Relative volume as a doubly stochastic binomial point process

Abstract

Relative intra-day cumulative volume is intra-day cumulative volume divided by final total volume. If intra-day cumulative volume is modeled as a Cox (doubly stochastic Poisson) point process, then using initial enlargement of filtration with the filtration of the Cox process enlarged by knowledge of final volume, it is shown that relative intra-day volume conditionally has a binomial distribution and is a novel generalization of a binomial point process: the doubly stochastic binomial point process. Re-scaling the intra-day traded volume to a relative volume between 0 (no volume traded) and 1 (daily trading completed) allows empirical intra-day volume distribution information for all stocks to be used collectively to estimate and identify the random intensity component of the doubly stochastic binomial point process and closely related Cox point process.