Asymptotic arbitrage and the APT with or without measure-theoretic structures

Abstract

We present a version of the APT based on an asset index set of an arbitrary infinite cardinality. Under assumptions due to Ross (1976, J. Econ. Theory 13, 341-360) and Chamberlain and Rothschild (1983, Econometrica 51, 1281-1303), we show that, in the absence of gains from asymptotic arbitrage, the square of the deviations of the individual rates of return from a factor-pricing formula sum to a finite number and that this absence, while sufficient, is not necessary for the formula to hold. We relate these results to recent work and explain, in particular, how a version of the APT exhibits several inconsistencies when the index set is the Lebesgue unit interval. Journal of Economic Literature Classification Numbers: G12, C60. © 2001 Academic Press.