On the statistics of scaling exponents and the multiscaling value at risk

Abstract

Research on scaling analysis in finance is vast and still flourishing. We introduce a novel statistical procedure based on the generalized Hurst exponent, the Relative Normalized and Standardized Generalized Hurst Exponent (RNSGHE), to robustly estimate and test the multiscaling property. 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 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 mainly 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.

The use of scaling properties to detect relevant changes in financial time series: a new visual warning tool

“…If B = 0, H q does not depend on q, i.e. H q = H for all q, hence the process is uniscaling, while if B = 0, the process is multiscaling [1,29,24,46]. For q = 1, the GHE is equivalent to the original Hurst exponent.…”

“…and conveys information on the span of the H q parameter. Conversely, the multiscaling curvature B is computed as the linear fit between q and H (θ) q , as described in Equation (3) ( [29,24]). If the process is uniscaling, both measures should be approximately zero as…”

“…Regarding the choice of the extreme values of q, we used two sets: For the multiscaling width W q 1 ,q 2 , we used a large span, q 1 = 0.1 and q 2 = 4, in order to capture the strong 'biasing' effect of the tails of the price change distributions as it has been reported elsewhere [24,29] for financial time-series. We want to include this "biased" version of the width in order to capture the dynamics of such bias and spot any transitions it may reveal in time.…”

“…It can also be caused by the endogeneity of markets for which a given order generates many other orders. The latter occurs especially in markets where algorithmic trading is prevalent [29]. However, scaling in a financial time series has also been shown to vary with time.…”

Forecasting Value-at-Risk in turbulent stock markets via the local regularity of the price process

“…Finally, it should be noticed that in recent years a substantial literature dealt with the Value-at-Risk in a multifractal context. For example, using simulation based on the Markov switching bivariate multifractal model, Calvet et al (2006) computed VaR in the US bond market and the exchange market for USD-AUD; using the multifractal random walk (MRW) and multifractal model of asset retruns (MMAR) respectively, Bacry et al (2008) and Batten et al (2014) measured VaR in exchange market; Bogachev and Bunde (2009) introduced a historical VaR estimation method considering multifractal property of data; Dominique et al (2011) find that the SP-500 Index is characterized by a high long-term Hurst exponent and construct a frequency-variation relationship that can be used as a practical guide to assess the Value-at-Risk; Lee et al (2016) introduce a VaR consistent with the multifractality of financial time series using the Multifractal Model of Asset Returns (MMAR); Brandi and Matteo (2021) propose a multiscaling consistent VaR using a Monte Carlo MRW simulation calibrated to the data. Anyway, the majority of such contributions follows the multifractal approach, which influences the regularity of the trajectories by acting on a proper time-change instead of calibrating the pointwise regularity itself.…”