Indication of multiscaling in the volatility return intervals of stock markets

Abstract

The distribution of the return intervals τ between price volatilities above a threshold height q for financial records has been approximated by a scaling behavior. To explore how accurate is the scaling and therefore understand the underlined non-linear mechanism, we investigate intraday datasets of 500 stocks which consist of the Standard & Poor's 500 index. We show that the cumulative distribution of return intervals has systematic deviations from scaling. We support this finding by studying the m-th moment µ m ≡ (τ / τ ) m 1/m , which show a certain trend with the mean interval τ . We generate surrogate records using the Schreiber method, and find that their cumulative distributions almost collapse to a single curve and moments are almost constant for most range of τ . Those substantial differences suggest that non-linear correlations in the original volatility sequence account for the deviations from a single scaling law. We also find that the original and surrogate records exhibit slight tendencies for short and long τ , due to the discreteness and finite size effects of the records respectively. To avoid as possible those effects for testing the multiscaling behavior, we investigate the moments in the range 10 < τ ≤ 100, and find the exponent α from the power law fitting µ m ∼ τ α has a narrow distribution around α = 0 which depend on m for the 500 stocks. The distribution of α for the surrogate records are very narrow and centered around α = 0. This suggests that the return interval distribution exhibit multiscaling behavior due to the non-linear correlations in the original volatility.

Multifactor analysis of multiscaling in volatility return intervals

“…(1), the different moments should be independent on τ and therefore the exponent δ should be 0. A significant δ suggests multiscaling, and the value of δ characterizes the strength of the multiscaling [24], thus we call δ multiscaling exponent [34]. We find that µ m changes systematically with τ .…”

“…The normalization constant c and the scaled parameter a depend on the exponent γ, and thus f (x) has only one free parameter [16,24]. When the record has no long-term correlations, the return intervals follow as expected an exponential distribution, i.e.…”

“…Market activity such as the intertrade time shows multiscaling in its distribution [28,29]. Recently we suggested that the return intervals distribution has multiscaling characteristics based on cumulative distributions and moments of scaled intervals for 500 constituents of the Standard & Poor's 500 index [24]. The following questions are, can we detect multiscaling for a broader market?…”

“…The temporal structure in volatilities is complex and still regarded as an open problem. Return interval τ , also called recurrence time or interspike interval, which is the time interval between two consecutive volatilities above a certain threshold q, provides a new approach to analyze long-term correlated time series [13,14,15,16,17,18,19,20,21,22,23,24]. Recent studies on financial markets [17,18,19,20,21] show that, for both daily and intraday data, i) the distribution of scaled interval τ / τ can be approximated by a single scaling function, where τ is the average of τ .…”

“…In a recent paper [24], indications of deviations from the scaling function Eq. (2) were observed for the return intervals.…”