Asymptotic Arbitrage Opportunities and Asset Market Equilibrium

In this paper, we show the reason why the absence of asymptotic arbitrage opportunities in the sense of convergence in quadratic mean (ACQM) as defined in Huberman (1982) is only a necessary condition for an asset market equilibrium. For certain classes of risk-averting investors, a portfolio that is not an ACQM may sometimes provide infinitely blissful gratification. These investors would relentlessly explore such a portfolio and cause market disequilibrium. Consequently, the APT that is based on Huberman’s concept of arbitrage is not a valid description of the no-arbitrage pricing relation as other types of asymptotic arbitrage opportunities may exist in the economy. To resolve this inconsistency, we replace Huberman’s concept of asymptotic arbitrage (convergence in quadratic mean) with that of convergence in probability. We show that if the idiosyncratic risks of the linear K-factor structure are weakly dependent (or, more precisely, the sequence of the idiosyncratic risks in a lacunary system of order p for some p >1), the absence of the arbitrage opportunities in the sense of convergence in probability (ACP) implies an approximate linear pricing relation which is consistent with an asset market equilibrium for a broader class of preferences. However, there are still circumstances in which the absence of the ACP is not compatible with the asset market equilibrium. Finally, we claim that the absence of "asymptotically exact" arbitrage opportunities is consistent with asset market equilibrium for all risk-averting investors.