Minimax theorems for American options in incomplete markets without time-consistency

TL;DR

This paper establishes minimax theorems for American options in incomplete markets without relying on time-consistency, linking the results to the path properties of density processes.

Contribution

It provides new sufficient conditions for minimax theorems that do not depend on stability under pasting or time-consistency, revealing unexpected connections.

Findings

Minimax theorems hold under new conditions without time-consistency

Connections between minimax results and density process path properties

Implications for arbitrage-free pricing in incomplete markets

Abstract

In this paper we give sufficient conditions guaranteeing the validity of the well-known minimax theorem for the lower Snell envelope with respect to a family of absolutely continuous probability measures. Such minimax results play an important role in the characterisation of arbitrage-free prices of American contingent claims in incomplete markets. Our conditions do not rely on the notions of stability under pasting or time-consistency and reveal some unexpected connection between the minimax result and the path properties of the corresponding density process.