Valuación de opciones europeas mediante procesos de Lévy exponenciales y transformada rápida de Fourier

In this paper, a new valuation methodology is tested as an alternative way to value European options by applying Fast Fourier Transform (FFT) on Lévy processes. It is shown how this new approach can be applied to the Variance-Gamma model by Madan, Carr and Chang (1998). This model has a closed form formula, which can be obtained, as in many other models, by partial differential equations or by probabilistic methods, but even these methodologies will not prove to be useful when we encounter ourselves with more complex models. Nevertheless, if we take into account the fact that there is a one-to-one relationship between a characteristic function of an exponential Lévy process and its probability density and that this cash flows are known in closed forms or can be expressed in terms of special mathematic functions, then it is possible to use this new methodology to obtain an alternate and maybe simpler valuation forms for models which do not have a closed form solution. Madan, Carr and Changs model is a particular case of general exponential Lévy processes. If there is a way to develop a new valuation method by using Fourier transform and this method proves to give same results as closed form solutions or by numerical method solutions, then we could be on our way to find simpler ways to price assets than using a traditional approach