Arbitrage Pricing Theory in Ergodic Markets

Traditional approaches to Arbitrage Pricing Theory (APT) propose a factor model, whereas empirical applications of APT nowadays are based on seemingly unrelated regression. I drop the factor model and assume only that the market is ergodic. This enables me to apply the theory of Hilbert spaces in a natural way. The expected return on any asset can always be approximated by an affine-linear function of its betas. We are even able to estimate the relative number of assets that violate the APT equation by observing the given expected returns and betas. The APT equation is essentially satisfied only if we do not omit any common risk whose market price differs from zero, provided there exists an arbitrarily large number of common risks for which the return equation is properly specified. I present a quite simple sufficient condition for the APT equation in its inexact form, and I show that the APT equation holds true in its exact form if and only if an equilibrium market is exhaustive. This means that the market participants must be able to replicate the betas and idiosyncratic risk of each asset by some strategy that diversifies away all approximation errors in the market