Joint analysis and estimation of stock prices and trading volume in Barndorff-Nielsen and Shephard stochastic volatility models

We introduce a variant of the Barndorff-Nielsen and Shephard stochastic volatility model where the non-Gaussian Ornstein-Uhlenbeck process describes some measure of trading intensity like trading volume or number of trades instead of unobservable instantaneous variance. We develop an explicit estimator based on martingale estimating functions in a bivariate model that is not a diffusion, but admits jumps. It is assumed that both the quantities are observed on a discrete grid of fixed width, and the observation horizon tends to infinity. We show that the estimator is consistent and asymptotically normal and give explicit expressions of the asymptotic covariance matrix. Our method is illustrated by a finite sample experiment and a statistical analysis of IBM™ stock from the New York Stock Exchange and Microsoft Corporation™ stock from Nasdaq during a history of five years.