Abstract
The goal of the paper is to show that some types of Levy processes such as the hyperbolic motion and the CGMY are particularly suitable for asset price modelling and option pricing. We wish to review some fundamental mathematic properties of Levy distributions, such as the one of infinite divisibility, and how they translate observed features of asset price returns. We explain how these processes are related to Brownian motion, the central process in finance, through stochastic time changes which can in turn be interpreted as a measure of the economic activity. Lastly, we focus on two particular classes of pure jump Levy processes, the generalized hyperbolic model and the CGMY models and report on the goodness of fit obtained both on stock prices and options prices.
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