Arbitrage & the Law of One Price

Identical payoffs must carry identical prices, or free money exists.

The idea

Theorem (The law of one price).

Two positions that pay exactly the same amount in every possible future state must cost the same today.

A violation hands out a certain profit. Suppose the two payoffs agree state by state but one position is cheaper. Buy the cheap one and sell the dear one at the same instant: the sale funds the purchase, so no capital is committed, and the payoffs cancel in every state, so no outcome can hurt you. The price gap is a sure gain. A trade with no capital, no risk, and a certain profit is an arbitrage.

Every model in this subject assumes that arbitrages do not exist, and the assumption enforces itself. Because the trade ties up no money, traders take it in whatever size the market will absorb, and their buying of the cheap side and selling of the dear side pulls the two prices together until the gap closes.

The law's working use is replication. To price a claim nobody quotes, build a portfolio from assets whose prices you do know that matches the claim's payoff state by state; the claim must cost what that portfolio costs. The argument never uses the probability of any state, or anyone's forecast of which one occurs.

Ways to work on it

• Walkthrough. Spot a riskless profit, state the law of one price precisely, and price a claim by replication.
• Practice. Lock in riskless profits from mismatched quotes and price simple claims by replication.
• Hardest. Solve for the replicating portfolio in a two-state world and read off the claim's price.

Not sure where to start? Take the ten-question placement test.