Modeling Financial Security Returns Using Lévy Processes

Lévy processes can capture the behaviors of return innovations on a full range of financial securities. Applying stochastic time changes to the Lévy processes randomizes the clock on which the processes run, thus generating stochastic volatilities and stochastic higher return moments. Therefore, with appropriate choices of Lévy processes and stochastic time changes, we can capture the return dynamics of virtually all financial securities. Furthermore, in contrast to the hidden factor approach, we can readily assign explicit economic meanings to each Lévy process component and its associated time change in the return dynamics. The explicit economic mapping not only facilitates the interpretation of existing models and their structural parameters, but also adds economic intuition and direction for designing new models capturing new economic behaviors. Finally, under this framework, the analytical tractability of a model for derivative pricing and model estimation originates from the tractability of the Lévy process specification and the tractability of the activity rate dynamics underlying the time change. Thus, we can design tractable models using any combination of tractable Lévy specifications and tractable activity rate dynamics. In this regard, we can incorporate and therefore encompass all tractable models in literature into our framework as building blocks. Examples include Brownian motions, compound Poisson jumps, and other tractable jump specifications like variance gamma, dampened power law, normal inverse Gaussian, and so on for Lévy processes, and affine, quadratic, and 3/2 processes for activity rate dynamics. In this chapter, I elaborate through examples on the generality of the framework in capturing the return behavior of virtually all financial securities, the explicit economic mapping that facilitates the interpretation and creation of new models capturing specific economic behaviors, and the tractability embedded in the framework for derivative pricing and model estimation. JEL Classifications: G10, G12, G13.