Multifactor analysis of multiscaling in volatility return intervals

Abstract

We study the volatility time series of 1137 most traded stocks in the US stock markets for the two-year period 2001-02 and analyze their return intervals τ , which are time intervals between volatilities above a given threshold q. We explore the probability density function of τ , P q (τ ), assuming a stretched exponential function, P q (τ ) ∼ e −τ γ . We find that the exponent γ depends on the threshold in the range between q = 1 and 6 standard deviations of the volatility. This finding supports the multiscaling nature of the return interval distribution. To better understand the multiscaling origin, we study how γ depends on four essential factors, capitalization, risk, number of trades and return. We show that γ depends on the capitalization, risk and return but almost does not depend on the number of trades. This suggests that γ relates to the portfolio selection but not on the market activity. To further characterize the multiscaling of individual stocks, we fit the moments of τ , µ m ≡ (τ / τ ) m 1/m , in the range of 10 < τ ≤ 100 by a power-law, µ m ∼ τ δ . The exponent δ is found also to depend on the capitalization, risk and return but not on the number of trades, and its tendency is opposite to that of γ. Moreover, we show that δ decreases with γ approximately by a linear relation. The return intervals demonstrate the temporal structure of volatilities and our findings suggest that their multiscaling features may be helpful for portfolio optimization.

First-Passage Time Properties of Correlated Time Series with Scale-Invariant Behavior and with Crossovers in the Scaling

“…The main statistical properties of the FPT are the functional form of its probability distribution p.`/ and its average length h`i. Empirical studies have reported a variety of forms for the probability distribution of FPT, including (1) pure exponential for random uncorrelated processes [2], (2) stretched exponential forms for a diverse group of natural and social complex systems ranging from neuron firing [3], climate fluctuations [4], or heartbeat dynamics [5], to Internet traffic [6,7] and stock market activity [8,9]; and (3) power-law form for certain on-off intermittency processes related to nonlinear electronic circuits [10] and anomalous diffusion [11][12][13][14]. Such diverse behavior is traditionally attributed to the specifics of the individual system.…”

Statistical Regularities of Seismic Noise

“…11), 15), 36) In after earthquake data, we note that the max amplitude varies in a very huge field. Therefore we test the relation between H and the max amplitude.…”

“…This result does not satisfy the linear correlation function β = 1−H as seen in financial data and human interaction data. 35), 36) Therefore there must be some differences between the seismic noise data and others, e.g. some assumptions needed for the relation of β = 1 − H might not be fulfilled.…”

“…Previous studies have indicated that the relation between the scaling in the growth rate and the long-range correlation is consistent with β = 1 − H in Human interaction 35) and Equity market activity. 36) This equation is based on a series of physical assumptions, such as stationarity of time series, power-law attenuation of autocorrelation, uniform distribution of Growth rate, etc, for details refer to Ref. 35).…”

“…To avoid huge fluctuations we define the growth rate at time t, g(t) as the logarithmic change of two consecutive cumulative values. 36) …”