Hyperfinite Asset Pricing Theory
We present a model of a financial market which unifies the capital-asset-pricing model (CAPM) of Sharpe-Lintner, and the arbitrage pricing theory (APT) of Ross. The model is based on a recent theory of hyperfinite processes, and it uncovers asset pricing phenomena which cannot be treated by classical methods, and whose asymptotic counterparts are not already, or even readily, apparent in the setting of a large but finite number of assets. In the model, an asset's unexpected return can be decomposed into a systematic and an unsystematic part, as in the APT, and the systematic part further decomposed leads to a pricing formula expressed in terms of a beta that is based on a specific index portfolio identifying essential risk, and constructed from factors and factor loadings that are endogenously extracted from the process of asset returns. Furthermore, the valuation formulas of the two individual theories imply, and are implied by, the pervasive economic principle of no arbitrage. Explicit formulas for the characterization, as well as conditions for the existence, of important portfolios are furnished. The hyperfinite factor model possesses an optimality property which justifies the use of a relatively small number of factors to describe the relevant correlational structures. The asymptotic implementability of the idealized limit model is illustrated by an interpretation of selected results for the large but finite setting.
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