Essays on the theory of arbitrage pricing

This dissertation consists of five essays on the theory of arbitrage pricing. The first essay derives the APT when the second central absolute moments (variances) of the assets returns do not exist. A bound on the pricing errors is obtained. This bound is similar to the bound obtained by Ross (1976) and Huberman (1982). More generally, it is shown that when idiosyncratic risks are not strongly dependent (or when the sequence of the idiosyncratic risks is a lacunary system), the approximate linear pricing relation still holds. It is demonstrated that the models in Huberman (1982), Ingersoll (1984) and Chamberlain and Rothschild (1983) are all special cases of this one. It is also proved that, under suitable assumptions on the linear factor structure, the 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 relation. Finally, It is established that the APT is still correct after any linear transformation as long as the product of the transformation matrix and its transpose is uniformly bounded. The second essay gives some examples which demonstrate that the portfolios satisfying the condition of no asymptotic arbitrage opportunities in the usual sense may actually create an infinite utility level for some utility-maximizer. The third essay examines the APT when differing definitions of the no-asymptotic-arbitrage and various assumptions about the idiosyncratic risks are used. This whole exercise is necessary since different definitions of the no-asymptotic-arbitrage condition usually work for different types of utility functions which is the main result of the second essay. In a Banach space setting, the fourth essay establishes the approximate linear pricing relation without resorting to notions other than linearity and bounded residual risks. The fifth essay studies the relations among mean-dispersion efficiency, the exact linear pricing, and mutual fund separation. The definition of dispersion depends on the context we are interested in. The relation between mimicking portfolios and exact arbitrage is also investigated in a way similar to Huberman, Kandel, and Stambaugh (1987)