Shuffling for understanding multifractality, application to asset price time series

Multifractal analysis has become a standard signal processing tool successfully used to model scale-free temporal dynamics in many applications, very different in nature. This is notably the case in financial engineering where, after Man-delbrot's seminal contributions, multifractal models have been used since the late 90ies to describe temporal fluctuations in asset prices. However, what exact features of temporal dynamics are actually encoded in multifractal properties remains generally only partially understood. In finance, notably, multifractality is associated to the burstiness of the returns, yet its relation to trends (signs of the returns) or volatility (modulus of the returns) remains unclear. Comparing the estimated multifractal properties of well-controlled synthetic multifractal processes to those of surrogate data, obtained by applying random permutations (shuffling) either to signs, or to modulus, or to both, of increments of original data, permits to better understand what aspects of temporal dynamics are captured by multifractality. The same procedure applied to a large dataset of asset prices entering the composition of the Eurostoxx600 index permits to evidence a simple and solid relation between multifractality and volatility as well as a weaker and complicated relation to returns.