Drift in Transaction-Level Asset Price Models

TL;DR

This paper investigates how drift, arising from nonzero mean shocks, influences the distribution and estimation of asset prices in transaction-level models driven by point processes, highlighting challenges in detecting true growth rates.

Contribution

It introduces a univariate pure-jump model linking drift to transaction counts, analyzing its impact on the limiting distribution and estimator convergence, and proposes a new ratio statistic for better inference.

Findings

Drift affects the limiting distribution of log prices, potentially making it non-Gaussian.

The rate of convergence of growth rate estimators can be altered by drift properties.

Standard hypothesis tests may be invalid due to the influence of durations on price dynamics.

Abstract

We study the effect of drift in pure-jump transaction-level models for asset prices in continuous time, driven by point processes. The drift is as-sumed to arise from a nonzero mean in the efficient shock series. It follows that the drift is proportional to the driving point process itself, i.e. the cumulative number of transactions. This link reveals a mechanism by which properties of intertrade durations (such as heavy tails and long memory) can have a strong impact on properties of average returns, thereby poten-tially making it extremely difficult to determine long-term growth rates or to reliably detect an equity premium. We focus on a basic univariate model for log price, coupled with general assumptions on the point process that are satisfied by several existing flexible models, allowing for both long mem-ory and heavy tails in durations. Under our pure-jump model, we obtain the limiting distribution for the suitably normalized log price. This limiting distribution need not be Gaussian, and may have either finite variance or infinite variance. We show that the drift can affect not only the limiting dis-tribution for the normalized log price, but also the rate in the corresponding normalization. Therefore, the drift (or equivalently, the properties of dura-tions) affects the rate of convergence of estimators of the growth rate, and can invalidate standard hypothesis tests for that growth rate. As a rem-edy to these problems, we propose a new ratio statistic which behaves more