In this study, we use the McKean's integral equation to evaluate the American option price for constant jump di

This thesis treats a range of stochastic methods with various applications, most notably in finance. It is comprised of five articles, and a summary of the key concepts and results these are built on. The first two papers consider a jump-to-default model, which is a model where some quantity, e.g. the price of a financial asset, is represented by a stochastic process which has continuous sample paths except for the possibility of a sudden drop to zero. In Paper I prices of European-type options in this model are studied together with the partial integro-differential equation that characterizes the price. In Paper II the price of a perpetual American put option in the same model is found in terms of explicit formulas. Both papers also study the parameter monotonicity and convexity properties of the option prices. The third and fourth articles both deal with valuation problems in a jump-diffusion model. Paper III concerns the optimal level at which to exercise an American put option with finite time horizon. More specifically, the integral equation that characterizes the optimal boundary is studied. In Paper IV we consider a stochastic game between two players and determine the optimal value and exercise strategy using an iterative technique. Paper V employs a similar iterative method to solve the statistical problem of determining the unknown drift of a stochastic process, where not only running time but also each observation of the process is costly.

Cette thèse examine les décisions financières à travers la théorie des options Américaines et Exotiques. Dans un premier temps, nous avons présenté une revue de la littérature sur les options de type Américain. La tarification de l’option Américaine standard d’achat est revisitée en vue de fournir les pré-requis. Dans l’étape suivante, un nouveau type de contrat d’option, appelé Strangle Euro-American ou Strangle Hybride, a été introduit. Des formules analytiques ont été fournies pour leurs prix ainsi que leurs paramètres de gestion. Une nouvelle méthode est proposée pour calculer les intégrales qui définissent les bornes d’exercice anticipé. Il a été démontré que cette méthode est efficiente, précise et rapide pour la tarification de tous les types de Strangle voir au delà. Puis, nous avons examiné les options Step de type Américain. Nous avons démontré que les propriétés des options d’achat "vanille" ne s’appliquent pas aux Step dans certaines situations. Les formules d’évaluation et des paramètres de gestion ont été déterminés. Et enfin, nous avons considéré l’évaluation d’une firme détenant simultanément une option d’abandon et une option d’expansion de ses activités selon des conditions du marché (favorables ou défavorables). Les seuils critiques de décision ont été obtenus. Des formules analytiques pour la valeur de la firme ont été obtenues. Des simulations illustrent le comportement de ces seuils critiques de décisions anticipées. / This thesis investigates the financial decisions through the theory of American and Exotic options. First, the literature on American-style derivatives is surveyed. The pricing of standard American call option in the early exercise premium representation is addressed in order to provide prerequisites for what follows. Second, a new variant of Strangle contracts, called Euro-American or Hybrid Strangles, is introduced and priced. Analytical formulas are provided for the prices of all these option contracts as well as their hedging parameters. A new quadrature is proposed to account for the systems of coupled integral equations that locate the early exercise boundaries. It is shown to be efficient, accurate, and fast for pricing all types of early exercisable strangles and more. Third, we examines the valuation of American Step options contract. The structures of the immediate exercise regions of the various contracts are identified. Typical properties of American vanilla calls are shown to fail in some cases. Formulas for prices and hedging parameters, for the American Step options, are derived. Finally, we consider the valuation of a firm holding simultaneously an option to expand and to abandon productions depending on the state of the market (good or bad) in a real option framework. Optimal decision levels are obtained. Analytical formulas for the firm’s value are provided. Numerical results document the behavior of the firm’s value and optimal exercise boundaries levels.