The inversion formula for conservative multifractal measures was unveiled mathematically a decade ago, which is however not well tested in real complex systems. In this Letter, we propose to verify the inversion formula using high-frequency turbulent financial data. We construct conservative volatility measure based on minutely S&P 500 index from 1982 to 1999 and its inverse measure of exit time. Both the direct and inverse measures exhibit nice multifractal nature, whose scaling ranges are not irrelevant. Empirical investigation shows that the inversion formula holds in financial markets.
This study investigates the volatility of daily Bitcoin returns and multifractal properties of the Bitcoin market by employing the rolling window method and examines relationships between the volatility asymmetry and market efficiency. Whilst we find an inverted asymmetry in the volatility of Bitcoin, its magnitude changes over time, and recently, it has become small. This asymmetric pattern of volatility also exists in higher frequency returns. Other measurements, such as kurtosis, skewness, average, serial correlation, and multifractal degree, also change over time. Thus, we argue that prope…
Bid-ask spread is taken as an important measure of the financial market liquidity. In this article, we study the dynamics of the spread return and the spread volatility of four liquid stocks in the Chinese stock market, including the memory effect and the multifractal nature. By investigating the autocorrelation function and the Detrended Fluctuation Analysis (DFA), we find that the spread return is lack of long-range memory, while the spread volatility is long-range time correlated. Moreover, by applying the Multifractal Detrended Fluctuation Analysis (MF-DFA), the spread return is observed t…
We study the problem of forecasting volatility for the multifractal random walk model. In order to avoid the ill posed problem of estimating the correlation length T of the model, we introduce a limiting object defined in a quotient space; formally, this object is an infinite range logvolatility. For this object and the non limiting object, we obtain precise prediction formulas and we apply them to the problem of forecasting volatility and pricing options with the MRW model in the absence of a reliable estimate of the average volatility and T.
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…
Recent studies have found that the log-volatility of asset returns exhibit roughness. This study investigates roughness or the anti-persistence of Bitcoin volatility. Using the multifractal detrended fluctuation analysis, we obtain the generalized Hurst exponent of the log-volatility increments and find that the generalized Hurst exponent is less than $1/2$, which indicates log-volatility increments that are rough. Furthermore, we find that the generalized Hurst exponent is not constant. This observation indicates that the log-volatility has multifractal property. Using shuffled time series of…
Asymmetric relationship between price and volatility is a prominent feature of the financial market time series. This paper explores the price-volatility nexus in cryptocurrency markets and investigates the presence of asymmetric volatility effect between uptrend (bull) and downtrend (bear) regimes. The conventional GARCH-class models have shown that in cryptocurrency markets, asymmetric reactions of volatility to returns differ from those of other traditional financial assets. We address this issue from a viewpoint of fractal analysis, which can cover the nonlinear interactions and the self-s…
We introduce tools for inference in the multifractal random walk introduced by Bacry et al. (2001). These tools include formulas for smoothing, filtering and volatility forecasting. In addition, we present methods for computing conditional densities for one- and multi-step returns. The inference techniques presented in this paper, including maximum likelihood estimation, are applied to data from the Oslo Stock Exchange, and it is observed that the volatility forecasts based on the multifractal random walk have a much richer structure than the forecasts obtained from a basic stochastic volatili…
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…
An empirical algorithm is used here to study the stochastic and multifractal nature of nonlinear time series. A parameter can be defined to quantitatively measure the deviation of the time series from a Wiener process so that the stochasticity of different time series can be compared. The local volatility of the time series under study can be constructed using this algorithm and the multifractal structure of the time series can be analyzed by using this local volatility. As an example, we employ this method to analyze financial time series from different stock markets. The result shows that wh…
We study the relation between serial correlation of financial returns and volatility at intraday level for the S&P500 stock index. At daily and weekly level, serial correlation and volatility are known to be negatively correlated (LeBaron effect). While confirming that the LeBaron effect holds also at intraday level, we go beyond it and, complementing the efficient market hyphotesis (for returns) with the heterogenous market hyphotesis (for volatility), we test the impact of unexpected volatility, defined as the part of volatility which cannot be forecasted, on the presence of serial correlati…
Multifractal analysis is one of the important approaches that enables us to measure the complexity of various data via the scaling properties. We compare the most common techniques used for multifractal exponents estimation from both theoretical and practical point of view. Particularly, we discuss the methods based on estimation of Rényi entropy, which provide a powerful tool especially in presence of heavy-tailed data. To put some flesh on bare bones, all methods are compared on various real financial datasets, including daily and high-frequency data.
In the past few decades considerable effort has been expended in characterizing and modeling financial time series. A number of stylized facts have been identified, and volatility clustering or the tendency toward persistence has emerged as the central feature. In this paper we propose an appropriately defined conditional probability as a new measure of volatility clustering. We test this measure by applying it to different stock market data, and we uncover a rich temporal structure in volatility fluctuations described very well by a scaling relation. The scale factor used in the scaling provi…
We investigate the large-volatility dynamics in financial markets, based on the minute-to-minute and daily data of the Chinese Indices and German DAX. The dynamic relaxation both before and after large volatilities is characterized by a power law, and the exponents $p_\pm$ usually vary with the strength of the large volatilities. The large-volatility dynamics is time-reversal symmetric at the time scale in minutes, while asymmetric at the daily time scale. Careful analysis reveals that the time-reversal asymmetry is mainly induced by exogenous events. It is also the exogenous events which driv…
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.
We introduce a family of random measures $M_{H,T} (d t)$, namely log S-fBM, such that, for $H>0$, $M_{H,T}(d t) = e^{ω_{H,T}(t)} d t$ where $ω_{H,T}(t)$ is a Gaussian process that can be considered as a stationary version of an $H$-fractional Brownian motion. Moreover, when $H \to 0$, one has $M_{H,T}(d t) \rightarrow {\widetilde M}_{T}(d t)$ (in the weak sense) where ${\widetilde M}_{T}(d t)$ is the celebrated log-normal multifractal random measure (MRM). Thus, this model allows us to consider, within the same framework, the two popular classes of multifractal ($H = 0$) and rough volatility (…
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 apply RMT, Network and MF-DFA methods to investigate correlation, network and multifractal properties of 20 global financial indices. We compare results before and during the financial crisis of 2008 respectively. We find that the network method gives more useful information about the formation of clusters as compared to results obtained from eigenvectors corresponding to second largest eigenvalue and these sectors are formed on the basis of geographical location of indices. At threshold 0.6, indices corresponding to Americas, Europe and Asia/Pacific disconnect and form different clusters b…
We present a comparative analysis of multifractal properties of financial time series built on stock indices from developing (WIG) and developed (S&P500) financial markets. It is shown how the multifractal image of the market is altered with the change of the length of time series and with the economic situation on the market. We emphasize that the proper adjustment of scaling range for multiscaling power laws is essential to obtain the multifractal image of time series. We analyze in this paper multifractal properties of real financial time series using Hölder $f(α)$ representation and multif…
We shortly review the statistical properties of the escape times, or hitting times, for stock price returns by using different models which describe the stock market evolution. We compare the probability function (PF) of these escape times with that obtained from real market data. Afterwards we analyze in detail the effect both of noise and different initial conditions on the escape time in a market model with stochastic volatility and a cubic nonlinearity. For this model we compare the PF of the stock price returns, the PF of the volatility and the return correlation with the same statistical…
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…
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.
This is a short review in honor of B. Mandelbrot's 80st birthday, to appear in W ilmott magazine. We discuss how multiplicative cascades and related multifractal ideas might be relevant to model the main statistical features of financial time series, in particular the intermittent, long-memory nature of the volatility. We describe in details the Bacry-Muzy-Delour multifractal random walk. We point out some inadequacies of the current models, in particular concerning time reversal symmetry, and propose an alternative family of multi-timescale models, intermediate between GARCH models and multif…
It is widely accepted that there is strong persistence in the volatility of financial time series. The origin of the observed persistence, or long-range memory, is still an open problem as the observed phenomenon could be a spurious effect. Earlier we have proposed the consentaneous model of the financial markets based on the non-linear stochastic differential equations. The consentaneous model successfully reproduces empirical probability and power spectral densities of volatility. This approach is qualitatively different from models built using fractional Brownian motion. In this contributio…
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…
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 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, bu…
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…
With the daily and minutely data of the German DAX and Chinese indices, we investigate how the return-volatility correlation originates in financial dynamics. Based on a retarded volatility model, we may eliminate or generate the return-volatility correlation of the time series, while other characteristics, such as the probability distribution of returns and long-range time-correlation of volatilities etc., remain essentially unchanged. This suggests that the leverage effect or anti-leverage effect in financial markets arises from a kind of feedback return-volatility interactions, rather than …
Bollerslev et al. (2006) study the cross-covariances for squared returns under the Heston (1993) stochastic volatility model. In order to obtain these cross-covariances the authors use an incorrect expression for the distribution of the squared returns. Here we will obtain the correct distribution of the squared returns and check that, under this new distribution, the result in Appendix A.2 in Bollerslev et al. (2006) still holds.
The volatility characterizes the amplitude of price return fluctuations. It is a central magnitude in finance closely related to the risk of holding a certain asset. Despite its popularity on trading floors, the volatility is unobservable and only the price is known. Diffusion theory has many common points with the research on volatility, the key of the analogy being that volatility is the time-dependent diffusion coefficient of the random walk for the price return. We present a formal procedure to extract volatility from price data, by assuming that it is described by a hidden Markov process …
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…
We consider a mean-reverting stochastic volatility model which satisfies some relevant stylized facts of financial markets. We introduce an algorithm for the detection of peaks in the volatility profile, that we apply to the time series of Dow Jones Industrial Average and Financial Times Stock Exchange 100 in the period 1984-2013. Based on empirical results, we propose a bivariate version of the model, for which we find an explicit expression for the decay over time of cross-asset correlations between absolute returns. We compare our theoretical predictions with empirical estimates on the same…
We investigate the historical volatility of the 100 most capitalized stocks traded in US equity markets. An empirical probability density function (pdf) of volatility is obtained and compared with the theoretical predictions of a lognormal model and of the Hull and White model. The lognormal model well describes the pdf in the region of low values of volatility whereas the Hull and White model better approximates the empirical pdf for large values of volatility. Both models fails in describing the empirical pdf over a moderately large volatility range.
In financial markets, greater volatility is usually considered synonym of greater risk and instability. However, large market downturns and upturns are often preceded by long periods where price returns exhibit only small fluctuations. To investigate this surprising feature, here we propose using the mean first hitting time, i.e. the average time a stock return takes to undergo for the first time a large negative or positive variation, as an indicator of price stability, and relate this to a standard measure of volatility. In an empirical analysis of daily returns for $1071$ stocks traded in t…
We perform a scaling analysis on NYSE daily returns. We show that volatility correlations are power-laws on a time range from one day to one year and, more important, that they exhibit a multiscale behaviour.
The multifractal behavior for tick data of prices is investigated in Korean financial market. Using the rescaled range analysis(R/S analysis), we show the multifractal nature of returns for the won-dollar exchange rate and the KOSPI. We also estimate the Hurst exponent and the generalized $q$th-order Hurst exponent in the unversal multifractal framework. Particularly, our financial market is a persistent process with long-run memory effects, and the statistical value of the Hurst exponents occurs the crossovers at charateristic time scales. It is found that the probability distribution of retu…
Statistical and multiscaling characteristics of WTI Crude Oil prices expressed in US dollar in relation to the most traded currencies as well as to gold futures and to the E-mini S$\&$P500 futures prices on 5 min intra-day recordings in the period January 2012 - December 2017 are studied. It is shown that in most of the cases the tails of return distributions of the considered financial instruments follow the inverse cubic power law. The only exception is the Russian ruble for which the distribution tail is heavier and scales with the exponent close to 2. From the perspective of multiscaling t…
The autocorrelation function of volatility in financial time series is fitted well by a superposition of several exponents. Such a case admits an explicit analytical solution of the problem of constructing the best linear forecast of a stationary stochastic process. We describe and apply the proposed analytical method for forecasting volatility. The leverage effect and volatility clustering are taken into account. Parameters of the predictor function are determined numerically for the Dow Jones 30 Industrial Average. Connection of the proposed method to the popular ARCH models is discussed.
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…
We apply the hybrid Monte Carlo (HMC) algorithm to the financial time sires analysis of the stochastic volatility (SV) model for the first time. The HMC algorithm is used for the Markov chain Monte Carlo (MCMC) update of volatility variables of the SV model in the Bayesian inference. We compute parameters of the SV model from the artificial financial data and compare the results from the HMC algorithm with those from the Metropolis algorithm. We find that the HMC decorrelates the volatility variables faster than the Metropolis algorithm. We also make an empirical analysis based on the Yen/Doll…
This paper investigates how the conditional quantiles of future returns and volatility of financial assets vary with various measures of ex-post variation in asset prices as well as option-implied volatility. We work in the flexible quantile regression framework and rely on recently developed model-free measures of integrated variance, upside and downside semivariance, and jump variation. Our results for the S&P 500 and WTI Crude Oil futures contracts show that simple linear quantile regressions for returns and heterogenous quantile autoregressions for realized volatility perform very well in …
This work is devoted to the study of modeling geophysical and financial time series. A class of volatility models with time-varying parameters is presented to forecast the volatility of time series in a stationary environment. The modeling of stationary time series with consistent properties facilitates prediction with much certainty. Using the GARCH and stochastic volatility model, we forecast one-step-ahead suggested volatility with +/- 2 standard prediction errors, which is enacted via Maximum Likelihood Estimation. We compare the stochastic volatility model relying on the filtering techniq…
A new space-time model for interacting agents on the financial market is presented. It is a combination of the Curie-Weiss model and a space-time model introduced by Järpe 2005. Properties of the model are derived with focus on the critical temperature and magnetization. It turns out that the Hamiltonian is a sufficient statistic for the temperature parameter and thus statistical inference about this parameter can be performed. Thus e.g. statements about how far the current financial situation is from a financial crisis can be made, and financial trading stability be monitored for detection of…
Using high frequency data, we have studied empirically the change of volatility, also called volatility derivative, for various time horizons. In particular, the correlation between the volatility derivative and the volatility realized in the next time period is a measure of the response function of the market participants. This correlation shows explicitly the heterogeneous structure of the market according to the characteristic time horizons of the differents agents. It reveals a volatility cascade from long to short time horizons, with a structure different from the one observed in turbulen…
Multifractal processes are a relatively new tool of stock market analysis. Their power lies in the ability to take multiple orders of autocorrelations into account explicitly. In the first part of the paper we discuss the framework of the Lux model and refine the underlying phenomenological picture. We also give a procedure of fitting all parameters to empirical data. We present a new approach to account for the effective length of power-law memory in volatility. The second part of the paper deals with the consequences of asymmetry in returns. We incorporate two related stylized facts, skewnes…
We simulate a series of daily returns from intraday price movements initiated by microstructure elements. Significant evidence is found that daily returns and daily return volatility exhibit first order autocorrelation, but trading volume and daily return volatility are not correlated, while intraday volatility is. We also consider GARCH effects in daily return series and show that estimates using daily returns are biased from the influence of the level of prices. Using daily price changes instead, we find evidence of a significant GARCH component. These results suggest that microstructure ele…
This article present a continuous cascade model of volatility formulated as a stochastic differential equation. Two independent Brownian motions are introduced as random sources triggering the volatility cascade. One multiplicatively combines with volatility; the other does so additively. Assuming that the latter acts perturbatively on the system, then the model parameters are estimated by application to an actual stock price time series. Numerical calculation of the Fokker--Planck equation derived from the stochastic differential equation is conducted using the estimated values of parameters.…
In this paper, we model the impact of oil price volatility on Tehranstock and industry indices in two periods of international sanctions and post-sanction. To analyse the purpose of study, we use Feed-forward neural net-works. The period of study is from 2008 to 2018 that is split in two periods during international energy sanction and post-sanction. The results show that Feed-forward neural networks perform well in predicting stock market and industry, which means oil price volatility has a significant impact on stock and industry market indices. During post-sanction and global financial cris…
In the canonical framework, we propose an alternative approach for the multifractal analysis based on the detrending moving average method (MF-DMA). We define a canonical measure such that the multifractal mass exponent $τ(q)$ is related to the partition function and the multifractal spectrum $f(α)$ can be directly determined. The performances of the direct determination approach and the traditional approach of the MF-DMA are compared based on three synthetic multifractal and monofractal measures generated from the one-dimensional $p$-model, the two-dimensional $p$-model and the fractional Bro…
A Nested Factor Model for Equity Markets: Reconciling Multifractal Stock Returns and Rough Index Volatilities
Ceremade, CNRS-UMR 7534, Université Paris-Dauphine PSL（巴黎第九大学）；Chair of Econophysics and Complex Systems, École polytechnique（巴黎综合理工学院）；LadHyX UMR CNRS 7646, École polytechnique（巴黎综合理工学院）；Capital Fund Management（资本基金管理公司）；Académie des Sciences（法兰西科学院）；SPE CNRS-UMR 6134, Université de Corse（科西嘉大学）
This thesis applies entropy as a model independent measure to address three research questions concerning financial time series. In the first study we apply transfer entropy to drawdowns and drawups in foreign exchange rates, to study their correlation and cross correlation. When applied to daily and hourly EUR/USD and GBP/USD exchange rates, we find evidence of dependence among the largest draws (i.e. 5% and 95% quantiles), but not as strong as the correlation between the daily returns of the same pair of FX rates. In the second study we use state space models (Hidden Markov Models) of volati…
In this paper we study the possible microscopic origin of heavy-tailed probability density distributions for the price variation of financial instruments. We extend the standard log-normal process to include another random component in the so-called stochastic volatility models. We study these models under an assumption, akin to the Born-Oppenheimer approximation, in which the volatility has already relaxed to its equilibrium distribution and acts as a background to the evolution of the price process. In this approximation, we show that all models of stochastic volatility should exhibit a scal…
