| 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 optimal from the standpoint of sequential minimax. The concept is that, given a path dependent-derivative, a multilinear superhedge (even the cheapest one) that was conceived 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 superhedging 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 approximation, 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 variations. 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 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 narrow 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. (Abstract shortened by ProQuest.). |
