Using Relative Returns to Accommodate Fat-Tailed Innovations in Processes and Option Pricing

Abstract

The canonical process used to described financial time series is based on a logarithmic random walk.Adding fat tail distributions for the innovations in this framework creates fundamental inconsistencies, essentially related to a diverging integral.We propose to use geometric processes and relative returns instead. This change solves four problems related to (1) the (infinite) values of statistical quantities for processes with fat-tail innovations, (2) the robustness of computations when dealing with large events (genuine or noise), (3) the negative skewness of returns in stocks, and (4) the pricing of options with heteroskedasticity and fat-tails. The mathematical properties of geometric processes is explored for set-ups of increasing complexity. The European option pricing framework is modified to use geometric processes for the underlying, allowing to incorporate naturally fat-tail innovations.