Levy Process Models for High Frequency Financial Data

In this paper we present parametric estimation of models for stock returns by describing price dynamic as the sum of two independent Levy components. The increments (moves) are viewed as discrete-time log price changes that follow an infinitely divisible distribution, i.e. stationary and independent price changes (zero drift) that follow a Levy-type distribution. We explore empirical plausibility of two parametric models: Jump-Diffusion (C-J) and pure jump model (TS-J). The first process describes dynamics of small frequent moves and is modeled by Brownian motion in C-J model and by tempered stable Levy process in TS-J model. The second process is responsible for big rare moves in asset prices and is modeled by compound Poisson process in both models. The estimation is performed via continuously updated GMM by matching the characteristic function implied by the model with the observed characteristic function. Using high frequency data on 13 stocks of different market capitalization for 2006-2008 sample period we find that C-J model performs well only for large cap stocks, while medium cap stock dynamics are captured by TS-J model. We also report evidence of positive relation between activity index of the process for stock returns and its frequency of trading.