Asymptotic Arbitrage Opportunities in Various Modes of Convergence and the Approximate Linear Pricing Relation in Asset Market

The three basic elements of the arbitrage pricing theory (APT) are the linear factor structure of asset returns, the nonexistence of asymptotic arbitrage opportunities, and the approximate linear pricing relation. This paper explores the necessary and sufficient conditions of the approximate linear pricing relation by systematically examining the associations among these three elements. The generalization evolves around various modes of stochastic convergence that characterize the nature of asymptotic arbitrage opportunities and around assorted assumptions about the idiosyncratic risks in the linear factor structure. This study is exhaustive in the sense that all modes of convergence are used in defining the asymptotic arbitrage opportunities. This study also allows researchers to know the trade-off between the linear factor structure and the no-asymptotic-arbitrage condition while keeping the approximate linear relation intact. Our generalization of the APT may enhance the understanding about the arbitrage pricing mechanism and the stochastic nature of the underlying economy.