Empty Promises and Arbitrage

Abstract

Analysis of absence of arbitrage normally ignores payoffs in states to which the agent assigns zero probability. We extend the fundamental theorem of asset pricing to the case of “no empty promises” in which the agent cannot promise arbitrarily large payments in some states. There is a superpositive pricing rule that can assign positive price to claims in zero probability states important to the market as well as assigning positive prices to claims in the states of positive probability. With continuous information arrival, no empty promises can be enforced by shutting down the agent’s subsequent investments once wealth hits zero. Dogmatic disagreements create arbitrage opportunities in competitive markets that are sufficiently complete. There may be someone in the economy who is certain (correctly or not on objective grounds) that gold prices will go up and someone else who believes there is a positive probability that gold prices will go down. The first person will have an arbitrage opportunity if at-the-money put options on gold have a positive price, while the second person will have an arbitrage opportunity if at-the-money put options on gold have a zero or negative price. Similarly, if some people are sure their favorite sport teams or horses will win but others are not so sure, any odds posted by a competitive bookmaker will imply arbitrage for one group or the other. 1 In practice, however, dogmatic differences in beliefs do not imply actual arbitrage that can be used to generate arbitrarily large profits because people have limited resources and cannot make “empty promises” of payments that exceed their own ability to pay in states the market cares about, even if those are states they personally believe to be impossible. The purpose of this article is to extend the study of the absence of arbitrage to situations in which no empty promises are permitted.