Market-Consistent Prices for Replicable Payoffs

One of the key tenets pervading much of mathematical finance is that it should not be possible to make a riskless profit, i.e., that a potential gain should always be balanced by a potential loss. There are two ways to ensure this. The first way is by prescribing the Law of One Price, which requires that all portfolios that replicate the same payoff have the same price. Indeed, agents could otherwise make an instantaneous riskless profit by short-selling the more expensive portfolio and buying the cheaper one. In a market where the Law of One Price holds it is possible to assign to every replicable payoff a market-consistent price in an unambiguous way. The resulting pricing rule is a linear functional defined on the marketed space. The second way of precluding riskless profits is by assuming the absence of arbitrage opportunities, i.e., the absence of portfolios with a nonzero positive payoff that have a nonpositive price. The absence of arbitrage opportunities is a stronger requirement than the Law of One Price and is equivalent to the strict positivity of the pricing rule.