Empty Promises and Arbitrage

Abstract

Analysis of absence of arbitrage normally ignores payoffs in a set of states to which the agent assigns zero probability. However, bankruptcy and limited resources make it impossible for a trader to make large promises in a state the trader believes to be impossible if the market considers that state important. The Fundamental Theorem of Asset Pricing states the equivalence of the absence of arbitrage, existence of a positive linear pricing rule, and existence of optimal demand for some agent who prefers more to less. This paper extends the Fundamental Theorem to the case of "no empty promises" in which the agent cannot promise the market arbitrarily large payments in some states. The main difference is that there is a "super-positive" pricing rule which can assign positive price to claims in zero probability states important to the market as well as assigning positive price to positive claims in the states of positive probability. With some additional structure in continuous time, the no-empty-promises condition can be enforced as a non-negative wealth constraint pathwise under the agent's beliefs.

JEL Classification: G12