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Item Essays on Universal Portfolios(2017-06) Garivaltis, AlexanderShow more This thesis has three chapters. Chapter 1 concentrates on a family of sequential portfolio selection algorithms called multilinear trading strategies. A multilinear strategy is characterized by the fact that its final wealth is linear separately in each period’s gross-return vector for the stock market. These strategies are simple, intuitive, and general enough for many purposes — and yet they retain a basic level of analytic and computational tractability. Thus, instead of the usual method of specifying his portfolio vector each period as a function of the return history, a trader can proceed differently. Rather, he selects a desired final wealth function (which, however, must be feasible) and works backward to recover the implied trading strategy. I show that the class of multilinear strategies is general enough for superhedging derivatives in discrete time. A superhedge for a derivative D is a self-financing trading strategy that guarantees to generate cash flows greater than or equal to those of the derivative in any outcome. In dominating D by a multilinear final wealth function, one is able to put upper bounds on the no-arbitrage price of D. This is relevant to realistic trading environments, which are hampered by transaction costs and the impossibility of continuous-time trading. Superhedging is a possible solution: the cost of the cheapest superhedge for D amounts to the greatest possible (model-independent) rational price for the derivative. Multilinear super-hedging amounts to interpolating D with a multilinear payoff, and then dynamically replicating the interpolating form. If D is a convex function separately of each period’s return vector, then there is a multilinear superhedge that is cheaper than any other (multilinear or not). For this reason, I give a detailed guide to the practical computation of multilinear strategies. The key requirement for tractibility is that the form (or derivative) be symmetric in the sense that its final wealth depend only on the numerical magnitudes of the return vectors x t , and not their order. For example, if the daily returns of the U.S. stock market before, during, and after the crash of 1929 were re-ordered in some way, the final wealth of a symmetric multilinear strategy would not have been affected. iv Chapter 1 concludes with an extensive study of the high-water mark of Cover’s theory of “universal portfolios.” Universal portfolios are best understood as superhedges (of varying efficiency) of a specific fictitious “lookback” derivative. The idea is this: a trader imagines a derivative D whose payoff represents the final wealth of a non-causal trading strategy, e.g. a trading strategy whose activities at t are in some way a function of the future path of stock prices. In the manner of Biff’s sports almanac, the payoff D has been rigged to “beat the market” by a significant margin. Obviously, the trader himself cannot use such a strategy: his behavior can be conditioned on the past, but not the future. However, what he can do is try to superhedge D. Cover found (1986, 1991, 1996, 1998) that D could be chosen so as to generate superhedges that (under some tacit restrictions on market behavior) de facto “beat the market asymptotically.” Any reasonably efficient superhedging strategy for this derivative will enjoy the asymptotic optimality property, and it turns out that there is a large collection of such strategies. The chapter then turns its attention to the question of just how long it takes to reach the asymptote, and what the practical consequences are of increasing the trading frequency. Chapter 2 studies a family of superhedging and trading strategies that are opti- mal from the standpoint of sequential minimax. The concept is that, given a path dependent-derivative, a multilinear superhedge (even the cheapest one) that was con- ceived at t = 0 will not necessarily make credible choices for all variations of market behavior. As the path of stock prices is slowly revealed to the trader, it (in everyday cases) becomes apparent that actual cost of superhedging will ultimately prove to be much lower than originally thought. This phenomenon is the result of the fact that su- perhedging ultimately hinges upon planning for a set of worst-case scenarios, albeit ones that will rarely occur in practice. When these worst cases fail to actually materialize, it has irrevocable consequences for the final payoff of the path-dependent derivative. A sophisticated superhedging strategy will exploit this to dynamically reduce the hedging cost. Instead of approximating D by a multilinear form and then hedging the approxima- tion, I explicitly calculate a backward induction solution from the end of the investment horizon. The superhedging strategies so-derived are the sharpest possible in all vari- ations. Universal portfolios are the major impetus for the technique, the point being to dynamically reduce the time needed to beat the market asymptotically. In addition v to their greater robustness, the sequential minimax trading strategies derived in the chapter are easier to calculate and implement than multilinear superhedges. This being done, I extend the trading model to account for leverage and a priori linear restrictions on the daily return vector in the stock market. In deriving a strategy that is robust to a smaller, more reasonable set of outcomes, the trader is able to use leverage in a reliable and perspicacious manner. In the sharpened model, the linear restrictions serve to nar- row the set of nature’s choices, while simulateneously allowing the trader the privilege of a richer set of (leveraged) strategies. To be specific, nature is required to choose the stock market’s return vector from a given cone, and the trader is allowed to pick any admissible (non-bankruptable) portfolio from the dual cone. a fortiori, this dynamic is guaranteed to increase the superhedging efficiency, sometimes substantially. This point is illustrated with many numerical examples. Again, the chapter studies the extent to which this trick reduces the time needed to beat the market. Chapter 2 concludes with a sequential minimax version of Cover’s (1996) universal portfolio with side information. In this environment, a discrete-valued signal (the “side information”) is available to the trader prior to each period’s trading session. The trader starts the game in total ignorance of the meaning of the signal, and he strives to interpret it in the most robust way possible. I provide a universal portfolio under “adversarial” signals whose performance guarantees are a significant refinement to those in Cover (1996). The idea is that a trader, making use of side information, should come to fear the possibility that nature chooses the signal maliciously, intending to create dynamic confusion vis-a-vis the exact meaning of the signal. This meaning is only ever revealed in hindsight, and the trader comes to regret the fact that he was ignorant of the most profitable interpretation of the signal. The trader plays to minimize this regret in the worst case. On account of the complicated environment, the implied optimum trading strategy is only practically computable for horizons on the order of 10-20 periods, and thus is suitable chiefly as, say, an annual trading model. Chapter 3 is a comprehensive study of universal sequential betting schemes, where the bets are placed on the outcomes of discrete events (colloquially called “horse races”). The Kelly horse race markets studied in the chapter get at the essential features that drive both the multilinear and sequential minimax universal portfolios. The chapter discusses the manner in which these two strategies particularize to one and the same vi thing under the Kelly horse race. In this connection, the two strategies just amount to the universal source code of Shtarkov (1987), suitably reinterpreted. The sharp performance of the minimax strategy is then compared to the horizon-free strategies that result from particularizing the “Dirichlet-weighted” (1996) universal portfolios and the “Empirical Bayes” (1986) portfolio. Careful attention is given to on-line computation of the universal bets, and several numerical visualizations and simulations are provided. The chapter ends with a sequential minimax refinement to the empirical Bayes stock portfolio. Whereas Cover (1986) is a direct instantiation of Blackwell’s (1956) geometric method for approaching a set of vector payoffs, the sequential minimax approach studied here is, on a fixed horizon, the most robust possible strategy for approaching the set. For convenient reference, a glossary of concepts and notation is given at the end of the thesis.Show more