Large deviation and self-similarity analysis of graphs: DAX stock prices

Two methods for analyzing graphs such as those occurring in the stock market, geographical profiles and rough surfaces, are investigated. They are based on different scaling laws for the distributions of jumps as a function of the lag. The first is a large deviation analysis, and the second is based on the concept of a self-similar process introduced by Mandelbrot and van Ness. We show that large deviation analysis does not apply to either the stock market nor fractional Brownian motion (H = 0.5). Instead the analysis based on self-similarity is applicable to both, and does indicate that especially the negative log-price fluctuations have a large degree of self-similarity. The latter analysis allows one to probe the degree of self-similarity of a process, beyond what is possible with the exponent H typically used to describe self-affine graphs.