Meziou, Asma (2006) Max-Plus decomposition of supermartingales and convex order. Application to American options and Portfolio insurance. PhD thesis CMAP, EP - CMAP Centre de Mathématiques Appliquées, EP/X p.261.
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Alternative Locations: http://www.imprimerie.polytechnique.fr/Theses/Files/Meziou.pdf
Abstract
We are concerned with a new type of supermartingale decomposition in the Max-Plus algebra, which essentially consists in expressing any quasi-left-continuous supermartingale of class (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. In fact, using the Max-Plus supermartingale decomposition, we suggest a new approach to the classic utility maximization problem with American constraints. To do so, we transform the problem into a constrained martingale one, whose aim is to dominate an obstacle, or equivalently its Snell envelope on every intermediate date. The optimization is performed with respect to the stochastic convex order on the terminal value, which avoids any arbitrary assumption regarding the form of the agent's utility function. The "Max-Plus martingale" is shown to be optimal and this is illustrated by an explicit example based on the geometric Brownian motion.
Furthermore, we exploit the links between the Azéma-Yor martingales and the Max-Plus decomposition to solve some portfolio optimization problems with state constraints and some ones related to perpetual American options. In particular, most of the classic results concerning the American boundaries of Lévy processes are shown in an elementary way.
The last chapter is devoted to the pricing of Swing options, using new numerical methods.
| Item Type: | PhD Thesis (PhD) |
|---|---|
| Thesis Supervisor: | El karoui, Nicole |
| Date: | November 2006 |
| Board of examiners: | Follmer, Hans and Gaussel, Nicolas and Gobet, Emmanuel and Jeanblanc-picque, Monique and Jouini, Elyes and Lamberton, Damien and Touzi, Nizar |
| Ecole Doctorale: | ED 447 ECOLE DOCTORALE DE L'ECOLE POLYTECHNIQUE |
| Discipline: | CMAP |
| Collection (Fonds): | EP/X |
| Institution: | EP/X |
| Department: | EP - CMAP Centre de Mathématiques Appliquées |
| Subjects: | 1. Mathematics and Applications |
| Uncontrolled Keywords: | Supermartingale decompositions, Max-Plus algebra, Running supremum process, American options, Optimal stopping, Lévy processes, Convex order, Martingale optimization with constraints, Portfolio insurance, Azéma-Yor martingales, Drawdown constraints, Décomposition de surmartingales, Algèbre Max-Plus, Running supremum, Options Américaines, Arrêt optimal, Processus de Lévy, Ordre convexe, Optimisation de martingales sous contraintes, Assurance de portefeuille, Martingales d'Azéma-Yor, Drawdown |
Table of content
- Introduction
- Algèbre Max-Plus
- Décomposition Max-Plus des surmartingales
- Généralisation du point de vue de Darling-Liggett-Taylor
- Existence de la décomposition Max-Plus des surmartingales
- Optimalité de la martingale Max-Plus pour l'ordre convexe
- Optimisation de portefeuille sous contraintes par rapport à l'ordre convexe
- Martingales d'Azéma-Yor et décomposition Max-Plus
- Numerical methods for the pricing of Swing options
| ID Code: | 2177 |
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| Deposited By: | Laurence Vidament |
| Deposited On: | 14 February 2007 |
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