ABSTRACT
This chapter provides a self-contained account on the characterization of L´evy processes and stochastic volatility models in modeling asset price dynamics. We start with review of the jump diffusion models and basic properties of L´evy processes. We then show the relation between infinitively divisible distribution and L´evy processes. The key prop- erties are the L´evy-Khintchine representation and L´evy-Itˆo decomposition theorem. We then introduce the notion of time change in infinite activity L´evy processes and show how to obtain several prototype L´evy models via subordination of a Brownian motion, like the Variance Gamma model and Normal Inverse Gaussian model. We illustrate the versatili- ty of applying time change via stochastic activity rate to generate stochastic volatility in exponential L´evy models. Lastly, we present analytic formulation of stochastic volatility models with jumps. In particular, we derive analytic formulas for the joint moment gen- erating functions for the affine stochastic volatility models and 3/2-model. [148 words]
