We argue that arbitrage-pricing theories (APT) imply the existence of a low-dimensional nonnegative nonlinear pricing kernel. In contrast to standard constructs of the APT, we do not assume a linear factor structure on the payoffs. This allows us to price both primitive and derivative securities. Semi-nonparametric techniques are used to estimate the pricing kernel and test the theory. Empirical results using size-based portfolio returns and yields on bonds reject the nested capital asset-pricing model and linear APT and support the nonlinear APT. Diagnostics show that the nonlinear model is more capable of explaining variations in small firm returns.