Relation between volatility correlations in financial markets and Omori processes occurring on all scales

Financial earthquakes: SARS-CoV-2 news shock propagation in stock and sovereign bond markets

The SARS-CoV-2 epidemics outbreak has shocked global financial markets, inducing policymakers to put in place unprecedented interventions to inject liquidity and to counterbalance the negative impact on worldwide financial systems. Through the lens of statistical physics, we examine the financial volatility of the reference stock and bond markets of the United States, United Kingdom, Spain, France, Germany and Italy to quantify the effects of country-specific socio-economic and political announcements related to the epidemics. Main results show that financial markets exhibit heterogeneous behaviours towards news on the epidemics, with the Italian and German bond markets responding with major delays to shocks. Additionally, credit markets tend to be slower than equity markets in adjusting prices after shocks, hence being slower at incorporating the effects of such news.

The impact of the SARS-CoV-2 pandemic on financial markets: a seismologic approach

This work investigates financial volatility cascades generated by SARS-CoV-2 related news using concepts developed in the field of seismology. We analyze the impact of socio-economic and political announcements, as well as of financial stimulus disclosures, on the reference stock markets of the United States, United Kingdom, Spain, France, Germany and Italy. We quantify market efficiency in processing SARS-CoV-2 related news by means of the observed Omori power-law exponents and we relate these empirical regularities to investors' behavior through the lens of a stylized Agent-Based financial market model. The analysis reveals that financial markets may underreact to the announcements by taking a finite time to re-adjust prices, thus moving against the efficient market hypothesis. We observe that this empirical regularity can be related to the speculative behavior of market participants, whose willingness to switch toward better performing investment strategies, as well as their degree of reactivity to price trend or mispricing, can induce long-lasting volatility cascades.

Power-law rheology controls aftershock triggering and decay

The occurrence of aftershocks is a signature of physical systems exhibiting relaxation phenomena. They are observed in various natural or experimental systems and usually obey several non-trivial empirical laws. Here we consider a cellular automaton realization of a nonlinear viscoelastic slider-block model in order to infer the physical mechanisms of triggering responsible for the occurrence of aftershocks. We show that nonlinear viscoelasticity plays a critical role in the occurrence of aftershocks. The model reproduces several empirical laws describing the statistics of aftershocks. In case of earthquakes, the proposed model suggests that the power-law rheology of the fault gauge, underlying lower crust, and upper mantle controls the decay rate of aftershocks. This is verified by analysing several prominent aftershock sequences for which the rheological properties of the underlying crust and upper mantle were established.

Self-similar aftershock rates

In many important systems exhibiting crackling noise-an intermittent avalanchelike relaxation response with power-law and, thus, self-similar distributed event sizes-the "laws" for the rate of activity after large events are not consistent with the overall self-similar behavior expected on theoretical grounds. This is particularly true for the case of seismicity, and a satisfying solution to this paradox has remained outstanding. Here, we propose a generalized description of the aftershock rates which is both self-similar and consistent with all other known self-similar features. Comparing our theoretical predictions with high-resolution earthquake data from Southern California we find excellent agreement, providing particularly clear evidence for a unified description of aftershocks and foreshocks. This may offer an improved framework for time-dependent seismic hazard assessment and earthquake forecasting.

Copulas and time series with long-ranged dependencies

We review ideas on temporal dependencies and recurrences in discrete time series from several areas of natural and social sciences. We revisit existing studies and redefine the relevant observables in the language of copulas (joint laws of the ranks). We propose that copulas provide an appropriate mathematical framework to study nonlinear time dependencies and related concepts-like aftershocks, Omori law, recurrences, and waiting times. We also critically argue, using this global approach, that previous phenomenological attempts involving only a long-ranged autocorrelation function lacked complexity in that they were essentially monoscale.

Reducing financial avalanches by random investments

Building on similarities between earthquakes and extreme financial events, we use a self-organized criticality-generating model to study herding and avalanche dynamics in financial markets. We consider a community of interacting investors, distributed in a small-world network, who bet on the bullish (increasing) or bearish (decreasing) behavior of the market which has been specified according to the S&P 500 historical time series. Remarkably, we find that the size of herding-related avalanches in the community can be strongly reduced by the presence of a relatively small percentage of traders, randomly distributed inside the network, who adopt a random investment strategy. Our findings suggest a promising strategy to limit the size of financial bubbles and crashes. We also obtain that the resulting wealth distribution of all traders corresponds to the well-known Pareto power law, while that of random traders is exponential. In other words, for technical traders, the risk of losses is much greater than the probability of gains compared to those of random traders.

Market dynamics immediately before and after financial shocks: Quantifying the Omori, productivity, and Bath laws

We study the cascading dynamics immediately before and immediately after 219 market shocks. We define the time of a market shock T{c} to be the time for which the market volatility V(T{c}) has a peak that exceeds a predetermined threshold. The cascade of high volatility "aftershocks" triggered by the "main shock" is quantitatively similar to earthquakes and solar flares, which have been described by three empirical laws-the Omori law, the productivity law, and the Bath law. We analyze the most traded 531 stocks in U.S. markets during the 2 yr period of 2001-2002 at the 1 min time resolution. We find quantitative relations between the main shock magnitude M≡log{10} V(T{c}) and the parameters quantifying the decay of volatility aftershocks as well as the volatility preshocks. We also find that stocks with larger trading activity react more strongly and more quickly to market shocks than stocks with smaller trading activity. Our findings characterize the typical volatility response conditional on M , both at the market and the individual stock scale. We argue that there is potential utility in these three statistical quantitative relations with applications in option pricing and volatility trading.

Quantitative law describing market dynamics before and after interest-rate change

We study the behavior of U.S. markets both before and after U.S. Federal Open Market Commission meetings and show that the announcement of a U.S. Federal Reserve rate change causes a financial shock, where the dynamics after the announcement is described by an analog of the Omori earthquake law. We quantify the rate n(t) of aftershocks following an interest-rate change at time T and find power-law decay which scales as n(t-T)∼(t-T)(-Ω) , with Ω positive. Surprisingly, we find that the same law describes the rate n'(|t-T|) of "preshocks" before the interest-rate change at time T . This study quantitatively relates the size of the market response to the news which caused the shock and uncovers the presence of quantifiable preshocks. We demonstrate that the news associated with interest-rate change is responsible for causing both the anticipation before the announcement and the surprise after the announcement. We estimate the magnitude of financial news using the relative difference between the U.S. Treasury Bill and the Federal Funds effective rate. Our results are consistent with the "sign effect," in which "bad news" has a larger impact than "good news." Furthermore, we observe significant volatility aftershocks, confirming a "market under-reaction" that lasts at least one trading day.

Statistical analysis of the overnight and daytime return

We investigate the two components of the total daily return (close-to-close), the overnight return (close-to-open), and the daytime return (open-to-close), as well as the corresponding volatilities of the 2215 New York Stock Exchange stocks for the 20 year period from 1988 to 2007. The tail distribution of the volatility, the long-term memory in the sequence, and the cross correlation between different returns are analyzed. Our results suggest that (i) the two component returns and volatilities have features similar to that of the total return and volatility. The tail distribution follows a power law for all volatilities, and long-term correlations exist in the volatility sequences but not in the return sequences. (ii) The daytime return contributes more to the total return. Both the tail distribution and the long-term memory of the daytime volatility are more similar to that of the total volatility, compared to the overnight records. In addition, the cross correlation between the daytime return and the total return is also stronger. (iii) The two component returns tend to be anticorrelated. Moreover, we find that the cross correlations between the three different returns (total, overnight, and daytime) are quite stable over the entire 20 year period.

Multifactor analysis of multiscaling in volatility return intervals

We study the volatility time series of 1137 most traded stocks in the U.S. stock markets for the two-year period 2001-2002 and analyze their return intervals tau , which are time intervals between volatilities above a given threshold q . We explore the probability density function of tau , P_(q)(tau) , assuming a stretched exponential function, P_(q)(tau) approximately e;(-tau;(gamma)) . We find that the exponent gamma 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 gamma depends on four essential factors, capitalization, risk, number of trades, and return. We show that gamma depends on the capitalization, risk, and return but almost does not depend on the number of trades. This suggests that gamma relates to the portfolio selection but not on the market activity. To further characterize the multiscaling of individual stocks, we fit the moments of tau , mu_(m) identical with(tautau);(m);(1m) , in the range of 10<tau< or =100 by a power law, micro_(m) approximately tau;(delta). The exponent delta 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 gamma . Moreover, we show that delta decreases with increasing gamma 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.

Serial correlation of detrended time series

A preliminary essential procedure in time series analysis is the separation of the deterministic component from the random one. If the signal is the result of superposing a noise over a deterministic trend, then the first one must estimate and remove the trend from the signal to obtain an estimation of the stationary random component. The errors accompanying the estimated trend are conveyed as well to the estimated noise, taking the form of detrending errors. Therefore the statistical errors of the estimators of the noise parameters obtained after detrending are larger than the statistical errors characteristic to the noise considered separately. In this paper we study the detrending errors by means of a Monte Carlo method based on automatic numerical algorithms for nonmonotonic trends generation and for construction of estimated polynomial trends alike to those obtained by subjective methods. For a first order autoregressive noise we show that in average the detrending errors of the noise parameters evaluated by means of the autocovariance and autocorrelation function are almost uncorrelated to the statistical errors intrinsic to the noise and they have comparable magnitude. For a real time series with significant trend we discuss a recursive method for computing the errors of the estimated parameters after detrending and we show that the detrending error is larger than the half of the total error.

Memory effects in the statistics of interoccurrence times between large returns in financial records

We study the statistics of the interoccurrence times between events above some threshold Q in two kinds of multifractal data sets (multiplicative random cascades and multifractal random walks) with vanishing linear correlations. We show that in both data sets the relevant quantities (probability density functions and the autocorrelation function of the interoccurrence times, as well as the conditional return period) are governed by power laws with exponents that depend explicitly on the considered threshold. By studying a large number of representative financial records (market indices, stock prices, exchange rates, and commodities), we show explicitly that the interoccurrence times between large daily returns follow the same behavior, in a nearly quantitative manner. We conclude that this kind of behavior is a general consequence of the nonlinear memory inherent in the multifractal data sets.

Indication of multiscaling in the volatility return intervals of stock markets

The distribution of the return intervals tau 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 nonlinear mechanism, we investigate intraday data sets of 500 stocks which consist of 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 micro_{m} identical with(tau/tau);{m};{1/m} , which show a certain trend with the mean interval tau . 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 ranges of tau . Those substantial differences suggest that nonlinear 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 tau , 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<tau< or =100 , and find that the exponent alpha from the power law fitting micro_{m} approximately tau;{alpha} has a narrow distribution around alpha not equal0 which depends on m for the 500 stocks. The distribution of alpha for the surrogate records are very narrow and centered around alpha=0 . This suggests that the return interval distribution exhibits multiscaling behavior due to the nonlinear correlations in the original volatility.