Growth Optimization in Stochastic Portfolio Theory with Applications to Robust Finance and Open Markets
Stochastic portfolio theory (SPT) is a financial framework with a large number d of stocks and the goal of modelling equity markets over long time horizons. This thesis concerns the study of growth optimization problems in the context of SPT in robust and constrained settings.
In Part I of the thesis we consider the problem of maximizing the asymptotic growth rate of an investor under drift uncertainty. As in the work of Kardaras and Robertson  we take as inputs (i) a Markovian volatility matrix c(x) and (ii) an invariant density p(x) for the market weights, but we additionally impose long-only constraints on the investor. Our principal contribution is proving a uniqueness and existence result for the class of concave functionally generated portfolios and developing a finite dimensional approximation, which can be used to numerically find the optimum. In addition to the general results outlined above, we propose the use of a broad class of models for the volatility matrix c(x), which can be calibrated to data and, under which, we obtain explicit formulas of the optimal unconstrained portfolio for any invariant density.
In Part II we propose a unified approach to several problems in SPT. Our approach combines open markets, where trading is constrained to the top N capitalized stocks as well as the market portfolio consisting of all d assets, with a parametric family of models which we call hybrid Jacobi processes. We provide a detailed analysis of ergodicity, particle collisions, and boundary attainment, and use these results to study the associated financial markets. Their properties include (1) stability of the capital distribution curve and (2) unleveraged and explicit growth optimal strategies. The sub-class of rank Jacobi models are additionally shown to (3) serve as the worst-case model for a robust asymptotic growth problem under model ambiguity and (4) exhibit stability in the large-d limit. Our definition of an open market is a relaxation of existing definitions which is essential to make the analysis tractable.
- Mathematical Sciences
- Doctor of Philosophy (PhD)