Trajectory Based Market Models: Evaluation of Minmax Price Bounds

Abstract

The paper studies sub and super-replication price bounds for contingent claims defined on general trajectory based market models. No prior probabilistic or topological assumptions are placed on the trajectory space which is of unrestricted cardinality. For a given option, there exists an interval bounding the set of possible fair prices; such interval exists under more general conditions than the usual no-arbitrage requirement. The paper develops a backward recursive method to evaluate the option bounds together with the associated hedging strategies; the global minmax optimization, defining the price interval, is reduced to a local minmax optimization via dynamic programming. Trajectory sets are introduced for which existing probabilistic and non-probabilistic market models are nested as particular cases. Several examples are presented, the effect of the presence of arbitrage on the price bounds is illustrated.