Trajectory based models. Evaluation of minmax pricing bounds

TL;DR

This paper introduces a trajectory-based approach to evaluate minmax bounds for option prices without probabilistic assumptions, using dynamic programming to compute bounds in general market models.

Contribution

It develops a recursive method to compute price bounds in trajectory models, extending non-probabilistic market models and analyzing arbitrage effects.

Findings

A backward recursive method for option bounds is proposed.

Price bounds are shown to exist under general conditions.

Examples illustrate the impact of arbitrage on 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, trading is assumed to take place at a finite number of occasions but not bounded in number nor necessarily equally spaced in time. 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; 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 non-probabilistic markets models are nested as a particular case. Several examples are presented, the effect of the presence of arbitrage on the price bounds is illustrated.