On portfolio construction through functional generation

Thesis

One of the main research questions in financial mathematics is that of portfolio construction: how should one systematically invest their wealth in a financial market? This problem has been tackled in numerous ways, typically through the modeling of market prices and the optimization of an investment objective. A recent approach to portfolio construction is that offered by Stochastic Portfolio Theory, in which a relatively general market model is assumed, and the portfolio selection criterion is to outperform a benchmark with probability one. In order to achieve this, Robert Fernholz developed the method of functional generation, which allows one to explicitly construct and study portfolios that depend deterministically on the currently observable prices.

The typical example of such a strategy is the diversity-weighted portfolio, which we extend in the first chapter of this work with a negative-parameter variation. We show that several modifications of this portfolio outperform the market index in theory, under certain assumptions on the market, and we perform an empirical study that confirms this.

In our second chapter, we develop a data-driven portfolio construction method that goes beyond functional generation, allowing for the inclusion of factors other than current prices. We empirically show that this Bayesian nonparametric approach, which utilizes Gaussian processes, leads to drastically improved performance compared to benchmark portfolios.

Next, we establish a formal equivalence between the method of functional generation and the mathematical field of optimal transport. Our results fortify known relations between the two, and extend this connection to additive functional generation, a recent variation of the method.

In Chapter 4, we apply our results to derive new properties and characterizations of functionally-generated wealth processes in very general market models. Finally, we develop methods for incorporating defaults into functional generation, improving its real-world implementability.

Authors: Vervuurt, A

• DOI: 10.5287/ora-2nxpedoqw
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Financial mathematics; Probability theory; Stochastic portfolio theory; Gaussian processes; Optimal transport
• Subjects: Mathematics