Physica A: Statistical Mechanics and its Applications
The moving averages demystified
Introduction
In Physics, theories are mainly motivated by observations. Conversely, experiments are set up in order to either confirm or infirm a theory. In Finance, the situation is quite different: there is a huge gap between econometry and empirical finance. Indeed, the hypothesis of a pure random stock market is almost taken for granted in econometry while it is definetely not considered as such in empirical finance. Econophysicists are trying to fill the above gap as Stanley said in his contribution “Can Physics contribute to Finance?” [1].
The question of the present contribution is quite the opposite: “Can Finance contribute to Physics?”. The answer of this question is undoubtedly YES. We will illustrate this answer in the particular case of a technical tool, the so-called moving averages.
Moving averages are common tools in Technical Analysis [2], [3]. By definition, a moving average at time t of a signal y iswhere T is the time interval over which the average is calculated. It is easy to show that if the trend of y(t) is positive, the moving average will be below y, while when the trend is negative.
Consider two different moving averages and characterized, respectively, by T1 and T2 intervals such that T2>T1. These moving averages are illustrated in Fig. 1 for the specific case of a typical financial time series, i.e. the evolution of Apple stock price from January 1st 1987 till December 31th 1996, and for the parameter values T1=50 and T2=200. The crossings of and coincide with drastic changes of the trend of y(t). If y(t) increases for a long period before decreasing rapidly, will cross from above. This event is called a “death cross” in empirical finance [2]. On the contrary, if crosses from below, the crossing point coincides with an upsurge of the signal y(t). This event is called a “gold cross”. Chartists often try to “extrapolate” the evolution of and expecting “gold” or “death” crosses. Most computers on trading places are equiped for performing this kind of analysis and forecasting [3]. Such curves are automatically displayed on stock charts on most trading softwares. Obviously, the positions of the crossing points are determined by the past history of the data and not to the future such that the forecasting in empirical finance as based on moving average “recipes” are far from a short-/long-range quantitative forecasting.
As physicists, we do not endorse these charting methods. Even though moving averages seem to be poor statistical measures, we will however see in this paper that they present some very practical interest for physicists and raise new questions in statistical physics.
Section snippets
Crossing points
The artificial time series used for the following demonstration within the successive random addition method originates in d=1 landscape profile construction. This method is also called “midpoint displacement” in the literature [4]. With this algorithm based on iterations, one generates a sequence of length N=2n+1 where n is an iteration number. At each iteration, one finds the intermediate positions (midpoints) of couples of neighboring points and calculates the values of the signal at the
Spectrum of moving averages
It is easy to show that the distance from the signal y to the moving average is proportional to the slope of the signal and proportional to the interval T. One such moving average is thus extracting some information about the trend of y over T. On this basis, we have developed a method that uses a larger set of moving averages in order to visualize different trends on any time scale.
The basic idea of the “spectrum of moving averages” is to fix the long-term period T2 to a high value. Then,
Conclusion
We have shown that a poor statistical tool though very commonly used in empirical finance can contribute to fundamental physics. Fractional Brownian motions have been considered. They lead to a non-trivial density of crossing points of moving averages. This has been used to develop some analysis techniques. A spectral transform has been shown to provide a useful technique for visualizing the trends and cycles on various length scales lying in such a signal.
Acknowledgements
NV is financially supported by the FNRS. A special grant from FNRS/LOTTO allowed to perform specific numerical work. Thanks to E. Labie for stimulating comments about moving averages. The referees are acknowledged for their constructive reports.
References (7)
- H.E. Stanley et al., Physica A 269 (1999) 156 [these...
- E. Labie, private...
- A.G. Ellinger, The Art of Investment, Bowers&Bowers, London,...
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