Context dependence and consistency in dynamic choice under uncertainty: the case of anticipated regret

We examine if and to what extent choice dispositions can allow dependence on contexts and maintain consistency over time, in a dynamic environment under uncertainty. We focus on a 'minimal' case of context dependence, opportunity dependence due to being affected by anticipated regret. There are two sources of potential inconsistency, one is arrival of information and the other is changing opportunities. First, we go over the general method of resolution of potential inconsistency, by taking any kinds of inconsistency as given constraints. Second, we characterize a class of choice dispositions that are consistent to information arrival but may be inconsistent to changing opportunities. Finally, we consider the full requirement of dynamic consistency and show that it necessarily implies independence of choice opportunities.

Essays on Universal Portfolios

University of Minnesota Ph.D. dissertation.June 2017. Major: Economics. Advisor: Aldo Rustichini. 1 computer file (PDF); xiii, 87 pages.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

Universal Codes from Switching Strategies

We discuss algorithms for combining sequential prediction strategies, a task which can be viewed as a natural generalisation of the concept of universal coding. We describe a graphical language based on Hidden Markov Models for defining prediction strategies, and we provide both existing and new models as examples. The models include efficient, parameterless models for switching between the input strategies over time, including a model for the case where switches tend to occur in clusters, and finally a new model for the scenario where the prediction strategies have a known relationship, and where jumps are typically between strongly related ones. This last model is relevant for coding time series data where parameter drift is expected. As theoretical ontributions we introduce an interpolation construction that is useful in the development and analysis of new algorithms, and we establish a new sophisticated lemma for analysing the individual sequence regret of parameterised models

Essays on Individual Decision Making under Risk and Uncertainty

This thesis aims to empirically test the validity of economic theories of the individual decision making under risk and uncertainty with a laboratory experiment. The first chapter outlines this thesis. The second chapter experimentally tests Manski's theory of satisficing (2017). He proposes solutions to two key questions: when should the decision-maker (DM) satisfice?; and how should the DM satisfice? The results show that some of Manski's proposition (those relating to the “how”) appear to be empirically valid while others (those relating to the “when”) are less so. The third chapter extends the findings from the previous chapter, mainly relating to “how to satisfice”. I propose an alternative story with a different assumption of the subjects' preference functional and of the payoff distribution. The results suggest that my alternative story appears to better-explain the subjects' behaviour than that of Manski's story. The fourth chapter explores the individual behaviour towards randomisation of the choice. I use the elicitation method that provides an additional option between two alternatives, namely “I am not sure what to choose” as an alternative of two standard options: "I choose A" or "I choose B". It gives a consequence where the subjects' payoff is determined by a randomisation of two alternatives through the flipping a coin. I propose four stories to account for the choice of this option. The results show that the most of the subjects either have strictly convex preferences with random risk attitude or simply cannot distinguish the two alternatives. The fifth chapter empirically tests Nicolosi's model (2018). He derives the optimal strategy for the fund manager under a specific payment contract and the investment environment. I compare his model with other strategies. The results show that Nicolosi's model receives strong empirical support to explain the subjects' behaviour

Minimum Description Length Model Selection - Problems and Extensions

The thesis treats a number of open problems in Minimum Description Length model selection, especially prediction problems. It is shown how techniques from the "Prediction with Expert Advice" literature can be used to improve model selection performance, which is particularly useful in nonparametric settings

Higher moment models for risk and portfolio management

This thesis considers specific topics related to the dynamic modelling and management of risk, with a particular emphasis on the generation of asymmetric and fat tailed behavior observed in practise. Specifically, extensions to the dynamics of the popular GARCH model, to capture time variation in higher moments, are considered in the univariate and multivariate context, with a special focus on the Generalized Hyperbolic distribution. In Chapter 1, I consider the extension of univariate GARCH processes with higher moment dynamics based on the Autoregressive Conditional Density model of Hansen (1994), with conditional distribution the Generalized Hyperbolic. The value of such dynamics are analyzed in the context of risk management, and the question of ignoring them discussed. In Chapter 2, I review some popular multivariate GARCH models with a particular emphasis on the dynamic correlation model of Engle (2002), and alternative distributions such those from the Generalized Asymmetric Laplace of Kotz, Kozubowski, and Podgorski (2001). In Chapter 3, I propose a multivariate extension to the Autoregressive Conditional Density model via the independence framework of the Generalized Orthogonal GARCH models, providing the first feasible model for large dimensional multivariate modelling of time varying higher moments. A comprehensive out-of- sample risk and portfolio management application provides strong evidence of the improvement over non time varying higher moments. Finally, in Chapter 4, I consider the benefits of active investing when the benchmark index is not optimally weighted. I investigate advances in the definition and use of risk measures in portfolio allocation, and propose certain simple solutions to challenges arising in the optimization of these measures. Combining the models discussed in the previous chapters, within a fractional programming optimization framework and using a range of popular risk measures, a large scale out-of-sample portfolio application on the point in time constituents of the Dow Jones Industrial Average is presented and discussed, with clear implications for active investing and benchmark policy choice

Antecipação na tomada de decisão com múltiplos critérios sob incerteza

Orientador: Fernando José Von ZubenTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: A presença de incerteza em resultados futuros pode levar a indecisões em processos de escolha, especialmente ao elicitar as importâncias relativas de múltiplos critérios de decisão e de desempenhos de curto vs. longo prazo. Algumas decisões, no entanto, devem ser tomadas sob informação incompleta, o que pode resultar em ações precipitadas com consequências imprevisíveis. Quando uma solução deve ser selecionada sob vários pontos de vista conflitantes para operar em ambientes ruidosos e variantes no tempo, implementar alternativas provisórias flexíveis pode ser fundamental para contornar a falta de informação completa, mantendo opções futuras em aberto. A engenharia antecipatória pode então ser considerada como a estratégia de conceber soluções flexíveis as quais permitem aos tomadores de decisão responder de forma robusta a cenários imprevisíveis. Essa estratégia pode, assim, mitigar os riscos de, sem intenção, se comprometer fortemente a alternativas incertas, ao mesmo tempo em que aumenta a adaptabilidade às mudanças futuras. Nesta tese, os papéis da antecipação e da flexibilidade na automação de processos de tomada de decisão sequencial com múltiplos critérios sob incerteza é investigado. O dilema de atribuir importâncias relativas aos critérios de decisão e a recompensas imediatas sob informação incompleta é então tratado pela antecipação autônoma de decisões flexíveis capazes de preservar ao máximo a diversidade de escolhas futuras. Uma metodologia de aprendizagem antecipatória on-line é então proposta para melhorar a variedade e qualidade dos conjuntos futuros de soluções de trade-off. Esse objetivo é alcançado por meio da previsão de conjuntos de máximo hipervolume esperado, para a qual as capacidades de antecipação de metaheurísticas multi-objetivo são incrementadas com rastreamento bayesiano em ambos os espaços de busca e dos objetivos. A metodologia foi aplicada para a obtenção de decisões de investimento, as quais levaram a melhoras significativas do hipervolume futuro de conjuntos de carteiras financeiras de trade-off avaliadas com dados de ações fora da amostra de treino, quando comparada a uma estratégia míope. Além disso, a tomada de decisões flexíveis para o rebalanceamento de carteiras foi confirmada como uma estratégia significativamente melhor do que a de escolher aleatoriamente uma decisão de investimento a partir da fronteira estocástica eficiente evoluída, em todos os mercados artificiais e reais testados. Finalmente, os resultados sugerem que a antecipação de opções flexíveis levou a composições de carteiras que se mostraram significativamente correlacionadas com as melhorias observadas no hipervolume futuro esperado, avaliado com dados fora das amostras de treinoAbstract: The presence of uncertainty in future outcomes can lead to indecision in choice processes, especially when eliciting the relative importances of multiple decision criteria and of long-term vs. near-term performance. Some decisions, however, must be taken under incomplete information, what may result in precipitated actions with unforeseen consequences. When a solution must be selected under multiple conflicting views for operating in time-varying and noisy environments, implementing flexible provisional alternatives can be critical to circumvent the lack of complete information by keeping future options open. Anticipatory engineering can be then regarded as the strategy of designing flexible solutions that enable decision makers to respond robustly to unpredictable scenarios. This strategy can thus mitigate the risks of strong unintended commitments to uncertain alternatives, while increasing adaptability to future changes. In this thesis, the roles of anticipation and of flexibility on automating sequential multiple criteria decision-making processes under uncertainty are investigated. The dilemma of assigning relative importances to decision criteria and to immediate rewards under incomplete information is then handled by autonomously anticipating flexible decisions predicted to maximally preserve diversity of future choices. An online anticipatory learning methodology is then proposed for improving the range and quality of future trade-off solution sets. This goal is achieved by predicting maximal expected hypervolume sets, for which the anticipation capabilities of multi-objective metaheuristics are augmented with Bayesian tracking in both the objective and search spaces. The methodology has been applied for obtaining investment decisions that are shown to significantly improve the future hypervolume of trade-off financial portfolios for out-of-sample stock data, when compared to a myopic strategy. Moreover, implementing flexible portfolio rebalancing decisions was confirmed as a significantly better strategy than to randomly choosing an investment decision from the evolved stochastic efficient frontier in all tested artificial and real-world markets. Finally, the results suggest that anticipating flexible choices has lead to portfolio compositions that are significantly correlated with the observed improvements in out-of-sample future expected hypervolumeDoutoradoEngenharia de ComputaçãoDoutor em Engenharia Elétric

Agrégation de prédicteurs pour des séries temporelles, optimalité dans un contexte localement stationnaire

This thesis regroups our results on dependent time series prediction. The work is divided into three main chapters where we tackle different problems. The first one is the aggregation of predictors of Causal Bernoulli Shifts using a Bayesian approach. The second one is the aggregation of predictors of what we define as sub-linear processes. Locally stationary time varying autoregressive processes receive a particular attention; we investigate an adaptive prediction scheme for them. In the last main chapter we study the linear regression problem for a general class of locally stationary processes.Cette thèse regroupe nos résultats sur la prédiction de séries temporelles dépendantes. Le document comporte trois chapitres principaux où nous abordons des problèmes différents. Le premier concerne l’agrégation de prédicteurs de décalages de Bernoulli Causales, en adoptant une approche Bayésienne. Le deuxième traite de l’agrégation de prédicteurs de ce que nous définissions comme processus sous-linéaires. Une attention particulaire est portée aux processus autorégressifs localement stationnaires variables dans le temps, nous examinons un schéma de prédiction adaptative pour eux. Dans le dernier chapitre nous étudions le modèle de régression linéaire pour une classe générale de processus localement stationnaires