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

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.Comment: Published in at http://dx.doi.org/10.1214/009117907000000222 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Numerical methods for the pricing of Swing options: a stochastic control approach

International audienceIn the natural gas market, many derivative contracts have a large degree of flexibility. These are known as Swing or Take-Or-Pay options. They allow their owner to purchase gas daily, at a fixed price and according to a volume of their choice. Daily, monthly and/or annual constraints on the purchased volume are usually incorporated. Thus, the valuation of such contracts is related to a stochastic control problem, which we solve in this paper using new numerical methods. Firstly, we extend the Longstaff–Schwarz methodology (originally used for Bermuda options) to our case. Secondly, we propose two efficient parameterizations of the gas consumption, one is based on neural networks and the other on finite elements. It allows us to derive a local optimal consumption law using a stochastic gradient ascent. Numerical experiments illustrate the efficiency of these approaches. Furthermore, we show that the optimal purchase is of bang-bang type

Decomposition Max-Plus des surmartingales et ordre convexe. Application aux options Americaines et a l'assurance de portefeuille.

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.Nous établissons une nouvelle décomposition des surmartingales, additive dans l'algèbre Max-Plus. Elle consiste essentiellement à exprimer toute surmartingale quasi-continue à gauche de la classe (D) comme une espérance conditionnelle d'un certain processus de running supremum. Comme application, nous montrons comment la décomposition Max-Plus permet en particulier de résoudre le problème Américain d'arrêt optimal sans avoir à calculer le prix de l'option. Ensuite, nous donnons quelques exemples illustratifs basés sur des processus de diffusion uni-dimensionnels. Une autre application intéressante concerne l'assurance de portefeuille. Nous proposons en effet une nouvelle approche au problème classique de maximisation d'utilité, avec garantie Américaine. Pour cela, nous nous ramenons à un problème général de martingales, sous contrainte de dominer un obstacle, ou de façon équivalente son enveloppe de Snell, à toute date intermédiaire. L'optimisation est relative à l'ordre convexe sur la valeur terminale, de manière à minimiser le rôle de la fonction d'utilité. Nous montrons l'optimalité de la "martingale Max-Plus" et nous traitons un exemple explicite dans le cadre d'un Brownien géométrique. Par ailleurs, nous exploitons les liens entre les martingales d'Azéma-Yor et la décomposition Max-Plus pour résoudre certains problèmes d'optimisation de portefeuille sous contraintes d'état et d'autres relatifs aux options Américaines perpétuelles. Nous retrouvons en particulier, d'une manière élémentaire, la plupart des résultats classiques sur les frontières Américaines de processus de Lévy. Le dernier chapitre propose de nouvelles méthodes numériques pour valoriser les contrats Swing

Décomposition Max-plus des surmartingales et ordre convexe. Application aux options américaines et à l'assurance de protefeuille

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