Robert Fernholz

Stochastic Portfolio Theory: an Overview

Abstract Stochastic Portfolio Theory is a flexible framework for analyzing portfolio behavior and equity market structure. This theory was introduced by Fernholz in the papers ( Journal of Mathematical Economics , 1999; Finance & Stochastics , 2001) and in the monograph Stochastic Portfolio Theory (Springer 2002). It was further developed by Fernholz , Karatzas and Kardaras ( Finance & Stochastics , 2005), Fernholz & Karatzas ( Annals of Finance , 2005), Banner , Fernholz and Karatzas ( Annals of Applied Probability , 2005), and Karatzas and Kardaras ( Finance & Stochastics , 2007). This theory is descriptive, as opposed to normative; it is consistent with observable characteristics of actual portfolios and markets, and it provides a theoretical tool which is useful for practical applications. As a theoretical tool, this framework offers fresh insights into questions of stock market structure and arbitrage, and can be used to construct portfolios with controlled behavior. As a practical tool, stochastic portfolio theory has been applied to the analysis and optimization of portfolio performance and has been the basis of successful investment strategies for over a decade.

Atlas models of equity markets

Atlas-type models are constant-parameter models of uncorrelated stocks for equity markets with a stable capital distribution, in which the growth rates and variances depend on rank. The simplest such model assigns the same, constant variance to all stocks; zero rate of growth to all stocks but the smallest; and positive growth rate to the smallest, the Atlas stock. In this paper we study the basic properties of this class of models, as well as the behavior of various portfolios in their midst. Of particular interest are portfolios that do not contain the Atlas stock.

Hybrid Atlas models

We study Atlas-type models of equity markets with local characteristics that depend on both name and rank, and in ways that induce a stable capital distribution. Ergodic properties and rankings of processes are examined with reference to the theory of reflected Brownian motions in polyhedral domains. In the context of such models we discuss properties of various investment strategies, including the so-called growth-optimal and universal portfolios.

On the diversity of equity markets

Abstract This is a study of the diversity of the distribution of capital in an equity market composed of stocks represented by continuous semimartingales. It is shown that for such a market, diversity is not a natural state and some mechanism such as dividend payments is needed to maintain it. Entropy is introduced as a measure of diversity and is used to study the compatibility of market diversity with capital market equilibrium. It is shown that dividend payments must be available to recycle capital from larger companies to smaller companies in order to maintain market diversity under conditions of capital market equilibrium.