Arbitrage and the law of one price

Definition Consider a one-period market with finite state space $\Omega = \{\omega_1, \ldots, \omega_m\}$ in which every state has positive physical probability, $\mathbb{P}(\omega_j) > 0$. There are $N$ traded instruments with time-$0$ price vector $c \in \mathbb{R}^N$ and terminal payoff matrix $A \in \mathbb{R}^{m \times N}$, whose entry $A_{ji}$ is the payment of instrument $i$ in state $\omega_j$, so states are rows and instruments are columns. A holdings vector $\theta \in \mathbb{R}^N$ costs $c^{\top}\theta$ at time $0$ and pays $A\theta$ at $T$. Unless a constraint is stated, every real $\theta$ is admissible, so borrowing and short selling are allowed at the quoted prices, with no fees, spreads or counterparty default. An arbitrage is an admissible $\theta$ with

and in addition

where inequalities between vectors hold entry by entry. So an arbitrage costs at most zero, never requires a payment at $T$, and either brings in money today or pays a positive amount in some state, which has positive probability. When a cash instrument with a positive price and a positive payment in every state is traded, investing the money received today in it turns a strategy of the first kind into one with zero cost and a payoff that is nonnegative in every state and positive in some state. The market is free of arbitrage when no admissible $\theta$ is an arbitrage.

Definition A market satisfies the law of one price if any two admissible portfolios with the same payoff in every state have the same cost,

In the frictionless one-period market of the previous card, no arbitrage implies the law of one price. To prove it, suppose $A\theta = A\phi$ and $c^{\top}\theta < c^{\top}\phi$. The strategy $\theta - \phi$ buys the cheaper portfolio and sells the dearer one. It costs $c^{\top}\theta - c^{\top}\phi < 0$ and pays $A\theta - A\phi = 0$ in every state, which is an arbitrage. The proof needs $\theta - \phi$ to be admissible, so it uses the freedom to sell. The converse fails, since the law of one price only compares portfolios with equal payoffs and says nothing about a portfolio that costs nothing and has a nonnegative, nonzero payoff.

Definition A payoff $X \in \mathbb{R}^m$ is attainable, or replicable, if some admissible $\theta$ satisfies $A\theta = X$, and any such $\theta$ is a replicating portfolio for $X$. Replication must match $X$ in every state, not only on average or in one scenario. If the law of one price holds, all replicating portfolios of $X$ have the same cost, so the price

belongs to the payoff and not to a particular portfolio. If a claim with payoff $X$ is also quoted at a price other than $\Pi(X)$, buying the cheaper of the claim and its replicating portfolio and selling the other is an arbitrage, provided both trades are admissible.

Definition The attainable payoffs form the subspace $M = \{A\theta : \theta \in \mathbb{R}^N\}$ of $\mathbb{R}^m$, the column space of $A$. Under the law of one price the price map $\Pi$ on $M$ is linear. If $A\theta = X$ and $A\phi = Y$, then $A(a\theta + b\phi) = aX + bY$ for all real $a$ and $b$, so

Linearity uses that holdings can be scaled by any real number and added, and that each instrument is bought and sold at one price. A stronger statement is the finite version of the fundamental theorem of asset pricing, stated here and proved later in the course. In this market, with every state of positive probability, there is no arbitrage if and only if there is a state-price vector $\pi \in \mathbb{R}^m$ with

Then $\Pi(X) = \pi^{\top}X$ for every attainable $X$, the strictly positive linear valuation operator of Lesson 1. When $M = \mathbb{R}^m$ the state prices are unique. Otherwise several state-price vectors price the traded instruments, and all of them give the same price to each attainable payoff.

Definition A payoff $X$ dominates a payoff $Y$ if $X(\omega_j) \ge Y(\omega_j)$ in every state, and strictly dominates it if in addition $X(\omega_j) > Y(\omega_j)$ in at least one state. In the frictionless one-period market without arbitrage, for attainable $X$ and $Y$,

and strict dominance gives $\Pi(X) > \Pi(Y)$. To prove it, suppose $\Pi(X) < \Pi(Y)$. Buying a replicating portfolio of $X$ and selling one of $Y$ costs $\Pi(X) - \Pi(Y) < 0$ and pays $X - Y \ge 0$, an arbitrage. If $X$ strictly dominates $Y$ and $\Pi(X) \le \Pi(Y)$, the same trade costs at most zero and pays $X - Y$, which is nonnegative and positive in a state of positive probability, again an arbitrage. Taking $Y = 0$ shows that an attainable payoff that is nonnegative and not zero has a positive price, the order property that strictly positive state prices express.

Definition A short-selling constraint forbids negative holdings in some instruments, so the admissible set becomes $\{\theta : \theta_i \ge 0 \text{ for each constrained } i\}$ instead of all of $\mathbb{R}^N$, and the definition of arbitrage applies only to admissible strategies. Borrowing is a negative holding of the cash instrument, so a borrowing limit is a constraint of the same kind. The proofs of the law of one price, of linear pricing and of dominance all sell a portfolio, so under a constraint their conclusions can fail in one direction, and no arbitrage then gives inequalities instead of equalities. Fees and bid-ask spreads have a similar effect, because they make the buying price and the selling price differ. The arithmetic of a strategy stays correct under a constraint. What changes is whether the strategy is allowed.