Waiting-times and returns in high-frequency financial data: an empirical study

Raberto, Marco; Scalas, Enrico; Mainardi, Francesco

doi:10.1016/s0378-4371(02)01048-8

Cited by 441 publications

(266 citation statements)

References 12 publications

(17 reference statements)

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“…(8) needed for Eq. (7), we used standard algorithms for E β (−t β ) [36,39,42], including the fast Fourier transform. In Fig.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The Mittag-Leffler distribution is an important example of fat-tailed waiting times; it arises as the natural survival probability leading to time-fractional diffusion equations. There is increasing evidence for physical phenomena [33,34,35] and human activities [36,37,38] that do not follow either exponential or, equivalently, Poissonian statistics. Equations (7) and (8) can be obtained by FourierLaplace transformation of the FDE, recalling the definition of the fractional derivatives used in Eq.…”

Section: B Fractional Diffusion Equationmentioning

confidence: 99%

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Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation

2008

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We present a numerical method for the Monte Carlo simulation of uncoupled continuous-time random walks with a Lévy α-stable distribution of jumps in space and a Mittag-Leffler distribution of waiting times, and apply it to the stochastic solution of the Cauchy problem for a partial differential equation with fractional derivatives both in space and in time. The one-parameter MittagLeffler function is the natural survival probability leading to time-fractional diffusion equations. Transformation methods for Mittag-Leffler random variables were found later than the well-known transformation method by Chambers, Mallows, and Stuck for Lévy α-stable random variables and so far have not received as much attention; nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the Mittag-Leffler distribution. Combining the two methods, we obtain an accurate approximation of space-and time-fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes.

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“…(8) needed for Eq. (7), we used standard algorithms for E β (−t β ) [36,39,42], including the fast Fourier transform. In Fig.…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: B Fractional Diffusion Equationmentioning

confidence: 99%

Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation

2008

Self Cite

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“…Several researchers have recently investigated the statistical properties of waiting times of high-frequency financial data [17][18][19][20][21][22][23][24], and Scalas et al [17][18][19][20][21] in particular have applied the theory of continuous time random walk (CTRW) to financial data. They also found that the waiting-time survival probability for high-frequency data of the 30 DJIA stocks is non-exponential [21].…”

Section: Introductionmentioning

confidence: 99%

Power law for the calm-time interval of price changes

Kaizoji

Kaizoji²

2004

Physica A: Statistical Mechanics and its Applications

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In this paper, we describe a newly discovered statistical property of time series data for daily price changes. We conducted quantitative investigation of the calm-time intervals of price changes for 800 companies listed in the Tokyo Stock Exchange, and for the Nikkei 225 index over a 27-year period from January 4, 1975 to December 28, 2001. A calm-time interval is defined as the interval between two successive price changes above a fixed threshold. We found that the calm-time interval distribution of price changes obeys a power law decay. Furthermore, we show that the power-law exponent decreases monotonically with respect to the threshold.

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“…However, on the other hand, recent empirical studies [3,4,5,6] observed that the waiting time distribution is non-exponential in different markets. Therefore, in order to understand market behavior quantitatively and systematically, it would be important to check validity of the exponential distribution hypothesis.…”

Section: Introductionmentioning

confidence: 98%

On the gap between an empirical distribution and an exponential distribution of waiting times for price changes in a financial market

Sazuka¹

2007

Physica A: Statistical Mechanics and its Applications

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We analyze waiting times for price changes in a foreign currency exchange rate. Recent empirical studies of high frequency financial data support that trades in financial markets do not follow a Poisson process and the waiting times between trades are not exponentially distributed. Here we show that our data is well approximated by a Weibull distribution rather than an exponential distribution in a non-asymptotic regime. Moreover, we quantitatively evaluate how much an empirical data is far from an exponential distribution using a Weibull fit. Finally, we discuss a phase transition between a Weibull-law and a power-law in the asymptotic long waiting time regime.

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Waiting-times and returns in high-frequency financial data: an empirical study

Cited by 441 publications

References 12 publications

Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation

Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation

Power law for the calm-time interval of price changes

On the gap between an empirical distribution and an exponential distribution of waiting times for price changes in a financial market

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