Max-Plus decomposition of supermartingales and convex order. Application to American options and portfolio insurance

TL;DR

This paper introduces a novel Max-Plus supermartingale decomposition method that simplifies solving American option and portfolio insurance problems by avoiding complex computations, with applications demonstrated on diffusion processes.

Contribution

It develops a new Max-Plus supermartingale decomposition technique and applies it to American options and portfolio insurance, providing solutions without traditional price calculations.

Findings

Max-Plus decomposition expresses supermartingales as supremum-based expectations.

The method simplifies American option optimal stopping problems.

Applications include portfolio insurance optimization under convex order.

Abstract

We are concerned with a new type of supermartingale decomposition in the Max-Plus algebra, which essentially consists in expressing any supermartingale of class $(\mathcal{D})$ as a conditional expectation of some running supremum process. As an application, we show how the Max-Plus supermartingale decomposition allows, in particular, to solve the American optimal stopping problem without having to compute the option price. Some illustrative examples based on one-dimensional diffusion processes are then provided. Another interesting application concerns the portfolio insurance. Hence, based on the ``Max-Plus martingale,'' we solve in the paper an optimization problem whose aim is to find the best martingale dominating a given floor process (on every intermediate date), w.r.t. the convex order on terminal values.