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

Nicole El Karoui and Asma Meziou

Papers from arXiv.org

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.

Date: 2008-04
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Published in Annals of Probability 2008, Vol. 36, No. 2, 647-697

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Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:0804.2561

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