Browsing by Subject "Hedging"
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Item Essays on Universal Portfolios(2017-06) Garivaltis, AlexanderThis 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.Item Evaluating weather derivatives and crop insurance for farm production risk management in Southern Minnesota.(2011-11) Chung, WonhoAgriculture is one of the most weather sensitive industries and weatherrelated risks are a major source of crop production risk exposure. One method of hedging the risk exposure has been through the use of crop insurance. However, the crop insurance market suffers from several problems of asymmetric information and systemic weather risk. Without government subsidies or reinsurance crop insurers would have to pass the cost of bearing the risk exposures to farmers. The rising cost of the federal crop insurance program has been an incentive for the government to seek alternative ways to reduce the cost. Weather derivatives have been suggested as a potential risk management tool to solve the problems. Previous studies have shown that weather derivatives are an effective means of hedging agricultural production risk. Yet, it is unclear what role weather derivatives will play as a risk management tool compared with the existing federal crop insurance program. This study compares the hedging cost and effectiveness of weather options with those of crop insurance for soybean and corn production in four counties of southern Minnesota. We calculate weather option premium by using daily simulation method and compare hedging effectiveness by several risk indicators: certainty equivalence, risk premium, Sharpe ratio, and value at risk. Our results show that the hedging effectiveness of using weather options is limited at the farm level compared with crop insurance products. This is because weather options insure against adverse weather events causing damage at the county level, while crop insurance protects farmers against the loss of their crops directly at the farm level as well as at the county level. Thus, individual farmers will continue to use crop insurance with government subsidy for their production risk management. However, we observe that the hedging effectiveness of using weather options increases as the level of spatial aggregation increases from farm level to county level to four-county aggregate level. This implies that the government as a reinsurer can reduce idiosyncratic yield risk by aggregating the individual risk exposures at the county or higher level, and hedge the remaining systemic weather risk by purchasing weather options in the financial market. As a result, weather derivatives could be used by the government as a hedging tool to reduce the social cost of the federal crop insurance program, since the government currently does not hedge their risk exposures in the program. Against our expectation based on the conventional wisdom, geographic basis risk is not significant in hedging our local weather risk with non-local exchange market weather options based on Minneapolis. It is likely due to the fact that the Midwest area including Minnesota has relatively homogeneous (or less variable) weather conditions and crop yields across the counties compared to other U.S. regions. The result indicates that we can hedge local weather risk with non-local exchange market weather derivatives in southern Minnesota. However, it should be applied cautiously to other locations, crops, or other types of weather derivatives, considering spatial correlation of weather variables between a specific farm location and a weather index reference point.Item Infinite-Horizon Optimal Hedging Under Cone Constraints(Center for Economic Research, Department of Economics, University of Minnesota, 1999-01) Huang, Kevin XiaodongWe address the issue of hedging in infinite horizon markets with a type of constraints that the set of feasible portfolio holdings forms a convex cone. We show that the minimum cost of hedging a liability stream is equal to its largest present value with respect to admissible stochastic discount factors, thus can be determined without finding an optimal hedging strategy. We solve for an optimal hedging strategy by solving a sequence of independent one-period hedging problems. We apply the results to a variety of trading restrictions and also show how the admissible stochastic discount factors can be characterized.