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

We propose a method to measure the Hurst exponents of financial time series. The scaling of the absolute moments against the aggregation horizon of real financial processes and of both uniscaling and multiscaling synthetic processes converges asymptotically towards linearity in log-log scale. In light of this we found appropriate a modification of the usual scaling equation via the introduction of a filter function. We devised a measurement procedure which takes into account the presence of the filter function without the need of directly estimating it. We verified that the method is unbiased within the errors by applying it to synthetic time series with known scaling properties. Finally we show an application to empirical financial time series where we fit the measured scaling exponents via a second or a fourth degree polynomial, which, because of theoretical constraints, have respectively only one and two degrees of freedom. We found that on our data set there is not clear preference between the second or fourth degree polynomial. Moreover the study of the filter functions of each time series shows common patterns of convergence depending on the momentum degree.

Figure 1

Blue solid line: numerical scaling of $E\left[\right|S_N\left|\right]$ for a tBM with $n=3$ [cf. Eq. (A3) with $q=1]$; black solid line: theoretical expectation in the continuous limit.

Figure 2

Blue solid line: numerical scaling of $E\left[\right|S_N\left|\right]$ for a shuffled MRW [cf. Eq. (A3) with $q=1]$; black solid line: theoretical expectation in the continuous limit.

Figure 3

(a) Integrated measured scaling [cf. Eq. (26)]; (b) in blue solid line $c\left({\tau }^{*}\right)$ and the maximum where the best parabolic fit is attained marked with a red or shaded dot [cf. Eqs. (28) and (29)]; (c) zoom of the behavior of $c\left({\tau }^{*}\right)$ around the maximum where the best parabolic fit is attained (red or shaded dot) [cf. Eqs. (28) and (29)]; (d) plain scaling in blue solid line and the asymptotic inferred scaling in red dashed line [cf. Eqs. (23) and (25)]; (e) filter function $g\left(x\right)$ in blue solid line and the zero level in red dashed line; (f) integrated filter function [cf. Eq. (27)].

Figure 4

Fitted measured scaling exponents for a realization of a MRW. Blue crosses: measured scaling exponents; red solid line: polynomial fit; black solid lines: $99\%$ confidence intervals of the values of the fitted curve.

Figure 5

Step-by-step application of the method for the scaling of $H\left(0.1\right)$ for AXP. (a) Integrated measured scaling [cf. Eq. (26))]; (b) in blue solid line $c\left({\tau }^{*}\right)$ and the maximum where the best parabolic fit is attained marked with a red or shaded dot [cf. Eqs. (28) and (29)]; (c) zoom of the behavior of $c\left({\tau }^{*}\right)$ around the maximum where the best parabolic fit is attained (red or shaded dot) [cf. Eqs. (28) and (29)]; (d) plain scaling in blue solid line and the asymptotic inferred scaling in red dashed line [cf. Eqs. (23) and (25)]; (e) filter function $g\left(x\right)$ in blue solid line and the zero level in red dashed line; (f) integrated filter function [cf. Eq. (27)].

Figure 6

Step-by-step application of the method for the scaling of $H\left(1\right)$ for AXP. (a) Integrated measured scaling [cf. Eq. (26)]; (b) in blue solid line $c\left({\tau }^{*}\right)$ and the maximum where the best parabolic fit is attained marked with a red or shaded dot [cf. Eqs. (28) and (29)]; (c) zoom of the behavior of $c\left({\tau }^{*}\right)$ around the maximum where the best parabolic fit is attained (red or shaded dot) [cf. Eqs. (28) and (29)]; (d) plain scaling in blue solid line and the asymptotic inferred scaling in red dashed line [cf. Eqs. (23) and (25)]; (e) filter function $g\left(x\right)$ in blue solid line and the zero level in red dashed line; (f) integrated filter function [cf. Eq. (27)].

Figure 7

Blue crosses: empirical scaling exponents; red solid line: polynomial fit; black solid lines: $99\%$ confidence intervals of the values of the fitted curve.

Figure 8

Fitted empirical scaling exponents for ABT time series. Blue crosses: empirical scaling exponents; red solid line: polynomial fit; black solid lines: $99\%$ confidence intervals of the values of the fitted curve.

Figure 9

Fitted empirical scaling exponents for ACN time series. Blue crosses: empirical scaling exponents; red solid line: polynomial fit; black solid lines: $99\%$ confidence intervals of the values of the fitted curve.

Figure 10

Fitted empirical scaling exponents for ADBE time series. Blue crosses: empirical scaling exponents; red solid line: polynomial fit; black solid lines: $99\%$ confidence intervals of the values of the fitted curve.

Figure 11

Fitted empirical scaling exponents for AIG time series. Blue crosses: empirical scaling exponents; red solid line: polynomial fit; black solid lines: $99\%$ confidence intervals of the values of the fitted curve.

Figure 12

Fitted empirical scaling exponents for AMD time series. Blue crosses: empirical scaling exponents; red solid line: polynomial fit; black solid lines: $99\%$ confidence intervals of the values of the fitted curve.

Figure 13

Fitted empirical scaling exponents for GOOGL time series. Blue crosses: empirical scaling exponents; red solid line: polynomial fit; black solid lines: $99\%$ confidence intervals of the values of the fitted curve.

Figure 14

Fitted empirical scaling exponents for HON time series. Blue crosses: empirical scaling exponents; red solid line: polynomial fit; black solid lines: $99\%$ confidence intervals of the values of the fitted curve.

Figure 15

Fitted empirical scaling exponents for MAR time series. Blue crosses: empirical scaling exponents; red solid line: polynomial fit; black solid lines: $99\%$ confidence intervals of the values of the fitted curve.

Figure 16

Fitted empirical scaling exponents for MMM time series. Blue crosses: empirical scaling exponents; red solid line: polynomial fit; black solid lines: $99\%$ confidence intervals of the values of the fitted curve.

Figure 17

Fitted empirical scaling exponents for PG time series. Blue crosses: empirical scaling exponents; red solid line: polynomial fit; black solid lines: $99\%$ confidence intervals of the values of the fitted curve.