Multifractal Random Walk Model Research Articles

Analysis of financial markets by coupled criticality approach

Analysis of the criticality in finance and economics is an important topic for scholars. The crises are conditions in which unpredictable behavior have been emerged. On the other hand, the contagion from one market to another one due to the complex nature of them is a crucial concept. Therefore, we should consider economies and financial markets as entangled systems that interact with each other, and studying one’s behavior without considering its interacting structures will not provide us with adequate information. In this paper, we have considered the critical states of some emerging and mature markets’ indices. S&P500 and Nikkei 225 as representatives of mature markets, and SSE Composite and TEDPIX as representatives of emerging markets have been analyzed. Coupled criticality emerges when two interacting systems reach their critical state due to their interactions. One of the efficient methods to study the coupled criticality is the bi-variate multifractal random walk model. This method is useful to assess the correlation of rare events and coupled criticality in entangled systems such as financial markets. The observations in this paper show that TEDPIX and S&P500 have higher critical states among others in most time lags by assessing criticality parameter explained in the model, and in longer time lags criticality parameter decreases in all indices. Coupled criticality parameter explained in the model is assessed between each index mentioned and TEDPIX. Results show that TEDPIX and S&P500 have the highest level of coupled criticality. To figure out how long-range time correlation between rare events in each index affect its critical behavior, the data of each index are shuffled, and the criticality parameter is calculated for the shuffled data. From the observations, it can be concluded that this matter has more substantial effect on mature markets. As it is known, shuffling removes the effect of the long-range time correlations. Therefore, the coupled criticality parameter is calculated for the shuffled S&P500 data and TEDPIX. Results show that the long-range correlation of S&P500 data has a significant effect on the coupled criticality behavior of S&P500 and TEDPIX.

Measuring informational efficiency of the European carbon market — A quantitative evaluation of higher order dependence

This paper introduces a new method for measuring nonlinear predictability in financial price changes: the so-called intermittency coefficient, a parameter of the multifractal random walk model by Bacry et al. (2001). As the intermittency coefficient can quantify the degree of nonlinear deviation from a random walk, we employ its estimates from financial data as a proxy for the loss of financial market efficiency. In addition, we propose a new statistical test of the random walk hypothesis. In an empirical application using data from the largest currently existing market for tradable pollution permits, the European Union Emissions Trading Scheme (EU ETS), we show that the degree of efficiency of this market remains largely unchanged over the period of observation 2008–2019. This suggests that the market has reached a mature state: informational efficiency in Phase III remains at a level comparable to Phase II. What is more, the EU ETS is found to be more efficient than the US stock market. This result, surprising as such, is largely attributable to the lower exposure to global economic shocks of the EU ETS.

Forecasting the variability of stock index returns with the multifractal random walk model for realized volatilities

We adapt the multifractal random walk model by Bacry et al. (2001) to realized volatilities (denoted RV-MRW) and take stock of recent theoretical insights on this model in Duchon et al. (2012) to derive forecasts of financial volatility. Moreover, we propose a new extension of the binomial Markov-switching multifractal (BMSM) model by Calvet and Fisher (2001) to the RV framework. We compare the predictive ability of the two against 10 classical and multifractal volatility models. Forecasting performance is evaluated out-of-sample based on the empirical MSE, MAE, and QLIKE criteria as well as using model confidence sets following the methodology of Hansen et al. (2011). Overall, our empirical study for 14 international stock market indices has a clear message: The RV-MRW is the best model throughout under the MAE criterion, i.e., for all indices and forecast horizons between one day and 100 days, with uniform results for forecast evaluations against both realized volatilities and squared returns or during tranquil/turbulent market periods. Based on the evaluation of MSE and QLIKE forecast errors, the RV-MRW, RV-BMSM, and RV-ARFIMA provide the most accurate forecasts during our tranquil sample from 2016–2018, where we can observe a transition from RV-MRW dominating long-term forecasts to RV-BMSM and RV-ARFIMA dominating in the short term. The new RV-BMSM takes the lead in phases of more turbulent market dynamics (sample 2010–2012), when it appears throughout in the 90% model confidence set at horizons ≤10 days and for 13 out of 14 indices at 20 days. These results are very promising if we consider that this is the first empirical application of the RV-MRW and RV-BMSM. Moreover, whereas RV-ARFIMA forecasts are often a time-consuming task, the RV-MRW stands out due to its fast execution and straightforward implementation.

A Multifractal Random-Walk Description of Atmospheric Turbulence: Small-Scale Multiscaling, Long-Tail Distribution, and Intermittency

The prevalent multifractal characteristics of turbulent velocity fluctuations in the atmosphere are important for estimating various wind effects in wind engineering. Here, the multifractal characteristics of turbulent velocity fluctuations, including the small-scale multiscaling, the long-tail distributions and the intermittency, are thoroughly investigated by using a high-frequency dataset of three-dimensional velocities (100 Hz) collected at three levels during one month. To reduce uncertainties in the estimate of multiscaling exponents, a new method, the sequential extended self-similarity, is proposed. Based on this method, we obtain the multiscaling exponents of qth-order moments of velocity increments as a function of q, that is the so-called multifractal spectrum. The multifractal random walk (MRW) model is then shown to describe the various multifractal spectra of turbulent velocity fluctuations. With the help of this model, we find a connection between the small-scale multiscaling and the long-tail distributions, which is generally observed in our dataset, again validating the MRW model. A non-linear multifractal spectrum is commonly considered to be related to the intermittency of turbulent velocity fluctuations at small scales and its curvature is usually used as a quantification of intermittency intensity. However, we suggest that models capturing the non-linear multifractal spectrum may fail to represent the long-tail distribution, which is a more direct quantification of intermittency. Finally, qualitative variations of validated indicators with specific boundary-layer parameters are investigated. Results show that the intermittency of turbulent velocity fluctuations is more relevant to the friction velocity, compared with the average wind speed, the average temperature, and the surface-layer stability.

How to Measure Financial Market Efficiency? A Multifractality-Based Quantitative Approach with an Application to the European Carbon Market

This paper proposes a new measure for the evaluation of financial market efficiency, the so-called intermittency coefficient. This is a multifractality measure that can quantify the deviation from a random walk within the framework of the multifractal random walk model by Bacry et al. (2001b). While the random walk corresponds to the most genuine form of market efficiency, the larger the value of the intermittency coefficient is, the more inefficient a market would be. In contrast to commonly used methods based on Hurst exponents, the intermittency coefficient is a more powerful tool due to its well-established inference apparatus based on the generalised method of moments estimation technique. In an empirical application using data from the largest currently existing market for tradable pollution permits, the European Union Emissions Trading Scheme, we show that this market becomes more efficient over time. In addition, the degree of market efficiency is overall similar to that for the US stock market; for one sub-period, the market efficiency is found to be higher. While the first finding is anticipated, the second finding is noteworthy, as various observers expressed concerns with regard to the information efficiency of this newly established artificial market.

Universal behavior of the interoccurrence times between losses in financial markets: independence of the time resolution.

We consider representative financial records (stocks and indices) on time scales between one minute and one day, as well as historical monthly data sets, and show that the distribution P(Q)(r) of the interoccurrence times r between losses below a negative threshold -Q, for fixed mean interoccurrence times R(Q) in multiples of the corresponding time resolutions, can be described on all time scales by the same q exponentials, P(Q)(r)∝1/{[1+(q-1)βr](1/(q-1))}. We propose that the asset- and time-scale-independent analytic form of P(Q)(r) can be regarded as an additional stylized fact of the financial markets and represents a nontrivial test for market models. We analyze the distribution P(Q)(r) as well as the autocorrelation C(Q)(s) of the interoccurrence times for three market models: (i) multiplicative random cascades, (ii) multifractal random walks, and (iii) the generalized autoregressive conditional heteroskedasticity [GARCH(1,1)] model. We find that only one of the considered models, the multifractal random walk model, approximately reproduces the q-exponential form of P(Q)(r) and the power-law decay of C(Q)(s).

Hilbert–Huang Transform based multifractal analysis of China stock market

In this paper, we employ the Hilbert–Huang Transform to investigate the multifractal character of Chinese stock market based on CSI 300 index. The measured Hilbert moment Lq(ω) shows a power-law behavior on the range 0.01<ω<0.1min−1, equivalent to a time scale range 10<τ<100min. The measured scaling exponents ζ(q) is convex with q and deviates from the value q/2, implying that the property of self-similarity is broken. Moreover, ζ(q) and the corresponding singularity spectrum D(h) can be described by a lognormal model with a Hurst number H=0.50 and an intermittency parameter μ=0.12. Our results suggest that the Chinese stock fluctuation might be captured well by a multifractal random walk model with a proper intermittency parameter.

Non-stationary multifractality in stock returns

We perform an extensive empirical analysis of scaling properties of equity returns, suggesting that financial data show time varying multifractal properties. This is obtained by comparing empirical observations of the weighted generalised Hurst exponent (wGHE) with time series simulated via Multifractal Random Walk (MRW) by Bacry et al. [E. Bacry, J. Delour, J.-F. Muzy, Multifractal random walk, Physical Review E 64 (2) (2001) 026103]. While dynamical wGHE computed on synthetic MRW series is consistent with a scenario where multifractality is constant over time, fluctuations in the dynamical wGHE observed in empirical data are not in agreement with a MRW with constant intermittency parameter. We test these hypotheses of constant multifractality considering different specifications of MRW model with fatter tails: in all cases considered, although the thickness of the tails accounts for most of the anomalous fluctuations of multifractality, it still cannot fully explain the observed fluctuations.

Market-Wide Price Co-movement around Crashes in the Tokyo Stock Exchange

As described in this paper, we study market-wide price co-movements around crashes by analyzing a dataset of high-frequency stock returns of the constituent issues of Nikkei 225 Index listed on the Tokyo Stock Exchange for the three years during 2007–2009. Results of day-to-day principal component analysis of the time series sampled at the 1 min time interval during the continuous auction of the daytime reveal the long range up to a couple of months significant auto-correlation of the maximum eigenvalue of the correlation matrix, which express the intensity of market-wide co-movement of stock prices. It also strongly correlates with the open-to-close intraday return and daily return of Nikkei 225 Index. We also study the market mode, which is the first principal component corresponding to the maximum eigenvalue, in the framework of Multi-fractal random walk model. The parameter of the model estimated in a sliding time window, which describes the covariance of the logarithm of the stochastic volatility, grows before almost all large intraday price declines of less than −5%. This phenomenon signifies the upwelling of the market-wide collective behavior before the crash, which might reflect a herding of market participants.

Modeling electricity spot prices using mean-reverting multifractal processes

We discuss stochastic modeling of volatility persistence and anti-correlations in electricity spot prices, and for this purpose we present two mean-reverting versions of the multifractal random walk (MRW). In the first model the anti-correlations are modeled in the same way as in an Ornstein–Uhlenbeck process, i.e. via a drift (damping) term, and in the second model the anti-correlations are included by letting the innovations in the MRW model be fractional Gaussian noise with H<1/2. For both models we present approximate maximum likelihood methods, and we apply these methods to estimate the parameters for the spot prices in the Nordic electricity market. The maximum likelihood estimates show that electricity spot prices are characterized by scaling exponents that are significantly different from the corresponding exponents in stock markets, confirming the exceptional nature of the electricity market. In order to compare the damped MRW model with the fractional MRW model we use ensemble simulations and wavelet-based variograms, and we observe that certain features of the spot prices are better described by the damped MRW model. The characteristic correlation time is estimated to approximately half a year.

FORECASTING VOLATILITY WITH THE MULTIFRACTAL RANDOM WALK MODEL

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 log volatility. For this object and the nonlimiting 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 σ and T.

The Scale-Invariant Brownian Motion Equation and the Lognormal Cascade in the Stock Market

A continuous-time scale-invariant Brownian motion (SIBM) stochastic equation is developed to investigate the dynamics of the stock market. The equation is used to solve the fat tail distribution of the stock universe and the DJIA time series. It is also used to model the volatility clustering in the DJIA time series. The equation is transformed from the Langevin equation into a fractal expression involving an infinite array of random walk. It predicts an elegant way of generating the skew form of the lognormal cascade distribution (Kolmogorov and Mandelbrot), which describes the static log-return distribution in the financial market as well as the velocity distribution in Largrangian turbulence. The higher order randomness (HORN) hypothesis is introduced as the stochastic source of the cascade distribution. A leakage term from HORN is introduced to model the covariance between large volatility and large negative return. A volatility model based on two SIBM processes is built to model the volatility autocorrelation. The volatility half-times of 20 days and 300 days are extracted from the DJIA data. The model generates the static log-return distributions from 10 days to 320 days that match the DJIA data satisfactorily. It also predicts an alternative interpretation of the volatility smile/skew observed in the options market. The relation between the SIBM model and the multifractal random walk model is examined, which yields a simplified SIBM model that could be quite useful in finance.

On the spectrum of multifractal diffusion process

A quasi-multifractal model of stochastic processes is considered. In contrast to the more widely known multifractal random walk model, it is free of such substantial drawbacks as infinite variance of the modeled processes and time-dependent increments. An analysis of a multifractal diffusion-type process is presented, including the moments of increments and local scaling exponents of the process. A quasi-multifractal spectrum of the process is computed. A focus is on two new concepts in the theory of multifractal processes: effective scaling exponent and quasi-multifractal spectrum of a process.

Algorithm for model validation: theory and applications.

Validation is often defined as the process of determining the degree to which a model is an accurate representation of the real world from the perspective of its intended uses. Validation is crucial as industries and governments depend increasingly on predictions by computer models to justify their decisions. We propose to formulate the validation of a given model as an iterative construction process that mimics the often implicit process occurring in the minds of scientists. We offer a formal representation of the progressive build-up of trust in the model. Thus, we replace static claims on the impossibility of validating a given model by a dynamic process of constructive approximation. This approach is better adapted to the fuzzy, coarse-grained nature of validation. Our procedure factors in the degree of redundancy versus novelty of the experiments used for validation as well as the degree to which the model predicts the observations. We illustrate the methodology first with the maturation of quantum mechanics as the arguably best established physics theory and then with several concrete examples drawn from some of our primary scientific interests: a cellular automaton model for earthquakes, a multifractal random walk model for financial time series, an anomalous diffusion model for solar radiation transport in the cloudy atmosphere, and a computational fluid dynamics code for the Richtmyer-Meshkov instability.

Importance of positive feedbacks and overconfidence in a self-fulfilling Ising model of financial markets

Following a long tradition of physicists who have noticed that the Ising model provides a general background to build realistic models of social interactions, we study a model of financial price dynamics resulting from the collective aggregate decisions of agents. This model incorporates imitation, the impact of external news and private information. It has the structure of a dynamical Ising model in which agents have two opinions (buy or sell) with coupling coefficients, which evolve in time with a memory of how past news have explained realized market returns. We study two versions of the model, which differ on how the agents interpret the predictive power of news. We show that the stylized facts of financial markets are reproduced only when agents are overconfident and mis-attribute the success of news to predict return to herding effects, thereby providing positive feedbacks leading to the model functioning close to the critical point. Our model exhibits a rich multifractal structure characterized by a continuous spectrum of exponents of the power law relaxation of endogenous bursts of volatility, in good agreement with previous analytical predictions obtained with the multifractal random walk model and with empirical facts.

Importance of Positive Feedbacks and Over-confidence in a Self-Fulfilling Ising Model of Financial Markets

Following a long tradition of physicists who have noticed that the Ising model provides a general background to build realistic models of social interactions, we study a model of financial price dynamics resulting from the collective aggregate decisions of agents. This model incorporates imitation, the impact of external news and private information. It has the structure of a dynamical Ising model in which agents have two opinions (buy or sell) with coupling coefficients which evolve in time with a memory of how past news have explained realized market returns. We study two versions of the model, which differ on how the agents interpret the predictive power of news. We show that the stylized facts of financial markets are reproduced only when agents are over-confident and mis-attribute the success of news to predict return to herding effects, thereby providing positive feedbacks leading to the model functioning close to the critical point. Our model exhibits a rich multifractal structure characterized by a continuous spectrum of exponents of the power law relaxation of endogenous bursts of volatility, in good agreement with previous analytical predictions obtained with the multifractal random walk model and with empirical facts.

Evidence of intermittent cascades from discrete hierarchical dissipation in turbulence

We present the results of a search of log-periodic corrections to scaling in the moments of the square of velocity increments taken as proxies of the energy dissipation rate in experiments at high Reynolds number (≈2500) of three-dimensional fully developed turbulence. A simple dynamical representation of the Richardson–Kolmogorov cartoon of a cascade shows that standard averaging techniques erase by their very construction the possible existence of log-periodic corrections to scaling associated with a discrete hierarchy. To remedy this drawback, we introduce a novel “canonical” averaging that we test extensively on synthetic examples constructed to mimic the interplay between a weak log-periodic component and rather strong multiplicative and phase noises. Tests are also presented on 1/ f 5/3 random-phase noise, on the multifractal random walk model, and on lattice and non-lattice multifractal models. Our extensive tests confirm the remarkable observation of statistically significant log-periodic corrections to scaling, with a preferred scaling ratio for length scales compatible with the value γ=2. A strong confirmation of this result is provided by the identification of up to five harmonics of the fundamental log-periodic undulations, associated with up to five levels of the underlying hierarchical dynamical structure. A natural interpretation of our results is that the Richardson–Kolmogorov mental picture of a cascade becomes a realistic description if one allows for intermittent births and deaths of discrete cascades at varying scales.