Machine Learning Algorithms with Applications in Finance

Online decision making and learning occur in a great variety of scenarios. The decisions involved may consist of stock trading, ad placement, route planning, picking a heuristic, or making a move in a game. Such scenarios vary also in the complexity of the environment or the opponent, the available feedback, and the nature of possible decisions. Remarkably, in the last few decades, the theory of online learning has produced algorithms that can cope with this rich set of problems. These algorithms have two very desirable properties. First, they make minimal and often worst-case assumptions on the nature of the learning scenario, making them robust. Second, their success is guaranteed to converge to that of the best strategy in a benchmark set, a property referred to as regret minimization. This work deals both with the general theory of regret minimization, and with its implications for pricing financial derivatives. One contribution to the theory of regret minimization is a trade-off result, which shows that some of the most important regret minimization algorithms are also guaranteed to have non-negative and even positive levels of regret for any sequence of plays by the environment. Another contribution provides improved regret minimization algorithms for scenarios in which the benchmark set of strategies has a high level of redundancy; these scenarios are captured in a model of dynamically branching strategies. The contributions to derivative pricing build on a reduction from the problem of pricing derivatives to the problem of bounding the regret of trading algorithms. They comprise regret minimization-based price bounds for a variety of financial derivatives, obtained both by means of existing algorithms and specially designed ones. Moreover, a direct method for converting the performance guarantees of general-purpose regret minimization algorithms into performance guarantees in a trading scenario is developed and used to derive novel lower and upper bounds on derivative prices.