Geometry and Optimization of Relative Arbitrage

Thesis (Ph.D.)--University of Washington, 2016-06

This thesis is devoted to the mathematics of volatility harvesting, the idea that extra portfolio growth may be created by systematic rebalancing. First developed by E. R. Fernholz in the late 90s and the early 2000s, stochastic portfolio theory provides a novel mathematical framework to analyze this phenomenon. A major result of the theory is the construction of portfolio strategies that outperform the market portfolio under realistic conditions. These portfolios are called relative arbitrage opportunities. In this thesis we adopt a discrete time, pathwise approach which reveals deep connections with optimal transport, nonparametric statistics and information geometry. Our main object of study is functionally generated portfolio, a family of volatility harvesting investment strategies with remarkable properties. This thesis consists of three parts. Part I gives a convex-analytic treatment of functionally generated portfolio and relates it with optimal transport theory. The optimal transport point of view provides the geometric structure required in order that a portfolio map is volatility harvesting. Part II turns to optimization of functionally generated portfolio. We introduce an optimization problem analogous to shape-constrained maximum likelihood density estimation. The Bayesian version of this problem leads naturally to an extension of T. M. Cover's universal portfolio and large deviations. Finally, in Part III we introduce and study the information geometry of exponentially concave functions, a deep and elegant geometric framework underlying the ideas of Part I. It extends the dually flat geometry of Bregman divergence studied by S. Amari and others, leading to plenty of problems for further study.