Generating distributions through convolution of characteristic functions

Financial data such as asset returns, exchange rates, or option prices cannot be modeled effectively by classical distributions such as the Gaussian. These types of data have probability density functions that are thick-tailed and negatively skewed. To account for these features, we propose a new method of generating classes of distribution functions through convolution of smooth and non-smooth characteristic functions where the smoothing parameter is used to control the thickness of the density tails. To illustrate the advantages of using such class of distributions, we consider special cases in which the smooth characteristic functions are of those of the uniform, the normal and the compact supported cosine distributions and the non-smooth is the characteristic function of the Cauchy distribution. As a comparison criterion between distributions, we use the Stiltjes-Hamburger conditions for moments’ existence and show how the proposed distributions outperform the Student and Pearson IV distributions, which are commonly used by financial engineers to model stock returns.