On Correlating Lévy Processes

Abstract

A relatively simple approach to correlating unit period returns of Levy processes is developed. We write the Levy process as a time changed Brownian motion and correlate the Brownian motions. It is shown that sample correlations understate the required correlation between the Brownian motions and we show how to correct for this. Pairwise tests illustrate the adequacy of the model and the significant improvement offered over the Gaussian alternative. We therefore advocate that the correlated time change model is a simple basic alternative to dependence modeling. From the perspective of explaining portfolio returns in higher dimensions we find adequacy for long-short portfolios. The long only portfolios appear to require a more complex modeling of dependency. We leave these questions for future research.