Exact arbitrage, well-diversified portfolios and asset pricing in large markets

For a market with an atomless continuum of assets, we formulate the intuitive idea of a “well-diversified” portfolio, and present a notion of “exact arbitrage”, strictly weaker than the more conventional notion of “asymptotic arbitrage”, and necessary and sufficient for the validity of an APT pricing formula. Our formula involves “essential” risk, one based on a specific index portfolio constructed from factors and factor loadings that are endogenously extracted to satisfy an optimality property involving a finite number of factors. We illustrate how our results can be translated to markets with a large but finite number of assets.