Order Flow, Transaction Clock, and Normality of Asset Returns

TLDR: In this article, it was shown that the cumulative number of trades is a better stochastic clock than the volume for generating virtually perfect normality in returns, and this clock can be modeled nonparametrically, allowing both the time-change and price processes to take the form of jump diffusions.

Abstract: The goal of this paper is to show that normality of asset returns can be recovered through a stochastic time change. Clark (1973) addressed this issue by representing the price process as a subordinated process with volume as the lognormally distributed subordinator. We extend Clark's results and find the following: (i) stochastic time changes are mathematically much less constraining than subordinators; (ii) the cumulative number of trades is a better stochastic clock than the volume for generating virtually perfect normality in returns; (iii) this clock can be modeled nonparametrically, allowing both the time-change and price processes to take the form of jump diffusions. The relations among trading volume, stock prices, and price volatility, the subject of empirical and theoretical studies over many years, have lately received renewed attention with the increased availability of high frequency data. A vast amount of research has focused on issues such as news arrivals, volume, and price changes or volatility moves, usually outside any framework of general or even partial equilibrium. Is the normality of returns-a key issue, for example, in the mean-variance paradigm for portfolio choice, or the recent study of the problems of risk management (e.g., in Value at Risk)-verified at any time horizon? The evidence accumulated from a number of studies that document the presence of leptokurtosis and skewness in the distribution of returns of a wide variety of financial assets suggests that the answer is no. Studies as early as, for example, Fama (1965), showed that daily returns are more long tailed than the normal density, with the distribution of returns approaching normality as the holding period is extended to one month. In the same manner, volatility smiles and other observed