The paper written by Thierry Ane and Helyette Geman, titled, “Order Flow, Transaction Clock, and Normality of Asset Returns” explores the concept of changing the “calendar time” of the asset return process to recover normality.

The idea that, “Calendar time might not be an appropriate measure of time in the financial markets”, has been explored in the past. Mandelbrot in 1963 published a paper that introduced a class of stable processes that could recover normality. The stable processes were nothing but Brownian motion processes subordinated by another stable process. In 1973, Clark published a paper in which he introduced a subordinator based on trading volume. However the presence of these two classic papers has not spawned further research. The investigation of specific subordinators that might recover asset return normality has been sparse in the finance literature

The authors of this paper test which of the following is a better subordinators :

  • Cumulated Traded Volume

  • Cumulated Number of Trades

The dataset is HFT data of two tech stocks Cisco and Intel.To begin with, a basic regression model is set to compare the performance of the volume and the number of trades in explaining changes in volatility. The model estimates show superiority of the number of trades over traded volume, in explaining volatility changes.

What are the authors searching for ? Basically they are on a quest of a subordinator that makes the asset returns appear Gaussian. Instead of assuming a specific distribution for the subordinator, the authors follow a non parametric approach. The search problem is cast as a convex optimization problem that involves the moment generating function of the asset returns process and various moments of the subordinator process. The results of optimization are various moments of the subordinator process. After obtaining various moment estimates, the authors compare these to the moments of the subordinator based on traded volume and number of trades. The empirical findings of the paper show that  cumulative number of trades is a better subordinator and one that can retrieve the normality of asset returns.  All the moments except the first moment of total number of trades match that of the optimization estimates. The authors compute the return series conditional on the number of traders, use a kernel density estimate and find that the returns are Gaussian.

The takeaway from the paper is that, by changing from “calendar time” to “total number of trades time”, the asset returns are Gaussian. Hence this stochastic change of time becomes very appealing to practitioners and traders.