Nonlinear Properties of Multifactor Financial Models

This paper provides a comprehensive analysis of the nonlinear properties of multifactor pricing models. Beginning with the generalized geometric Brownian motion, we develop a method whereby the log returns of a set of d-assets or portfolios admit a scale mixture model. This is followed by an analytical study on the conditional behavior of a subset of assets given another subset. Expressions for the first two conditional moments are provided under the scale mixture family. The regression equation when the scaling variable is constant (unity) corresponds with the renowned APT. Computable conditional moment expressions for the scaling variable are derived under both inverse gamma and gamma distributions. These moment equations are nonlinear in parameters, apart from containing the usual linear terms under the APT. We then apply the above nonlinear methodology to the log asset returns of four major companies in the U.S. stock market.