《Fractal Geometry of Financial Time Series》.pdf

Appeared in: Fractals Vol. 3, No. 3, pp. 609-616 (1995), and in: Fractal Geometry and Analysis, The Mandelbrot Festschrift, Cura¸cao 1995, World Scientific (1996) FractalGeometryofFinancialTimeSeries CarlJ.G.Evertsz Center for Complex Systems and Visualization, University of Bremen FB III, Box 330 440, D-28334 Bremen, Germany Abstract– A simple quantitative measure of the self-similarity in time-series in general and in the stock market in particular is the scaling behavior of the absolute size of the jumps across lags of size k. A stronger form of self-similarity entails not only that this mean absolute value, but also the full distributions of lag-k jumps have a scaling behavior characterized by the above Hurst exponent. In 1963 Benoit Mandelbrot showed that cotton prices have such a strong form of (distributional) self-similarity, and for the first time introduced L´evy’s stable random variables in the modeling of price records. This paper discusses the analysis of the self-similarity of high-frequency DEM-USD exchange rate records and the 30 main German stock price records. Distributional self- similarity is found in both cases and some of its consequences are discussed. 1 Introduction Self-similarity[1] in financial price records manifests itself in the virtual impossibility to distin- guish a daily price record from, say, a monthly, when the axis are not labeled. Figure 1 illustrates this phenomenon for the German DAX composite index. The left figure plots the logarithm of the daily closing prices of the index over the period 1986 to 1993.