[Paper Review] Waiting-time distribution for a stock-market index

Jae Wook Lee, Kyoung Eun Lee+1 more|ArXiv.org|Aug 30, 2005

Complex Systems and Time Series Analysis3 citations

TL;DR

This study investigates the waiting-time distribution (WTD) of absolute returns in the KOSPI stock index using high-frequency data (1992–1999). It reveals a power-law decay in the WTD with two scaling regimes separated by a crossover time ≈200 minutes, where the exponent β₂ remains nearly constant across thresholds but decreases with increasing return time Δt, indicating long-range correlations and scaling behavior in market volatility clusters.

ABSTRACT

We investigate the waiting-time distribution of the absolute return in the Korean stock-market index KOSPI. We define the waiting time as a time interval during which the normalized absolute return remains continuously below a threshold $r_c$. Through an exponential bin plot, we observe that the waiting-time distribution shows power-law behavior, $p_f (t) \sim t^{-β}$, for a range of threshold values. The waiting-time distribution has two scaling regimes, separated by the crossover time $t_c \approx 200$ min. The power-law exponents of the waiting-time distribution decrease when the return time $Δt$ increases. In the late-time regime, $t > t_c$, the power-law exponents are independent of the threshold to within the error bars for fixed return time.

Motivation & Objective

• To examine the statistical properties of waiting times between volatility bursts in the KOSPI index.
• To determine whether the waiting-time distribution (WTD) of absolute returns exhibits power-law scaling, indicating long-range correlations in market activity.
• To investigate how the scaling exponents of the WTD depend on the return time Δt and threshold rc.
• To distinguish between correlated market dynamics and uncorrelated stochastic behavior by comparing original and shuffled data.

Proposed method

• Defined the normalized absolute return r(t) as |g(t) − ⟨g(t)⟩| / σ(Δt), where g(t) is the logarithmic return over time interval Δt.
• Introduced the waiting time as the interval between consecutive crossings of a threshold rc by the absolute return, representing calm periods between volatility bursts.
• Applied exponential binning to improve statistical resolution in the tail of the WTD, enabling accurate estimation of power-law exponents.
• Used least-squares fitting on log-log plots of exponentially binned histograms to extract scaling exponents β₁ and β₂ for early and late-time regimes.
• Compared original data with 100× shuffled data to isolate the role of temporal correlations in generating power-law behavior.
• Scaled the WTD by the mean waiting time ⟨T⟩ to test for universal scaling across different thresholds.

Experimental results

Research questions

• RQ1 Does the waiting-time distribution of absolute returns in the KOSPI index exhibit power-law scaling?
• RQ2 Are there distinct scaling regimes in the waiting-time distribution, and what is the nature of the crossover time?
• RQ3 How do the power-law exponents of the waiting-time distribution depend on the return time Δt and threshold rc?
• RQ4 To what extent are the observed scaling behaviors due to temporal correlations in the return time series?
• RQ5 Is the late-time scaling exponent β₂ independent of the threshold rc, as expected for universal scaling?

Key findings

• The waiting-time distribution (WTD) of absolute returns in the KOSPI index exhibits power-law behavior p_f(t) ∼ t^−β for t > t_c, with a crossover time t_c ≈ 200 minutes.
• For t < t_c, the WTD has a power-law exponent β₁ ≈ 1.48(3), while for t > t_c, the exponent β₂ is nearly independent of the threshold rc, with β₂ ≈ 2.0 for Δt = 1 min.
• The late-time exponent β₂ decreases with increasing return time Δt, dropping to 1.58 at Δt = 60 min and 1.42 at Δt = 600 min.
• The WTD for shuffled data follows an exponential distribution, confirming that power-law scaling arises from temporal correlations in the original time series.
• Scaling the WTD by the mean waiting time ⟨T⟩ collapses the curves across different thresholds, indicating universal scaling in the late-time regime.
• The fat-tailed distribution of absolute returns (with power-law exponent α ≈ 3.06 for Δt = 1 min) contributes to the observed WTD scaling, likely due to herding and nonlinear market dynamics.