A new elementary geometric approach to option pricing bounds in discrete time models

• A new geometric approach to determine the option pricing bounds is provided.
• The methodology reduces to a (sequence of) convex hull problem.
• Barycentric coordinates play the role of a martingale measure that needs however not be equivalent to the historical one.
• We show how to obtain barycentric coordinates that play the role of an equivalent measure in a multi-asset framework.

The aim of this paper is to provide a new straightforward measure-free methodology based on convex hulls to determine the no-arbitrage pricing bounds of an option (European or American). The pedagogical interest of our methodology is also briefly discussed. The central result, which is elementary, is presented for a one period model and is subsequently used for multiperiod models. It shows that a certain point, called the forward point, must lie inside a convex polygon. Multiperiod models are then considered and the pricing bounds of a put option (European and American) are explicitly computed. We then show that the barycentric coordinates of the forward point can be interpreted as a martingale pricing measure. An application is provided for the trinomial model where the pricing measure has a simple geometric interpretation in terms of areas of triangles. Finally, we consider the case of entropic barycentric coordinates in a multi asset framework.