Analysis of Financial Time Series, Third Edition

5.7 Bivariate Models for Price Change and Duration

In this section, we introduce a model that considers jointly the process of price change and the associated duration. As mentioned before, many intraday transactions of a stock result in no price change. Those transactions are highly relevant to trading intensity, but they do not contain direct information on price movement. Therefore, to simplify the complexity involved in modeling price change, we focus on transactions that result in a price change and consider a price change and duration (PCD) model to describe the multivariate dynamics of price change and the associated time duration.

We continue to use the same notation as before, but the definition is changed to transactions with a price change. Let ti be the calendar time of the ith price change of an asset. As before, ti is measured in seconds from midnight of a trading day. Let be the transaction price when the ith price change occurred and Δti = ti − ti−1 be the time duration between price changes. In addition, let Ni be the number of trades in the time interval (ti−1, ti) that result in no price change. This new variable is used to represent trading intensity during a period of no price change. Finally, let Di be the direction of the ith price change with Di = 1 when price goes up and Di = − 1 when the price comes down, and let Si be the size of the ith price change measured ...