A model-free no-arbitrage price bound for variance options

In the framework of Galichon, Henry-Labordère and Touzi, we consider the model-free no-arbitrage bound of variance option given the marginal distributions of the underlying asset. We first make some approximations which restrict the computation on a bounded domain. Then we propose a gradient projection algorithm together with a finite difference scheme to approximate the bound. The general convergence result is obtained. We also provide a numerical example on the variance swap option.Variance option ; model-free price bound ; gradient projection algorithm.

Monotonicity condition for the θ\theta-scheme for diffusion equations

We derive the necessary and sufficient condition for the $L^{\infty}-$monotonicity of finite difference $\theta$-scheme for a diffusion equation. We confirm that the discretization ratio $\Delta t = O(\Delta x^2)$ is necessary for the monotonicity except for the implicit scheme. In case of the heat equation, we get an explicit formula, which is weaker than the classical CFL condition.

A model-free no-arbitrage price bound for variance options

International audienceIn the framework of Galichon, Henry-Labordère and Touzi, we consider the model-free no-arbitrage bound of variance option given the marginal distributions of the underlying asset. We first make some approximations which restrict the computation on a bounded domain. Then we propose a gradient projection algorithm together with a finite difference scheme to approximate the bound. The general convergence result is obtained. We also provide a numerical example on the variance swap option.Dans le cadre de Galichon, Henry-Labordère et Touzi, nous considérons la borne sans arbitrage, indépendante d'un modèle, étant donné la distribution marginale du sous-jacent. Nous restreignons d'abord le calcul à un domaine borné. Puis nous proposons un algorithme de gradient avec projection, combiné à un schéma de différences finies, pour approcher la borne. Nous obtenons un résultat général de convergence, puis traitons un exemple numérique d'option sur swap

Rates of convergence of Newton type methods for variational inequalities and nonlinear programming

Projet PROMATHThis paper presents some new results in the theory of Newton type methods for variational inequalities and their application to nonlinear programming. A condition of semi-stability is shown to ensure the quadratic convergence of Newton's method and the superlinear convergence of some quasi-Newton algorithms, provided the sequence defined by the algorithm exists and converges. A partial extension of these results to nonsmooth function is given. The second part of the paper considers some particular variationnal inequalities with unknowns {x, l) generalizing optimality systems. Here only the question of superlinear convergence of {xk} is considered. Some necessary or sufficient conditions are given. Applied to some quasi-Newton algorithms they allow to obtain the superlinear convergence of {xk}. The application of the previous results to nonlinear programming allows to strenghten the know results, the main point being a characterization of the superlinear convergence of {xk} assuming a weak second-order condition without strict complementary

Optimization of running strategies based on anaerobic energy and variations of velocity

Parallel sessionInternational audience1 Keller's model 2 Variable energy recreation 3 Bounding the derivative of

Pseudopower expansion of solutions of generalized equations and constrained optimization problems

Projet PROMATHWe compute the solution of a strongly regular perturbed generalized equations as the sum of a speudopower expansion, i.e. the expansion at order k is the solution of the generalized equation expanded at order k and thus depends itself on the perturbation parameter [??]. In the polyhedral case, the pseudopower expansion reduces to a classical Taylor expansion. For constrained optimization problems with strongly regular solution, we check that the quadratic growth condition holds and that, at least locally, solutions of the problem and solutions of the associated optimality system coincide. In the special case of a finite number of inequality constraints, the solution and the Lagrange multiplier can be expanded in Taylor series if the gradients of the active constraints are linearly independent. If the data are analytic, the solution and the multiplier are analytic functions in [??] provided that some strong second order condition holds

Application d'une nouvelle classe de lagrangiens augmentés en contrôle optimal de systèmes distribués

Ce rapport présente une nouvelle méthode de résolution des problèmes de contröle optimal de systèmes distribués. On suppose le critère convexe et l'équation d'état affine par rapport à la paire (état, contrôle)

Deux etudes en programmation non lineaire

La première étude concerne le comportement des solutions locales lorsqu'on perturbe un problème d'optimisation non convexe. On y donne des conditions sous lesquelles une solution isolée bifurque en un nombre fini de solutions, qu'on peut effectivement calculer. La seconde étude établit l'augmentabilité et la pénalizabilité exacte au voisinage de solutions locales vérifiant des conditions suffisantes (faibles) du deuxième ordre

On an algorithm for optimal control using Pontryagin's maximum principle

Y. Sakawa and Y. Shindo recently proposed an algorithm to solve open-loop optimal control problems using Pontryagin's maximum principl

Analysis and control of a non-linear parabolic unstable system

This paper is concerned with a non-linear evolutive system of diffusion-reaction typ