Model for interevent times with long tails and multifractality in human communications: An application to financial trading

Social, technological, and economic time series are divided by events which are usually assumed to be random, albeit with some hierarchical structure. It is well known that the interevent statistics observed in these contexts differs from the Poissonian profile by being long-tailed distributed with resting and active periods interwoven. Understanding mechanisms generating consistent statistics has therefore become a central issue. The approach we present is taken from the continuous-time random-walk formalism and represents an analytical alternative to models of nontrivial priority that have been recently proposed. Our analysis also goes one step further by looking at the multifractal structure of the interevent times of human decisions. We here analyze the intertransaction time intervals of several financial markets. We observe that empirical data describe a subtle multifractal behavior. Our model explains this structure by taking the pausing-time density in the form of a superstatistics where the integral kernel quantifies the heterogeneous nature of the executed tasks. A stretched exponential kernel provides a multifractal profile valid for a certain limited range. A suggested heuristic analytical profile is capable of covering a broader region.

Figure 1

(Color online) The pausing-time distribution for different cases as a function of the $t∕{\tau }_0$. The dashed line provides the Poisson case as given by Eq. (4). The dotted line is the PTD, Eq. (5), when $\rho \left(\epsilon \right)$ is uniform (${\tau }_0=1$, $\Delta =3$, $\beta =1$). The solid line represents the PTD, Eq. (8), when $\rho \left(\epsilon \right)$ follows the Laplace density (7) [$\sigma =2$ and $\beta =1$; hence, the power-law exponent is $3∕2$; cf. Eq. (9)].

Figure 2

(Color online) The normalized moment $⟨t^q⟩∕\Gamma (1+q)$ as a function of the order $q$ from six empirical financial data sets in a semilogarithmic scale. Solid lines correspond to regressions with functions of the form ${\tau }^q$ for $q$ values between 10 and 20.

Figure 3

(Color online) The ratio ${\varphi }_i\left(q\right)$ defined in Eq. (30) as a function of the order $q$ from six empirical financial data sets in a semilogarithmic scale. Observe the merging of all data sets for $q>10$.

Figure 4

(Color online) The function $f_{\mathrm{F}}\left(q\right)$ defined in Eq. (31) as a function of the order $q$ from six empirical financial data sets. Observe again the merging of all data sets for $q>10$.

Figure 5

(Color online) The emphasized plot presented in the semilogarithmic scale, taken from Fig. 2, from six empirical financial data sets for $⟨t^q⟩∕\Gamma (1+q)$. Solid lines provide two different theoretical predictions for the DAX index and Telefonica (TEF) according to multifractal formula (26) with parameters given by Table .

Figure 6

(Color online) The function $f_{\mathrm{MF}}\left(q\right)$ defined in Eq. (33) as a function of the scaled moment order $b{q}^{1∕(\alpha -1)}$ from six empirical financial data sets. Observe again the merging of all data sets for small $q$ to linear curve with slope equals 1.

Figure 7

(Color online) The normalized moment $⟨t^q⟩∕\Gamma (1+q)$ as a function of the order $q$ from Telefonica stock and DAX index in a semilogarithmic scale. Solid lines correspond to regressions assuming the heuristic multifractal (HMF) formula (34) with parameters from Table , while dashed lines show again the multifractal (MF) model with an stretched exponential superstatistics (26) with parameters from Table . The inset focuses in on a shorter domain of $q$ values.

Figure 8

(Color online) The function $f_{\mathrm{HMF}}\left(q\right)$ defined in Eq. (36) as a function of the order $b_1{q}^{1∕(\alpha -1)}$ from six empirical financial data sets using parameters of Table . We finally observe the merging of all data sets for $0<q<20$ along a curve of the form $1-\exp (-x)$ where $x=b_1{q}^{1∕(\alpha -1)}$.

Figure 9

(Color online) The transformed $q$ moment given by Eq. (37) as a function of $\ln L=b$ when $q=2$ (bottom line) and $q=5$ and $q=15$ (line above) and for six empirical financial data. Solid lines verify linear dependence across different markets.

Figure 10

(Color online) The sojourn probability $\Psi \left(t\right)$ of the Telefonica (TEF) stock. The solid line provides the numerical computation of $\Psi \left(t\right)$ for the MF model as given by Eq. (40) with parameters $b=0.12$, $\alpha =1.6$, and $\ln \left({\tau }_0\right)+\mu \beta =1.7$. Dashed and dotted lines are, respectively, the fits with the $q$ exponential (with $m=0.94\pm 0.07$ and $q=1.3\pm 0.2$) and the Weibull probabilities (with $a=1.53\pm 0.02$ and $c=0.459\pm 0.004$).