, $\theta = A^{-1}X$ and $\pi = (A^{\top})^{-1}c$, and every claim gets the single value $\pi^{\top}X$. When $\operatorname{rank} A < K$ and the dual system has a solution, it has infinitely many, because $A^{\top}$ then has a nonzero null space. These solutions agree on every attainable claim, but they do not all agree on a claim outside $\operatorname{Col} A$, which therefore has no single price fixed by replication. That case is treated in Market completeness and nonunique prices.

Definition Let $\pi$ be a state-price vector and let $\mathbb{P}$ give every state a probability $p_j > 0$. The pricing kernel, also called the stochastic discount factor, is the random variable $m$ with $m(\omega_j) = \pi_j / p_j$. For every claim $X$,

so $V_0 = \mathbb{E}^{\mathbb{P}}[mX]$ for every attainable claim. Applied to the cash account this gives $\mathbb{E}^{\mathbb{P}}[m] = 1/R$, and therefore

The kernel depends on the market prices and on the chosen $\mathbb{P}$, it is positive in every state exactly when the state prices are, and the division needs $p_j > 0$ in every state. Discounting the physical mean payoff at the cash return gives the price of an attainable claim exactly when its covariance with $m$ is zero.