On the Topic of Portfolio Optimization

This dissertation explores the use of information entropy as a risk measure for the purpose of investment portfolio optimization and selection. First, we present an improved method of applying entropy as a risk in portfolio optimization. A new family of portfolio optimization problems called the return-entropy portfolio optimization (REPO) is introduced that simplifies the computation of portfolio entropy using a combinatorial approach. REPO addresses five main practical concerns with traditional mean-variance portfolio optimization (MVPO), by using a mean-entropy objective function instead of the mean-variance objective function used in MVPO. REPO also simplifies the portfolio entropy calculation by utilizing combinatorial generating functions in the optimization objective function. Next, we extend the REPO approach to the optimization problem for assets with discrete distributed returns, such as those from a Bernoulli distribution like binary options. Under a discrete probability distribution, portfolios of binary options can be viewed as repeated short-term investments with an optimal buy/sell strategy or general betting strategy. Portfolio selection under this setting can be formulated as a new optimization problem called discrete entropic portfolio optimization (DEPO). DEPO creates optimal portfolios for discrete return assets based on expected growth rate and relative entropy. We show how a portfolio of binary options provides an ideal general setting for this kind of portfolio selection. Finally, we introduce a further generalized portfolio selection method called generalized entropic portfolio optimization (GEPO). GEPO extends DEPO to include intervals of continuous returns, with direct application to a wide range of options strategies. This lays the groundwork for an adaptable optimization framework that can accommodate a wealth of option portfolios. These options strategies exhibit mixed returns: a combination of discrete and continuous returns with performance best measured by portfolio growth rate, making entropic portfolio optimization an ideal method for option portfolio selection. GEPO provides the mathematical tools to select efficient option portfolios based on their growth rate and relative entropy.