Modeling financial returns by discrete stable distributions

This paper develops a new approach to modeling financial returns by introducing discrete stable distributions. It is well known that the financial returns are not normally distributed, extremal events occur more often than the Gaussian distribution suggests. Already in the sixties Benoit Mandelbrot suggested a hypothesis that returns follow a stable Paretian law. Inspired by the discrete nature of prices appearing on the markets we model the financial returns by discrete analogues of absolutely continuous stable distributions. The known discrete stability of random variables on N is generalized to the case of random variables on Z. We give brief introduction to the theory of discrete stability on Z, show connection of discrete stable random variables to their absolutely continuous counterparts and focus mainly on methods of estimation of parameters of these distributions from the real data of financial returns.