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Theses Doctoral

Topics in Stochastic Portfolio Theory: Pathwise Generation of Trading Strategies, and Portfolio Theory in Open Markets

Kim, Donghan

This thesis generalizes stochastic portfolio theory in two different aspects. The first part demonstrates the functional generation of portfolios in a pathwise way. This notion of functional generation of portfolios was first introduced by E.R. Fernholz, to construct a variety of portfolios solely in the terms of the individual companies' market weights. I. Karatzas and J. Ruf developed recently another approach to the functional construction of portfolios, which leads to very simple conditions for strong relative arbitrage with respect to the market. Both of these notions of functional portfolio generation are generalized in a pathwise, probability-free setting; portfolio generating functions, possibly less smooth than twice-differentiable, involve the current market weights, as well as additional bounded-variation functionals of past and present market weights. This generalization leads to a wider class of functionally-generated portfolios than was heretofore possible to analyze, and to improved conditions for outperforming the market portfolio over suitable time-horizons.

The second part develops portfolio theory in open markets. An open market is a subset of the entire equity market, composed of a certain fixed number of top-capitalization stocks. Though the number of stocks in open market is fixed, the constituents of the market change over time as each company's rank by its market capitalization fluctuates. When one is allowed to invest also in money market, an open market resembles the entire 'closed' equity market in the sense that most of the results that are valid for the entire market, continue to hold when investment is restricted to the open market. One of these results is the equivalence of market viability (lack of arbitrage) and the existence of num\'eraire portfolio (portfolio which cannot be outperformed). When access to the money market is prohibited, the class of portfolios shrinks significantly in open markets. In such a case, we discuss the Capital Asset Pricing Model, how to construct functionally-generated portfolios, and the concept of universal portfolio in open market setting.

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More About This Work

Academic Units
Mathematics
Thesis Advisors
Karatzas, Ioannis
Degree
Ph.D., Columbia University
Published Here
July 13, 2020