Agent-Based Models for Two Stocks with Superhedging

An agent-based modelling methodology for the joint price evolution of two stocks is put forward. The method models future multidimensional price trajectories reflecting how a class of agents rebalance their portfolios in an operational way by reacting to how stocks' charts unfold. Prices are expressed in units of a third stock that acts as numeraire. The methodology is robust, in particular, it does not depend on any prior probability or analytical assumptions and it is based on constructing scenarios/trajectories. A main ingredient is a superhedging interpretation that provides relative superhedging prices between the two modelled stocks. The operational nature of the methodology gives objective conditions for the validity of the model and so implies realistic risk-rewards profiles for the agent's operations. Superhedging computations are performed with a dynamic programming algorithm deployed on a graph data structure. Null subsets of the trajectory space are directly related to arbitrage opportunities (i.e. there is no need for probabilistic considerations) that may emerge during the trajectory set construction. It follows that the superhedging algorithm handles null sets in a rigorous and intuitive way. Superhedging and underhedging bounds are kept relevant to the investor by means of a worst case pruning method and, as an alternative, a theory supported pruning that relies on a new notion of small arbitrage.