On the statistics of scaling exponents and the Multiscaling Value at Risk

Abstract: Scaling and multiscaling financial time series have been widely studied in the literature. The research on this topic is vast and still flourishing. One way to analyze the scaling properties of time series is through the estimation of their scaling exponents, that are recognized as being valuable measures to discriminate between random, persistent, and anti-persistent behaviors in these time series. In the literature, several methods have been proposed to study the multiscaling property. In this paper, we use the generalized Hurst exponent (GHE) tool and we propose a novel statistical procedure based on GHE which we name Relative Normalized and Standardized Generalized Hurst Exponent (RNSGHE). This method is used to robustly estimate and test the multiscaling property and, together with a combination of t-tests and F-tests, serves to discriminate between real and spurious scaling. Furthermore, we introduce a new tool to estimate the optimal aggregation time used in our methodology which we name Autocororrelation Segmented Regression. We numerically validate this procedure on simulated time series by using the Multifractal Random Walk (MRW) and we then apply it to real financial data. We present results for times series with and without anomalies and we compute the bias that such anomalies introduce in the measurement of the scaling exponents. We also show how the use of proper scaling and multiscaling can ameliorate the estimation of risk measures such as Value at Risk (VaR). Finally, we propose a methodology based on Monte Carlo simulation, which we name Multiscaling Value at Risk (MSVaR), that takes into account the statistical properties of multiscaling time series. We show that by using this statistical procedure in combination with the robustly estimated multiscaling exponents, the one year forecasted MSVaR mimics the VaR on the annual data for the majority of the stocks analyzed.