Non-Probabilistic Supermartingales.Trajectorial Models For Two Stocks

The dissertation develops mathematical results and specific empirical models in the context of a trajectorial/non-probabilistic setting. This framework starts with scenarios for the future prices of a group of assets and allows for the presence of hedging portfolios. The setting relies on few assumptions of the evolving prices and, in this way, allows for general algorithmic constructions closely linked to empirical characteristics of the evolving assets. Chapters 1 and 2 of the dissertation concentrate on purely mathematical developments. We prove the following well known probabilistic results on non-negative supermartingales in our non-probabilistic trajectorial frame work: Doob’s optional stopping theorem, Doob’s pointwise convergence theorem and Dubin’s inequality. Our results provide evidence that the trajectorial framework embodies the analogous probabilistic notions but re-interpreted in terms of superhedging prices in a context of hedging and trading. Chapters 3 and 4 of the dissertation focus on the construction of trajectory models with the intent to provide concrete examples illustrating the scope of the trajectorial framework. We put forward a self-consistent point of view which we term operational indicating that all steps used in the algorithmic construction of the models have an empirical meaning. The centerpiece of the construction is an agent that reacts to unfolding chart data in a prescribed way by rebalancing their portfolio, composed of a given asset and a numeraire asset, with the intent of super hedging a target asset. The end result permits our agent to gauge the target market asset’s price relative to her operations. The dissertation shows by example that the proposed models have risk-reward characteristics that are relevant and of interest to such an agent. The data structure of the proposed trajectory spaces are coded as a graph data structure along with a multiprocessing implementation needed to test the behaviour of the models as more historical data is aggregated.