Évaluation d'options "vanilles" et "digitales" dans le modèle de marché à intervalles

Abstract : We investigate a finite horizon minimax differential game and multistage game, arising in an european options pricing problem. The differential game proceeds in a 3D plus time space. Its main features are, on the one hand, to mix a continuous and an impulse control for the same player and, on the other hand, in the case of digital options, to have a discontinuous terminal cost function. This minimax game results from a robust control approach on the admissible set of underlying stock price trajectories, without endowing this set with any probability law, as prescribed by the hypothesis of an interval model market. From the mathematical finance view point, our objective is to develop a consistent option pricing theory, both in continuous and discrete trading within the same market model, with transaction costs. We obtain the premium as the Value function and the hedging strategy as the optimal feedback strategy. In the continuous trading framework, we develop a detailed analysis of the extremal trajectories of the impulsive qualitative game with target at horizon time, via tools of the Isaacs-Breakwell Theory (semipermeability). This analysis allows us to completely solve the problem. The solution involves many singularities: dispersal manifold, equivocal junctions, and a focal manifold, of dimension 2, with jump trajectories. Moreover we obtain a representation theorem for the Value function in terms of a solution of a set of two linear coupled PDEs. We end up the investigation of the continuous theory by a more classical analytic verification, showing that the Value we exhibit via the representation theorem is a viscosity solution of the Isaacs equation of a free end-time game without impulse control which has the same Value as our differential game. In the discrete trading framework, we use a classical dynamic programming approach. It yields a representation theorem similar to that of the continuous theory, which provides a fast algorithm to compute numerically both the premium and the trading strategy. We also prove the convergence of the Value function of the multistage game toward that of the continuous time game when the time step vanishes, for both the vanilla and digital options. Hence the fast algorithm also gives us a good approximation of the solution (premium and trading strategy) of the continuous time theory. We conclude this dissertation by a discussion of the relevance of our result in the mathematical finance framework, including, a study of the robustness of the hedging strategy to violations of the hypothesis of the market model. Discussing the strengths and the weaknesses of our theory as compared to the Black and Scholes theory, we stress that we do not claim any kind of overall superiority of our theory, but only that this is a possible approach that, with large transaction costs and/or in discrete trading, may be an interesting alternative. This in spite of a major weakness arising from the incompleteness of our market model.
Document type :
Mathématiques [math]. Université Nice Sophia Antipolis, 2008. Français

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Submitted on : Friday, February 26, 2010 - 2:07:25 PM
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  • HAL Id : tel-00460176, version 1



Stéphane Thiery. Évaluation d'options "vanilles" et "digitales" dans le modèle de marché à intervalles. Mathématiques [math]. Université Nice Sophia Antipolis, 2008. Français. <tel-00460176>




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