Simulating Multifractal Signals for Risk Assessment

Abstract

Many data sets collected from physical processes or human engineered systems exhibit self-similar properties that are best understood from the perspective of multifractals. These signals fail to satisfy the mathematical definition of stationarity and are therefore incompatible with Gaussian-based analysis. Efficient algorithms for analyzing the multifractal properties exist, but there is a need to simulate signals that exhibit the same multifractal spectrum as an empirical data set. The following work outlines two different algorithms for simulating multifractal signals and addresses the strengths and weaknesses of each approach. We introduce a procedure for fitting the parameters of a multifractal spectrum to one extracted empirically from data and illustrate how the algorithms can be employed to simulate potential future paths of a multifractal process. We illustrate the procedure using a high-frequency sample of IBM’s stock price and demonstrate the utility of simulating multifractals in risk management.