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The definition of time is still an open question when one deals with high-frequency time series. If time is simply the calendar time, prices can be modeled as continuous random processes and values resulting from transactions or given quotes are discrete samples of this underlying dynamics. On the contrary, if one takes the business time point of view, price dynamics is a discrete random process, and time is simply the ordering according to which prices are quoted in the market. In this paper, we suggest that the business time approach is perhaps a better way of modeling price dynamics than calendar time. This conclusion comes from testing probability densities and conditional variances predicted by the two models against the experimental ones. The data set we use contains the DEM/USD exchange quotes provided to us by Olsen & Associates during a period of one year from January to December 1998. In this period, 1,620,843 quotes entries in the EFX system were recorded.

In the high-frequency arena there are two mainstreams about modeling the stochastic properties of quotes. The first approach is to consider quotations as sampled values of an underlying continuous-time random process [1] and [2]. Sampling is itself a random operation, thus introducing a twofold uncertainty in the price determination [3] and [4]. In this framework, time in the model flows continuously, and is called calendar time. In the second approach, quoted prices are modeled through a discrete-time stochastic process [5]; in this setting, time is just the natural total order relation among quotations, and it is isomorphic with the set of non-negative integers (time being 0, the time associated to the first considered quotation). This is the business time approach, and randomness only enters in the determination of prices. It should be pointed out, however, that the waiting times between two quotes are also random quantities, but they are assumed to not contribute to the price determination process. Whether a calendar-time or a business-time framework should be adopted in modeling the stochastic nature of financial quotes has been a longly debated issue by the finance research community, and it clearly depends on many factors, like, for example, (a) adherence to the physical behavior of reported prices, (b) usefulness in terms of a theory to be developed, and (c) last but not least, a matter of taste. See, for example [6], [7] and [8]. In this paper, we suggest that business time is perhaps a better tool for modeling the asset dynamics than calendar time. In order to support our claim, we consider (1) returns corresponding to a given calendar time lag and any business time lag, (2) returns corresponding to the same calendar time lag but having a fixed business time lag. We find out that their statistical properties are different consistently with the business hypothesis and inconsistently with the calendar one. In practice, we estimate some variances and some probability densities whose behavior is different in the two scenarios. The data set we use contains the DEM/USD exchange quotes taken from Reuters’ EFX pages (the data set having been supplied by Olsen & Associates) during a period of 1 year from January to December 1998. In this period, 1,620,843 quotes entries in the EFX system were recorded. The data set provides a continuously updated sequence of bid and asks exchange quotation pairs from individual institutions whose names and locations are also recorded. The reason for using FX data is that this market is not subject to any working time restriction; in fact, it is open 24 h a day, seven days a week. This is in contrast to stock markets, where artificial time regulation would have made it more difficult, if not impossible, to find out the results outlined in this paper.

In this paper, we suggest that the business time approach is perhaps a better way of modeling price dynamics than calendar time. In order to derive some insight from data we neglect possible autocorrelation between returns and possible autocorrelation between lags assuming implicitly that they would only give a second-order correction to our findings. With this simplification, our results altogether seem to provide enough evidence for the rejection of hypothesis H1 (calendar time model) and the acceptance of hypothesis H2 (business time model). Nevertheless, it should be noticed that hypothesis H1 assumes that the sampling process is independent of the price evolution. Therefore, our results do not rule out the continuous time model, but rather they show that the continuous time model would require correlations between processes M and S in order to fit the data. The deep reason of the behavior we point out in this paper is that, when an asset (at least a forex asset) is not traded, the prices evolution is slow while the evolution is fast when the asset is heavily traded. A faster evolution corresponds to a larger volatility in calendar time [11] and [12]; therefore, one could even maintain the calendar point of view, but in this case it should accept a seasonal modulation of volatility. The fact that the evolution of a price is slow when there are few transactions is very well known to practitioners, but it is still not accepted in its extremal consequence that prices are frozen when assets are not traded at all. This is because this behavior is in contrast to the stock market experience where opening prices are different from previous night closing prices. Nevertheless, the difference between the two markets is not astonishing if one thinks that the stock market is artificially time regulated, while the forex exchange market is an over the counter (OTC) market not subject to any time restriction.