From tick data to semimartingales

Tick-by-tick asset price data exhibit a number of empirical regularities, including discreteness, long periods where prices are flat, periods of price moves of alternating plus and minus one tick, periods of rapid successive price moves of the same sign, and others. This paper proposes a framework to examine whether and how these microscopic features of the tick data are compatible with the typical macroscopic continuous-time models, based on Itô semimartingales, that are employed to represent asset prices. We construct in particular tick-by-tick models that deliver by scaling macroscopic semimartingale models with stochastic volatility and jumps.