Measuring multiscaling in financial time-series

Abstract

We discuss the origin of multiscaling in financial time-series and investigate how to best quantify it. Our methodology consists in separating the different sources of measured multifractality by analysing the multi/uni-scaling behaviour of synthetic time-series with known properties. We use the results from the synthetic time-series to interpret the measure of multifractality of real log-returns timeseries. The main finding is that the aggregation horizon of the returns can introduce a strong bias effect on the measure of multifractality. This effect can become especially important when returns distributions have power law tails with exponents in the range [2,5]. We discuss the right aggregation horizon to mitigate this bias.

Time-dependent scaling patterns in high frequency financial data

“…The origin of the scaling laws could be attributed to the autocorrelation of the process, the presence of heavy tails and the non-stationarity of the time series. It is beyond the purpose of this paper to investigate this aspect (which is discussed in [22] by using a different approach). Our aim here was to uncover non-stationary scaling patterns which we showed to be significant, reproducible and characteristic of specific stock markets.…”

“…Another self-similar process with stationary and independent increments but with heavy-tailed infinite variance distribution is the α-stable Lévy motion [19]. Empirical estimates of the scaling properties are affected by both the autocorrelation between the increments, as in the case of FBM, and the high variability, as in stochastic processes whose increments are independent and heavy-tailed (Lévy processes) [21,22].…”

Asymptotic scaling properties and estimation of the generalized Hurst exponents in financial data

“…However, as shown in Ref. [42], the estimation of the scaling exponents turned out to be strongly biased. Convergence issues arise for both power law-tailed and autocorrelated discrete processes, for both synthetic and real data.…”

“…In a previous paper [42], solving an ongoing debate in the literature (see, for example, Refs. [43][44][45]), it has been clarified that the true source of the multifractal behavior found in empirical financial time series is their causal structure.…”

“…In particular, already for processes with i.i.d. increments but with power law tails in their distribution with exponents between two and five, which is the range empirically observed [46], due to the slow convergence of the Central Limit Theorem, the small aggregation horizon is affected by strong biases [42]. In light of this, we now face the problem of building up an estimation procedure able to address these issues and to reduce as much as possible these biases by proposing a reliable proxy of the asymptotic multifractal behavior of real financial time series.…”