Fractal Geometry of Financial Time Series

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 that not only 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 Lévy’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.