The inherently stochastic nature of economic phenomena that evolve continuously at multiple frequencies – such as stock exchange, but also energy feedstock prices – poses difficulties and challenges to their systematic study, but also renders traditional deterministic analysis tools inadequate, especially inasmuch as quantitative understanding of intrinsic structure is an issue. Benoit Mandelbrot (1963), in a seminar paper, discovered the surprising fact that the seemingly random signals of cotton prices, when appropriately examined by means of advanced statistical techniques, possess inherent structure – the latter can be quantitatively characterized by means of both conventional statistical metrics (e.g. kurtosis) as well as novel indices (e.g. drift exponent). Essentially, Mandelbrot's fractal theory is impressively powerful and accurate for the purpose of price series analysis, because it allows for the study of statistical distributions that have a well-defined range of mean variation, but also infinite variance variation: this corresponds very well to the aforementioned stochastic signals, which are essentially bounded for a given period of study (thus finite mean variation), yet evolve across orders of frequency (thus infinite variance).

Time series data often arise when monitoring industrial processes or tracking corporate business metrics; in this case, we are interested in the evolution of crude oil prices over several decades. Its definitive characteristic is that time series analysis accounts for the fact that data points taken over time may have an internal structure (such as autocorrelation, trend or seasonal variation) that should be accounted for, and can actually be analyzed quantitatively. Methods and tools include concepts such as stationarity and seasonality, and tools such as autocorrelation function, single and multiple exponential smoothing, fractal dimension and intrinsic time.

In many statistical autocorrelation studies of intra-day financial time series data, it has been identified (Muller et al., 1995) that the absolute prices of price changes behave like the fractional noise of Mandelbrot and Van Ness (Mandelbrot, 1963). The mean absolute price change yields a linear correlation to the analysis time interval size, in logarithmic coordinates, effectively indicating that price changes obey a scaling law – and, while the frequencies involved in the phenomenon span several orders of magnitude, they can be conclusively and comprehensively identified and studied.

This paper presents of historical price data for various crude oil types (Brent, Saudi, Iranian, Texas ), applying different degrees of time resolution; results are interrelated in order to identify patterns and analyze variation timescales. A specific target of this study is to investigate the presence of fractal properties. In particular, the hypothesis that the mean size of the absolute values of price changes follows a “fractal” scaling law (a power law) as a function of the analysis time interval (the latter has been considered in this study as an independently varying parameter, ranging from a few minutes up to a year). The analysis reveals some interesting trends, which are useful for understanding the seasonality but also the intrinsic structure of crude oil markets.

REFERENCES

1. Mandelbrot, “The variation of certain speculative prices”, Journal of Business 36: 394 (1963).

2. Mandelbrot and Taylor, “On the distribution of stock prices differences”, Operations Research 15: 1057 (1967).

3. Mandelbrot and Van Ness., “Fractional Brownian motions, fractional noises and applications”, SIAM Review 10: 422 (1968).

4. Muller, Dacorogna, Davé, Pictet, Olsen, Ward, “Fractals and intrinsic time – A challenge to econometricians”, Technical Report UAM.1993-08-16, The O&A Research Group (1993).

5. Mandelbrot and Hudson, “The (mis)behavior of markets: A fractal view of risk, ruin and reward”, Basic Books (2004).