ARBITRAGE PRICING THEORY IN ERGODIC MARKETS

Abstract

Traditional approaches to Arbitrage Pricing Theory (APT) propose a factor model, but empirical applications of APT are, nowadays, 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 and we are able to estimate the relative number of assets that violate the APT equation by taking the expected returns and betas in the market into account. I present a simple sufficient condition for the APT equation in its inexact form. Further, I show that the APT equation holds true in its exact form if and only if an equilibrium market is exhaustive, which means that it must be possible to replicate the betas and idiosyncratic risk of each asset by some strategy that diversifies away all approximation errors in the market.