In this paper we will try to assess the multifractality displayed by the high-frequency returns of Madrid's Stock Exchange IBEX35 index. A Multifractal Detrended Fluctuation Analysis shows that this index has a wide singularity spectrum which is most likely caused by its long memory. Our findings also show that this long-memory can be considered as the superposition of a high-frequency component (related to the daily cycles of arrival of information to the market), over a slowly-varying component that reverberates for long periods of time and which shows no apparent relation with human economic cycles. This later component is therefore postulated to be endogenous to market's dynamics and to be also the most probable source of some of the stylized facts commonly associated with financial time series.
In this paper, we use the generalized Hurst exponent approach to study the multi- scaling behavior of different financial time series. We show that this approach is robust and powerful in detecting different types of multiscaling. We observe a puzzling phenomenon where an apparent increase in multifractality is measured in time series generated from shuffled returns, where all time-correlations are destroyed, while the return distributions are conserved. This effect is robust and it is reproduced in several real financial data including stock market indices, exchange rates and interest rates. …
Many financial variables are found to exhibit multifractal nature, which is usually attributed to the influence of temporal correlations and fat-tailedness in the probability distribution (PDF). Based on the partition function approach of multifractal analysis, we show that there is a marked finite-size effect in the detection of multifractality, and the effective multifractality is the apparent multifractality after removing the finite-size effect. We find that the effective multifractality can be further decomposed into two components, the PDF component and the nonlinearity component. Referr…
An average instantaneous cross-correlation function is introduced to quantify the interaction of the financial market of a specific time. Based on the daily data of the American and Chinese stock markets, memory effect of the average instantaneous cross-correlations is investigated over different price return time intervals. Long-range time-correlations are revealed, and are found to persist up to a month-order magnitude of the price return time interval. Multifractal nature is investigated by a multifractal detrended fluctuation analysis.
Multifractality is ubiquitously observed in complex natural and socioeconomic systems. Multifractal analysis provides powerful tools to understand the complex nonlinear nature of time series in diverse fields. Inspired by its striking analogy with hydrodynamic turbulence, from which the idea of multifractality originated, multifractal analysis of financial markets has bloomed, forming one of the main directions of econophysics. We review the multifractal analysis methods and multifractal models adopted in or invented for financial time series and their subtle properties, which are applicable t…
We report evidence of a deep interplay between cross-correlations hierarchical properties and multifractality of New York Stock Exchange daily stock returns. The degree of multifractality displayed by different stocks is found to be positively correlated to their depth in the hierarchy of cross-correlations. We propose a dynamical model that reproduces this observation along with an array of other empirical properties. The structure of this model is such that the hierarchical structure of heterogeneous risks plays a crucial role in the time evolution of the correlation matrix, providing an int…
We discuss the origin of multiscaling in financial time-series and investigate how to best quantify it. Our methodology consists in separating the different sources of measured multifractality by analysing the multi/uni-scaling behaviour of synthetic time-series with known properties. We use the results from the synthetic time-series to interpret the measure of multifractality of real log-returns time-series. The main finding is that the aggregation horizon of the returns can introduce a strong bias effect on the measure of multifractality. This effect can become especially important when retu…
Multiplicative random cascade model naturally reproduces the intermittency or multifractality, which is frequently shown among hierarchical complex systems such as turbulence and financial markets. As described herein, we investigate the validity of a multiplicative hierarchical random cascade model through an empirical study using financial data. Although the intermittency and multifractality of the time series are verified, random multiplicative factors linking successive hierarchical layers show strongly negative correlation. We extend the multiplicative model to incorporate an additional s…
Detailed study of multifractal characteristics of the financial time series of asset values and of its returns is performed using a collection of the high frequency Deutsche Aktienindex data. The tail index ($α$), the Renyi exponents based on the box counting algorithm for the graph ($d_q$) and the generalized Hurst exponents ($H_q$) are computed in parallel for short and daily return times. The results indicate a more complicated nature of the stock market dynamics than just consistent multifractal.
We perform a systematic investigation on the components of the empirical multifractality of financial returns using the daily data of Dow Jones Industrial Average from 26 May 1896 to 27 April 2007 as an example. The temporal structure and fat-tailed distribution of the returns are considered as possible influence factors. The multifractal spectrum of the original return series is compared with those of four kinds of surrogate data: (1) shuffled data that contain no temporal correlation but have the same distribution, (2) surrogate data in which any nonlinear correlation is removed but the dist…
The concept of multifractality offers a powerful formal tool to filter out multitude of the most relevant characteristics of complex time series. The related studies thus far presented in the scientific literature typically limit themselves to evaluation of whether or not a time series is multifractal and width of the resulting singularity spectrum is considered a measure of the degree of complexity involved. However, the character of the complexity of time series generated by the natural processes usually appears much more intricate than such a bare statement can reflect. As an example, based…
The statistical properties of the multipliers of the absolute returns are investigated using one-minute high-frequency data of financial time series. The multiplier distribution is found to be independent of the box size $s$ when $s$ is larger than some crossover scale, providing direct evidence of the existence of scale invariance in financial data. The multipliers with base $a=2$ are well approximated by a normal distribution and the most probable multiplier scales as a power law in respect to the base $a$. We unravel that the volatility multipliers possess multifractal nature which is indep…
We investigate multifractality in the Korean stock-market index KOSPI. The generalized $q$th order height-height correlation function shows multiscaling properties. There are two scaling regimes with a crossover time around $t_c =40$ min. We consider the original data sets and the modified data sets obtained by removing the daily jumps, which occur due to the difference between the closing index and the opening index. To clarify the origin of the multifractality, we also smooth the data through convolution with a Gaussian function. After convolution we observe that the multifractality disappea…
There is more and more empirical evidence that multifractality constitutes another and perhaps the most significant financial stylized fact. A realistic model of the financial dynamics should therefore incorporate this effect. The most promising in this respect is the Multifractal Model of Asset Returns (MMAR) introduced by Mandelbrot in which multifractality is carried by time deformation. In our study we focus on the Lux extension to MMAR and empirical data from Warsaw Stock Exchange. We show that this model is able to reproduce relevant aspects of the real stock market dynamics.
Recently the statistical characterizations of financial markets based on physics concepts and methods attract considerable attentions. We used two possible procedures of analyzing multifractal properties of a time series. The first one uses the continuous wavelet transform and extracts scaling exponents from the wavelet transform amplitudes over all scales. The second method is the multifractal version of the detrended fluctuation analysis method (MF-DFA). The multifractality of a time series we analysed by means of the difference of values singularity stregth as a suitable way to characterise…
Complex systems are composed of mutually interacting components and the output values of these components are usually long-range cross-correlated. We propose a method to characterize the joint multifractal nature of such long-range cross correlations based on wavelet analysis, termed multifractal cross wavelet analysis (MFXWT). We assess the performance of the MFXWT method by performing extensive numerical experiments on the dual binomial measures with multifractal cross correlations and the bivariate fractional Brownian motions (bFBMs) with monofractal cross correlations. For binomial multifr…
This paper is trying to unveil general statistical characteristic of financial; time series data that is subjected to several financial time series data present in Indonesia, e.g. individual index such as stock price of PT. TELKOM, stock price of PT HM SAMPOERNA, and compiled stock price index (Jakarta Stock Exchange Index). Characteristics that we try to analyze are volatility clustering, truncated Levy distribution, and multifractality feature. This analysis is directed for further works of research in making Indonesian artificial stock exchange that gives representation of micro structure o…
A new branch based on Markov processes is developing in the recent literature of financial time series modeling. In this paper, an Indexed Markov Chain has been used to model high frequency price returns of quoted firms. The peculiarity of this type of model is that through the introduction of an Index process it is possible to consider the market volatility endogenously and two very important stylized facts of financial time series can be taken into account: long memory and volatility clustering. In this paper, first we propose a method for the optimal determination of the state space of the …
It is shown phenomenologically that the fractional derivative $ξ=D^αu$ of order $α$ of a multifractal function has a power-law tail $\propto |ξ| ^{-p_\star}$ in its cumulative probability, for a suitable range of $α$'s. The exponent is determined by the condition $ζ_{p_\star} = αp_\star$, where $ζ_p$ is the exponent of the structure function of order $p$. A detailed study is made for the case of random multiplicative processes (Benzi {\it et al.} 1993 Physica D {\bf 65}: 352) which are amenable to both theory and numerical simulations. Large deviations theory provides a concrete criterion, whi…
We present some stylized facts exhibited by the time series of returns of the Mexican Stock Exchange Index (IPC) and compare them to a sample of both developed (USA, UK and Japan) and emerging markets (Brazil and India). The period of study is 1997-2011. The stylized facts are related mostly to the probability distribution func- tion and the autocorrelation function (e.g. fat tails, non-normality, volatility cluster- ing, among others). We find that positive skewness for returns in Mexico and Brazil, but not in the rest, suggest investment opportunities. Evidence of nonlinearity is also docume…
We extend existing models in the financial literature by introducing a cluster-derived canonical vine (CDCV) copula model for capturing high dimensional dependence between financial time series. This model utilises a simplified market-sector vine copula framework similar to those introduced by Heinen and Valdesogo (2008) and Brechmann and Czado (2013), which can be applied by conditioning asset time series on a market-sector hierarchy of indexes. While this has been shown by the aforementioned authors to control the excessive parameterisation of vine copulas in high dimensions, their models ha…
In this paper, a statistical analysis of high frequency fluctuations of the IPC, the Mexican Stock Market Index, is presented. A sample of tick-to-tick data covering the period from January 1999 to December 2002 was analyzed, as well as several other sets obtained using temporal aggregation. Our results indicates that the highest frequency is not useful to understand the Mexican market because almost two thirds of the information corresponds to inactivity. For the frequency where fluctuations start to be relevant, the IPC data does not follows any alpha-stable distribution, including the Gauss…
We analyze the financial crash in 2008 for different financial markets from the point of view of log-periodic function model. In particular, we consider Dow Jones index, DAX index and Hang Seng index. We shortly discuss the possible relation of the theory of critical phenomena in physics to financial markets.
We discuss the statistical properties of index returns in a financial market just after a major market crash. The observed non-stationary behavior of index returns is characterized in terms of the exceedances over a given threshold. This characterization is analogous to the Omori law originally observed in geophysics. By performing numerical simulations and theoretical modelling, we show that the nonlinear behavior observed in real market crashes cannot be described by a GARCH(1,1) model. We also show that the time evolution of the Value at Risk observed just after a major crash is described b…
We propose an approach to explain fluctuations in time intervals of financial markets data from the view point of the Gini index. We show the explicit form of the Gini index for a Weibull distribution which is a good candidate to describe the first passage time of foreign exchange rate. The analytical expression of the Gini index gives a very close value with that of empirical data analysis.
Traditional sentiment construction in finance relies heavily on the dictionary-based approach, with a few exceptions using simple machine learning techniques such as Naive Bayes classifier. While the current literature has not yet invoked the rapid advancement in the natural language processing, we construct in this research a textual-based sentiment index using a well-known pre-trained model BERT developed by Google, especially for three actively trading individual stocks in Hong Kong market with at the same time the hot discussion on Weibo.com. On the one hand, we demonstrate a significant e…
At present, cryptocurrencies have become a global phenomenon in financial sectors as it is one of the most traded financial instruments worldwide. Cryptocurrency is not only one of the most complicated and abstruse fields among financial instruments, but it is also deemed as a perplexing problem in finance due to its high volatility. This paper makes an attempt to apply machine learning techniques on the index and constituents of cryptocurrency with a goal to predict and forecast prices thereof. In particular, the purpose of this paper is to predict and forecast the close (closing) price of th…
In this manuscript we present a comprehensive study on the multifractal properties of high-frequency price fluctuations and instantaneous volatility of the equities that compose Dow Jones Industrial Average. The analysis consists about quantification of dependence and non-Gaussianity on the multifractal character of financial quantities. Our results point out an equivalent influence of dependence and non-Gaussianity on the multifractality of time series. Moreover, we analyse l-diagrams of price fluctuations. In the latter case, we show that the fractal dimension of these maps is basically inde…
In this paper we study the high frequency dynamic of financial volumes of traded stocks by using a semi-Markov approach. More precisely we assume that the intraday logarithmic change of volume is described by a weighted-indexed semi-Markov chain model. Based on this assumptions we show that this model is able to reproduce several empirical facts about volume evolution like time series dependence, intra-daily periodicity and volume asymmetry. Results have been obtained from a real data application to high frequency data from the Italian stock market from first of January 2007 until end of Decem…
In this paper, a statistical analysis of log-return fluctuations of the IPC, the Mexican Stock Market Index is presented. A sample of daily data covering the period from $04/09/2000-04/09/2010$ was analyzed, and fitted to different distributions. Tests of the goodness of fit were performed in order to quantitatively asses the quality of the estimation. Special attention was paid to the impact of the size of the sample on the estimated decay of the distributions tail. In this study a forceful rejection of normality was obtained. On the other hand, the null hypothesis that the log-fluctuations a…
An analysis of the stylized facts in financial time series is carried out. We find that, instead of the heavy tails in asset return distributions, the slow decay behaviour in autocorrelation functions of absolute returns is actually directly related to the degree of clustering of large fluctuations within the financial time series. We also introduce an index to quantitatively measure the clustering behaviour of fluctuations in these time series and show that big losses in financial markets usually lump more severely than big gains. We further give examples to demonstrate that comparing to conv…
We develop a new stock market index that captures the chaos existing in the market by measuring the mutual changes of asset prices. This new index relies on a tensor-based embedding of the stock market information, which in turn frees it from the restrictive value- or capitalization-weighting assumptions that commonly underlie other various popular indexes. We show that our index is a robust estimator of the market volatility which enables us to characterize the market by performing the task of segmentation with a high degree of reliability. In addition, we analyze the dynamics and kinematics …
In this paper we propose a bivariate generalization of a weighted indexed semi-Markov chains to study the high frequency price dynamics of traded stocks. We assume that financial returns are described by a weighted indexed semi-Markov chain model. We show, through Monte Carlo simulations, that the model is able to reproduce important stylized facts of financial time series like the persistence of volatility and at the same time it can reproduce the correlation between stocks. The model is applied to data from Italian stock market from 1 January 2007 until the end of December 2010.
We investigate the variety of a portfolio of stocks in normal and extreme days of market activity. We show that the variety carries information about the market activity which is not present in the single-index model and we observe that the variety time evolution is not time reversal around the crash days. We obtain the theoretical relation between the square variety and the mean return of the ensemble return distribution predicted by the single-index model. The single-index model is able to mimic the average behavior of the square variety but fails in describing quantitatively the relation be…
In this paper, we studied the dynamics of the log-return distribution of the Korean Composition Stock Price Index (KOSPI) from 1992 to 2004. Based on the microscopic spin model, we found that while the index during the late 1990s showed a power-law distribution, the distribution in the early 2000s was exponential. This change in distribution shape was caused by the duration and velocity, among other parameters, of the information that flowed into the market.
The relaxation dynamics of aftershocks after large volatility shocks are investigated based on two high-frequency data sets of the Shanghai Stock Exchange Composite (SSEC) index. Compared with previous relevant work, we have defined main financial shocks based on large volatilities rather than large crashes. We find that the occurrence rate of aftershocks with the magnitude exceeding a given threshold for both daily volatility (constructed using 1-minute data) and minutely volatility (using intra-minute data) decays as a power law. The power-law relaxation exponent increases with the volatilit…
Researchers have studied the first passage time of financial time series and observed that the smallest time interval needed for a stock index to move a given distance is typically shorter for negative than for positive price movements. The same is not observed for the index constituents, the individual stocks. We use the discrete wavelet transform to illustrate that this is a long rather than short time scale phenomenon -- if enough low frequency content of the price process is removed, the asymmetry disappears. We also propose a new model, which explain the asymmetry by prolonged, correlated…
The importance of considering the volumes to analyze stock prices movements can be considered as a well-accepted practice in the financial area. However, when we look at the scientific production in this field, we still cannot find a unified model that includes volume and price variations for stock assessment purposes. In this paper we present a computer model that could fulfill this gap, proposing a new index to evaluate stock prices based on their historical prices and volumes traded. Besides the model can be considered mathematically very simple, it was able to improve significantly the per…
In this paper we study BSE Index financial time series for fractal and multifractal behaviour. We show that Bombay stock Exchange (BSE)Index time series is mono-fractal and can be represented by a fractional Brownian motion.
Log-normal continuous random cascades form a class of multifractal processes that has already been successfully used in various fields. Several statistical issues related to this model are studied. We first make a quick but extensive review of their main properties and show that most of these properties can be analytically studied. We then develop an approximation theory of these processes in the limit of small intermittency $λ^2\ll 1$, i.e., when the degree of multifractality is small. This allows us to prove that the probability distributions associated with these processes possess some very…
Université de Montreal（蒙特利尔大学）；Canada Mortgage and Housing Corporation（加拿大抵押贷款和住房公司）
We propose a unified multi-tasking framework to represent the complex and uncertain causal process of financial market dynamics, and then to predict the movement of any type of index with an application on the monthly direction of the S&P500 index. our solution is based on three main pillars: (i) the use of transfer learning to share knowledge and feature (representation, learning) between all financial markets, increase the size of the training sample and preserve the stability between training, validation and test sample. (ii) The combination of multidisciplinary knowledge (Financial economi…
Following the thermodynamic formulation of multifractal measure that was shown to be capable of detecting large fluctuations at an early stage, here we propose a new index which permits us to distinguish events like financial crisis in real time . We calculate the partition function from where we obtain thermodynamic quantities analogous to free energy and specific heat. The index is defined as the normalized energy variation and it can be used to study the behavior of stochastic time series, such as financial market daily data. Famous financial market crashes - Black Thursday (1929), Black Mo…
We investigated the critical dynamics on the daily Taiwan stock exchange index (TSE) from 1971 to 2005, and the 5-min intraday data from 1996 to 2005. A global persistence exponent $θ_{p}$ was defined for non-equilibrium critical phenomena \cite{Janssen,Majumdar}, and describing dynamic behavior in an economic index \cite{Zheng}. In recent numerical analysis studies of literatures, it is illustrated that the persistence probability has a universal scaling form $P(t) \sim t^{-θ_{p}}$ \cite{Zheng1}. In this work, we analyzed persistence properties of universal scaling behavior on Taiwan financia…
We present a systematic algorithm testing for the existence of collective self-organization in the behavior of agents in social systems, with a concrete empirical implementation on the Dow Jones Industrial Average index (DJIA) over the 20th century and on Hong Kong Hang Seng composite index (HSI) since 1969. The algorithm combines ideas from critical phenomena, the impact of agents' expectation, multi-scale analysis and the mathematical method of pattern recognition of sparse data. Trained on the three major crashes in DJIA of the century, our algorithm exhibits a remarkable ability for genera…
Recently, to account for low-frequency market dynamics, several volatility models, employing high-frequency financial data, have been developed. However, in financial markets, we often observe that financial volatility processes depend on economic states, so they have a state heterogeneous structure. In this paper, to study state heterogeneous market dynamics based on high-frequency data, we introduce a novel volatility model based on a continuous Ito diffusion process whose intraday instantaneous volatility process evolves depending on the exogenous state variable, as well as its integrated v…
We propose a simple stochastic volatility model which is analytically tractable, very easy to simulate and which captures some relevant stylized facts of financial assets, including scaling properties. In particular, the model displays a crossover in the log-return distribution from power-law tails (small time) to a Gaussian behavior (large time), slow decay in the volatility autocorrelation and multiscaling of moments. Despite its few parameters, the model is able to fit several key features of the time series of financial indexes, such as the Dow Jones Industrial Average, with a remarkable a…
A non-parametric method for ranking stock indices according to their mutual causal influences is presented. Under the assumption that indices reflect the underlying economy of a country, such a ranking indicates which countries exert the most economic influence in an examined subset of the global economy. The proposed method represents the indices as nodes in a directed graph, where the edges' weights are estimates of the pair-wise causal influences, quantified using the directed information functional. This method facilitates using a relatively small number of samples from each index. The ind…
We perform wavelet decomposition of high frequency financial time series into large and small time scale components. Taking the FTSE100 index as a case study, and working with the Haar basis, it turns out that the small scale component defined by most ($\simeq$ 99.6%) of the wavelet coefficients can be neglected for the purpose of option premium evaluation. The relevance of the hugely compressed information provided by low-pass wavelet-filtering is related to the fact that the non-gaussian statistical structure of the original financial time series is essentially preserved for expiration times…
