An optimal prediction problem in financial modelling

Abstract

The subject of the present thesis is an problem concerning the ultimate maximum of a stable Levy process over a finite interval of time. Such optimal prediction problems are of both theoretical and practical interest, in particular they have applications in finance. For instance, suppose that an investor has a long position in one financial asset, whose price is modelled by some stochastic process. The investor's objective is to determine a best moment at which to close out the position and to sell the asset at the highest possible price. This decision must be based on continuous observations of the asset price performance and only on the information accumulated to date. Hence, the investor should use a (forecasting) of the future evolution of the price of the financial security. We examine this problem in the case where the asset price is modelled by a Levy process. Indeed, during the last several years, the application of Levy processes in the modelling financial asset returns has become one of the active research directions in quantitative finance. Thus, this thesis contains suitable new results concerning Levy processes. We derive the law of the supremum process associated with a strictly stable Levy process with no negative jumps which is not a subordinator. We note that the latter problem dates back to 1973. In particular, we show that the probability density function of the supremum process can be expressed using an explicit power series representation or via an integral representation. We also derive the infinitesimal generator of the reflected process associated with a general strictly stable Levy process. Throughout this thesis, we apply the theory of stopping, the methods of fractional differential calculus, and some results from fluctuation theory. Implementing these theories in the context of Levy processes requires the development of specific analytical results. In the case where the asset price is modelled by a spectrally positive stable Levy process, we describe the strategy under certain conditions on the model parameters. The strategy is of the following form: the investor must stop the observation of the price process and sell the asset as soon as the associated reflected process crosses for the first time a particular stopping boundary. We also provide numerical estimates and simulation examples of the results obtained by using this strategy.