Competitive algorithms for online conversion problems with interrelated prices

The classical uni-directional conversion algorithms are based on the assumption that prices are arbitrarily chosen from the fixed price interval[m, M] where m and M represent the estimated lower and upper bounds of possible prices 0<m<M. The estimated interval is erroneous and no attempts are made by the algorithms to update the erroneous estimates. We consider a real world setting where prices are interrelated, i.e., each price depends on its preceding price. Under this assumption, we drive a lower bound on the competitive ratio of randomized non-primitive algorithms. Motivated by the fixed and erroneous price bounds, we present an update model that progressively improves the bounds. Based on the update model, we propose a non-preemptive reservations price algorithm RP* and analyze it under competitive analysis. Finally, we report the findings of an experimental study that is conducted over the real world stock index data. We observe that RP* consistently outperforms the classical algorithm

Optimal Online Two-Way Trading with Bounded Number of Transactions

We consider a two-way trading problem, where investors buy and sell a stock whose price moves within a certain range. Naturally they want to maximize their profit. Investors can perform up to k trades, where each trade must involve the full amount. We give optimal algorithms for three different models which differ in the knowledge of how the price fluctuates. In the first model, there are global minimum and maximum bounds m and M. We first show an optimal lower bound of φ (where φ= M/ m) on the competitive ratio for one trade, which is the bound achieved by trivial algorithms. Perhaps surprisingly, when we consider more than one trade, we can give a better algorithm that loses only a factor of φ2 / 3 (rather than φ) per additional trade. Specifically, for k trades the algorithm has competitive ratio φ(2k+1)/3. Furthermore we show that this ratio is the best possible by giving a matching lower bound. In the second model, m and M are not known in advance, and just φ is known. We show that this only costs us an extra factor of φ1 / 3, i.e., both upper and lower bounds become φ(2k+2)/3. Finally, we consider the bounded daily return model where instead of a global limit, the fluctuation from one day to the next is bounded, and again we give optimal algorithms, and interestingly one of them resembles common trading strategies that involve stop loss limits

Kompetitive Algorithmen für den Börsenhandel

In dieser Arbeit wird das Online-Conversion Problem untersucht. Dieses beschäftigt sich mit der Konvertierung von Vermögen von einer Anlage in eine andere Anlage. Die unterschiedlichen Ausprägungen des Online-Conversion Problems werden mathematisch dargestellt und voneinander abgegrenzt. In der Literatur gibt es zur Lösung des Online-Conversion Problems einige Handelsalgorithmen, die auf Basis von Vergangenheitsdaten Entscheidungen treffen. Diese Algorithmen stellen jedoch zumeist Heuristiken ohne eine Garantie hinsichtlich der Lösungsgüte dar. Andere Handelsalgorithmen sind sogenannte kompetitive Algorithmen. Für diese Algorithmen kann durch eine theoretische Analyse eine Lösungsgarantie bestimmt werden. In dieser Arbeit werden sowohl theoretische als auch empirische Methoden zur Evaluation von Handelsalgorithmen erläutert. Diese Arbeit präsentiert zudem einen kompetitiven Algorithmus, der - zumindest für spezifische Parameter - die bisher beste Lösungsgarantie für eine konkrete Ausprägung des Online-Conversion Problems bietet. Zudem wird der Algorithmus auch hinsichtlich seiner empirischen PerformanzmitzweiausgewähltenkompetitivenAlgorithmensowiezweiinderPraxis häufigverwendetenHeuristikenverglichen.ImRahmenderempirischenAnalysewird zudem der Einfluss von Transaktionskosten sowie unterschiedlicher Handelszeiträume untersucht. Die Ergebnisse der empirischen Analyse zeigen auf, dass der vorgestellte Algorithmus je nach verwendetem Testdesign in der Lage ist, signifikant höhere Ergebnisse zu erzielen als ausgewählte Algorithmen.In this thesis, the online conversion problem is considered. A player has to convert wealth from one asset into another and at the same time he wants to maximize his terminal wealth. Several variants of the online conversion problem can be found in the literature. For each variant of the online conversion problem, a mathematical description is given. Based on the mathematical descriptions, the differences between the variants of the online conversion problem are considered. In practice, the problem is solved with trading algorithms based on historical data. Nevertheless, the majority of algorithms which can be found in practice and in the literature are heuristics where the development of the terminal wealth is not bounded. Other algorithms, with theoretical bound, are called competitive algorithms. In this thesis, empirical and theoretical methods to evaluate trading algorithms are explained. A new competitive algorithm for a specific variant of the online conversion problem is given. Based on a theoretical analysis, the algorithm achieves for specific given input parameters the best theoretical bound of the solution compared to all other competitive algorithms known so far. In addition, the algorithm is compared empirically with two other competitive algorithms and two heuristics which are often used in practice. The impact of transactions costs and the length of the trading period on the results are investigated. The empirical analysis shows, the new algorithm presented in this thesis, can achieve significantly better results than the chosen benchmarks