A General Theory of Arbitrage Pricing: When the Idiosyncratic Risks are Dependent and their Second Moments Do Not Exist
In this paper, we generalize the Arbitrage Pricing Theory (APT) to incorporate the cases where the idiosyncratic risks of the factor model are dependent and/or the second central absolute moments (variances) of the assets returns do not exist. A bound on the pricing errors, similar to the one derived in Ross (1976) and Huberman (1982), is derived in our generalized framework. Specifically, it is shown that as long as the idiosyncratic risks are weakly dependent (or when the sequence of the idiosyncratic risks is a lacunary system), the approximate linear pricing relation holds in the absence of "arbitrage" in the sense of convergence in pth mean (ACPM). It can be demonstrated that the models in Huberman (1982), Ingersoll (1984) and Chamberlain and Rotschild (1983) are all special cases of this version of the APT. It is also established that, under suitable assumptions of the linear factor structure, the approximate linear pricing relation implies the nonexistence of asymptotic arbitrage opportunities. Thus the no-asymptotic-arbitrage position is a necessary and sufficient condition for the approximate linear pricing relation.
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