Are volatility correlations in financial markets related to Omori processes occurring on all scales?

Abstract

We analyze the memory in volatility by studying volatility return intervals, defined as the time between two consecutive fluctuations larger than a given threshold, in time periods following stock market crashes. Such an aftercrash period is characterized by the Omori law, which describes the decay in the rate of aftershocks of a given size with time t by a power law with exponent close to 1. A shock followed by such a power law decay in the rate is here called Omori process. Studying several aftercrash time series, we show that the Omori law holds not only after significant market crashes, but also after ``intermediate shocks''. Moreover, we find self-similar features in the volatility. Specifically, within the aftercrash period there are smaller shocks that themselves constitute Omori processes on smaller scales, similar to the Omori process after the large crash. We call these smaller shocks subcrashes, which are followed by their own aftershocks. We also find similar Omori processes after intermediate crashes in time regimes without a large market crash. By appropriate detrending we remove the influence of the crashes and subcrashes from the data, and find that this procedure significantly reduces the memory in the records. Our results are consistent with the hypothesis that the memory in volatility is related to Omori processes present on different time scales.