Scaling Properties of Financial Time Series

This book first critisizes standard financial theory.The focus will be on return distributions, the efficient market hypothesis and the independence of returns. Part two gives the intuition to look at markets in a different way. Namely the one proposed by Benoit Mandelbrot who has shown that nature itself can often be described with fractals. In the folowing, the relationship between fractal power laws und scaling will be explained. The main part focuses on the estimation of the tail index as a scaling parameter with the help of three different techniques: 1. OLS regression on a log-log plot, 2. Hill estimator and 3. the alpha exponent within the stable distribution. The next section shows a different power law exponent and will be used to test for long-memory effects (i.e. nonperiodical cycles) in return distributions. This exponent is the well known Hurst exponenent and will be compared to Andrew Lo\'s test statistic. The last section concludes and emphasizes that we are far from return predictabilty and should not try to explain marktet movements with fundamental\"causes\". However, power laws provide a way to describe financial markets.