Dynamics of real financial markets: A reply to Frank’s comment

This reply addresses the assertion in the comment of T.D. Frank [T.D. Frank, Physica A 387 (2008) 773] on our paper [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343] that the approach to modeling financial markets that we propose is unrealistic. In our paper, we considered variable diffusion processes that have a diffusion coefficient that varies with both position (return in finance) and time, and used them to show that measuring a Hurst exponent H≠1/2 in a time series does not necessarily imply correlations between increments. We also proposed that such a variable diffusion process is the underlying stochastic process governing the dynamics of financial markets. Frank asserts that this is unrealistic because variable diffusion processes with H≠1/2 are driven with a “force” that varies in time as a power law. He claims, instead, that markets obey nonextensive thermostatistics. We discuss evidence from a recently published empirical study of the Euro-Dollar exchange rate [K.E. Bassler, J.L. McCauley, G.H. Gunaratne, PNAS 104 (2007) 17287] that shows that the market can be described with a variable diffusion process, but is inconsistent with nonextensive thermostatistics. This evidence demonstrates that our modeling approach is realistic and accurate.     

Tail distribution of index fluctuations in World markets

We have investigated the tail distribution of the daily fluctuations in 202 different indices in the stock markets of 59 countries for the time span of the last 20 years. Power law, log-normal, Weibull, exponential and power law with exponential cutoff distributions are considered as possible candidates for the tail distribution of the normalized returns. It is found that the power exponent depends strongly on the choice of the tail threshold and a sizeable number of indices can be better fitted by a distribution function other than the power law at the region that has power law exponent of 3. Also, we have found that the power exponent is not an indicator of the maturity of the market.     

Multifractal analysis on international crude oil markets based on the multifractal detrended fluctuation analysis

The multifractal nature of WTI and Brent crude oil markets is studied employing the multifractal detrended fluctuation analysis. We find that two crude oil markets become more and more efficient for long-term and two Gulf Wars cannot change time scale behavior of crude oil return series. Considering long-term influence caused by Gulf Wars, we find such “turning windows” in generalized Hurst exponents obtained from three periods divided by two Gulf Wars so that WTI and Brent crude oil returns possess different properties above and below the windows respectively. Comparing with the results obtained from three periods we conclude that, before the First Gulf War, international crude oil markets possessed the highest multifractality degree, small-scope fluctuations presented the strongest persistence and large-scope fluctuations presented the strongest anti-persistence. We find that, for two Gulf Wars, the first one made a greater impact on international oil markets; for two markets, Brent was more influenced by Gulf Wars. In addition, we also verified that the multifractal structures of two markets’ indices are not only mainly attributed to the broad fat-tail distributions and persistence, but also affected by some other factors.     

Stylized facts in internal rates of return on stock index and its derivative transactions

Universal features in stock markets and their derivative markets are studied by means of probability distributions in internal rates of return on buy and sell transaction pairs. Unlike the stylized facts in normalized log returns, the probability distributions for such single asset encounters incorporate the time factor by means of the internal rate of return, defined as the continuous compound interest. Resulting stylized facts are shown in the probability distributions derived from the daily series of TOPIX, S & P 500 and FTSE 100 index close values. The application of the above analysis to minute-tick data of NIKKEI 225 and its futures market, respectively, reveals an interesting difference in the behavior of the two probability distributions, in case a threshold on the minimal duration of the long position is imposed. It is therefore suggested that the probability distributions of the internal rates of return could be used for causality mining between the underlying and derivative stock markets. The highly specific discrete spectrum, which results from noise trader strategies as opposed to the smooth distributions observed for fundamentalist strategies in single encounter transactions may be useful in deducing the type of investment strategy from trading revenues of small portfolio investors.     

Forecasting volatility in Shanghai and Shenzhen markets based on multifractal analysis

This paper analyzes the multifractality in Shanghai and Shenzhen stock markets using multifractal spectrum analysis and multifractal detrended fluctuation analysis. We find that the main source of multifractality is long-range correlations of large and small fluctuations. Then, we introduce a multifractal volatility measure (MV) and find that by taking MV as daily conditional volatility, the simulated series displayed similar “stylized facts” to the original daily return series. By capturing the dynamics of MV using the ARFIMA model, we find that the out-of-sample forecasting performance of the ARFIMA-MV model is better than some GARCH-class models and the ARFIMA-RV model under some criteria of loss function.     

Crossover Phenomena in Detrended Fluctuation Analysis Used in Financial Markets

A systematic analysis of Shanghai and Japan stock indices for the period of Jan. 1984 to Dec. 2005 is performed. After stationarity is verified by ADF （Augmented Dickey-Fuller） test, the power spectrum of the data exhibits a power law decay as a whole characterized by 1/f^β processes with possible long range correlations. Subsequently, by using the method of detrended fluctuation analysis （DFA） of the general volatility in the stock markets, we find that the long-range correlations are occurred among the return series and the crossover phenomena exhibit in the results obviously.Further, Shanghai stock market shows long-range correlations in short time scale and shows short-range correlations in long time scale. Whereas, for Japan stock market, the data behaves oppositely absolutely. Last, we compare the varying of scale exponent in large volatility between two stock markets. All results obtained may indicate the possibility of characteristic of multifractal scaling behavior of the financial markets.     

Probability of large movements in financial markets

Based on empirical financial time series, we show that the “silence-breaking” probability follows a super-universal power law: the probability of observing a large movement is inversely proportional to the length of the on-going low-variability period. Such a scaling law has been previously predicted theoretically [R. Kitt, J. Kalda, Physica A 353 (2005) 480], assuming that the length-distribution of the low-variability periods follows a multi-scaling power law.     

Modeling interactions of trading volumes in financial dynamics

A dynamic herding model with interactions of trading volumes is introduced. At time t, an agent trades with a probability, which depends on the ratio of the total trading volume at time t−1 to its own trading volume at its last trade. The price return is determined by the volume imbalance and number of trades. The model can reproduce the power-law distributions of the trading volume, number of trades and price return, and the probable relation between them. The exponents are tunable by adjusting the values of the parameters, but show slight deviation from those revealed in empirical studies. Moreover, the time series generated are long-range correlated. We demonstrate that the results are rather robust, and do not depend on the particular form of the trading probability.     

Impact of monetary policy changes on the Chinese monetary and stock markets

The impact of monetary policy changes on the monetary market and stock market in China is investigated in this study. The changes of two major monetary policies, the interest rate and required reserve ratio, are analyzed in a study period covering seven years on the interbank monetary market and Shanghai stock market. We find that the monetary market is related to the macro economy trend and we also find that the monetary change surprises both of lowering and raising bring significant impacts to the two markets and the two markets respond to the changes differently. The results suggest that the impact of fluctuations is much larger for raising policy changes than lowering changes in the monetary market on policy announcing and effective dates. This is consistent with the “sign effect”, i.e. bad news brings a greater impact than good news. By studying the event window of each policy change, we also find that the “sign effect” still exists before and after each change in the monetary market. A relatively larger fluctuation is observed before the event date, which indicates that the monetary market might have a certain ability to predict a potential monetary change, while it is kept secret by the central bank before official announcement. In the stock market, we investigate how the returns and spreads of the Shanghai stock market index respond to the monetary changes. Evidences suggest the stock market is influenced but in a different way than the monetary market. The climbing of returns after the event dates for the lowering policy agrees with the theory that lowering changes can provide a monetary supply to boost the market and drive the stock returns higher but with a delay of 2 to 3 trading days on average. While in the bear market, the lowering policy brings larger volatility to the market on average than the raising ones. These empirical findings are useful for policymakers to understand how monetary policy changes impact the monetary and stock markets especially in an emerging market like China where the economy is booming and the policy changes impact the markets as surprises by the central bank without a pre-decided schedule. This is totally different from previous studies on FED, which follows pre-decided schedules for monetary policy changes.     

Tick size and stock returns

Tick size is an important aspect of the micro-structural level organization of financial markets. It is the smallest institutionally allowed price increment, has a direct bearing on the bid-ask spread, influences the strategy of trading order placement in electronic markets, affects the price formation mechanism, and appears to be related to the long-term memory of volatility clustering. In this paper we investigate the impact of tick size on stock returns. We start with a simple simulation to demonstrate how continuous returns become distorted after confining the price to a discrete grid governed by the tick size. We then move on to a novel experimental set-up that combines decimalization pilot programs and cross-listed stocks in New York and Toronto. This allows us to observe a set of stocks traded simultaneously under two different ticks while holding all security-specific characteristics fixed. We then study the normality of the return distributions and carry out fits to the chosen distribution models. Our empirical findings are somewhat mixed and in some cases appear to challenge the simulation results.     

Wavepacket dynamics, quantum reversibility, and random matrix theory

We introduce and analyze the physics of “driving reversal” experiments. These are prototype wavepacket dynamics scenarios probing quantum irreversibility. Unlike the mostly hypothetical “time reversal” concept, a “driving reversal” scenario can be realized in a laboratory experiment, and is relevant to the theory of quantum dissipation. We study both the energy spreading and the survival probability in such experiments. We also introduce and study the “compensation time” (time of maximum return) in such a scenario. Extensive effort is devoted to figuring out the capability of either linear response theory or random matrix theory (RMT) to describe specific features of the time evolution. We explain that RMT modeling leads to a strong non-perturbative response effect that differs from the semiclassical behavior.     

Financial Return Distributions: Past,Present, and COVID-19

We analyze the price return distributions of currency exchange rates, cryptocurrencies, and contracts for differences (CFDs) representing stock indices, stock shares, and commodities. Based on recent data from the years 2017–2020, we model tails of the return distributions at different time scales by using power-law, stretched exponential, and q-Gaussian functions. We focus on the fitted function parameters and how they change over the years by comparing our results with those from earlier studies and find that, on the time horizons of up to a few minutes, the so-called “inverse-cubic power-law” still constitutes an appropriate global reference. However, we no longer observe the hypothesized universal constant acceleration of the market time flow that was manifested before in an ever faster convergence of empirical return distributions towards the normal distribution. Our results do not exclude such a scenario but, rather, suggest that some other short-term processes related to a current market situation alter market dynamics and may mask this scenario. Real market dynamics is associated with a continuous alternation of different regimes with different statistical properties. An example is the COVID-19 pandemic outburst, which had an enormous yet short-time impact on financial markets. We also point out that two factors—speed of the market time flow and the asset cross-correlation magnitude—while related (the larger the speed, the larger the cross-correlations on a given time scale), act in opposite directions with regard to the return distribution tails, which can affect the expected distribution convergence to the normal distribution.     

A long-range memory stochastic model of the return in financial markets

We present a nonlinear stochastic differential equation (SDE) which mimics the probability density function (PDF) of the return and the power spectrum of the absolute return in financial markets. Absolute return as a measure of market volatility is considered in the proposed model as a long-range memory stochastic variable. The SDE is obtained from the analogy with an earlier proposed model of trading activity in the financial markets and generalized within the nonextensive statistical mechanics framework. The proposed stochastic model generates time series of the return with two power law statistics, i.e., the PDF and the power spectral density, reproducing the empirical data for the one-minute trading return in the NYSE.     

The correlation length of commodity markets 1. Empirical evidence

It is a common belief nowadays that the world economy is fairly well “integrated”. Yet, this belief often turns out to be in contradiction with empirical evidence. As a matter of fact the way distant markets interact is a question that has largely been ignored by economists. In this series of two papers we examine the role that space, that is to say geographical distance, plays in the economics of commodity markets. The first of these papers presents the empirical evidence while the second develops a theoretical framework. The empirical enquiry discloses several noteworthy features, e.g. (i) with respect to spatial interaction there is a sharp contrast between stock markets and commodity markets. While there is almost perfect spatial arbitrage in the first case, this is not true for commodity markets. (ii) In spite of their chaotic behavior in the course of time commodity prices display well defined spatial patterns, (iii) as in statistical physics and fluid dynamics interactions can be described in terms of correlation length. The correlation length of a set of markets is seen to increase along with the number of transactions; it also increases when transport costs decline as was the case during the “transportation revolution” of the mid-nineteenth century. Using the notion of correlation length one is able to give a quantitative meaning to the otherwise ill-defined concept of market integration. Received 17 May 1999 and Received in final form 31 May 1999     

Nonextensive statistical features of the Polish stock market fluctuations

The statistics of return distributions on various time scales constitutes one of the most informative characteristics of the financial dynamics. Here, we present a systematic study of such characteristics for the Polish stock market index WIG20 over the period 04.01.1999–31.10.2005 for the time lags ranging from 1min$1\min$ up to 1 h. This market is commonly classified as emerging. Still on the shortest time scales studied we find that the tails of the return distributions are consistent with the inverse cubic power law, as identified previously for majority of the mature markets. Within the time scales studied, a quick and considerable departure from this law towards a Gaussian can however be traced. Interestingly, all the forms of the distributions observed can be comprised by the single q-Gaussians which provide a satisfactory and at the same time compact representation of the distribution of return fluctuations over all magnitudes of their variation. The corresponding nonextensivity parameter q was found to systematically decrease when increasing the time scales. The temporal correlations quantified here in terms of multifractality provide further arguments in favor of nonextensivity.     

Quantum phase properties associated to solvable quantum systems using the nonlinear coherent states approach

In this paper we study the quantum phase properties of “nonlinear coherent states” and “solvable quantum systems with discrete spectra” using the Pegg-Barnett formalism in a unified approach. The presented procedure will then be applied to few special solvable quantum systems with known discrete spectrum as well as to some new classes of nonlinear oscillators with particular nonlinearity functions. Finally the associated phase distributions and their nonclasscial properties such as the squeezing in number and phase operators have been investigated, numerically.     

A contribution to the systematics of stochastic volatility models

We systematically compare several classes of stochastic volatility models of stock market fluctuations. We show that the long-time return distribution is either Gaussian or develops a power-law tail, while the short-time return distribution has generically a stretched-exponential form, but can also assume an algebraic decay, in the family of models which we call “GARCH” type. The intermediate regime is found in the exponential Ornstein-Uhlenbeck process. We also calculate the decay of the autocorrelation function of volatility.     

Statistical properties of daily ensemble variables in the Chinese stock markets

We study dynamical behavior of the Chinese stock markets by investigating the statistical properties of daily ensemble return and variety defined, respectively, as the mean and the standard deviation of the ensemble daily price return of a portfolio of stocks traded in China's stock markets on a given day. The distribution of the daily ensemble return has an exponential form in the center and power-law tails, while the variety distribution is lognormal in the bulk followed by a power-law tail for large variety. Based on detrended fluctuation analysis, R/S analysis and modified R/S analysis, we find evidence of long memory in the ensemble return and strong evidence of long memory in the evolution of variety.     

Asymmetric responses of international stock markets to trading volume

The major goal of this paper is to examine the hypothesis that stock returns and return volatility are asymmetric, threshold nonlinear, functions of change in trading volume. A minor goal is to examine whether return spillover effects also display such asymmetry. Employing a double-threshold GARCH model with trading volume as a threshold variable, we find strong evidence supporting this hypothesis in five international market return series. Asymmetric causality tests lend further support to our trading volume threshold model and conclusions. Specifically, an increase in volume is positively associated, while decreasing volume is negatively associated, with the major price index in four of the five markets. The volatility of each series also displays an asymmetric reaction, four of the markets display higher volatility following increases in trading volume. Using posterior odds ratio, the proposed threshold model is strongly favored in three of the five markets, compared to a US news double threshold GARCH model and a symmetric GARCH model. We also find significant nonlinear asymmetric return spillover effects from the US market.