A multifractal approach for stock market inefficiency

In this paper, the multifractality degree in a collection of developed and emerging stock market indices is evaluated. Empirical results suggest that the multifractality degree can be used as a quantifier to characterize the stage of market development of world stock indices. We develop a model to test the relationship between the stage of market development and the multifractality degree and find robust evidence that the relationship is negative, i.e., higher multifractality is associated with a less developed market. Thus, an inefficiency ranking can be derived from multifractal analysis. Finally, a link with previous volatility time series results is established.     

Measuring capital market efficiency: Global and local correlations structure

We introduce a new measure for capital market efficiency. The measure takes into consideration the correlation structure of the returns (long-term and short-term memory) and local herding behavior (fractal dimension). The efficiency measure is taken as a distance from an ideal efficient market situation. The proposed methodology is applied to a portfolio of 41 stock indices. We find that the Japanese NIKKEI is the most efficient market. From a geographical point of view, the more efficient markets are dominated by the European stock indices and the less efficient markets cover mainly Latin America, Asia and Oceania. The inefficiency is mainly driven by a local herding, i.e. a low fractal dimension.     

Forbidden patterns, permutation entropy and stock market inefficiency

In this paper we introduce two new quantifiers for the stock market inefficiency: the number of forbidden patterns and the normalized permutation entropy. They are model-independent measures, thus they have more general applicability. We find robust evidence that degree of market inefficiency is positively correlated with the number of forbidden patterns and negatively correlated with the permutation entropy. Our empirical results suggest that these two physical tools are useful to discriminate the stage of stock market development and can be easily implemented.     

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.     

Analysis of binarized high frequency financial data

A non-trivial probability structure is evident in the binary data extracted from the up/down price movements of very high frequency data such as tick-by-tick data for USD/JPY. In this paper, we analyze the Sony bank USD/JPY rates, ignoring the small deviations from the market price. We then show there is a similar non-trivial probability structure in the Sony bank rate, in spite of the Sony bank rate's having less frequent and larger deviations than tick-by-tick data. However, this probability structure is not found in the data which has been sampled from tick-by-tick data at the same rate as the Sony bank rate. Therefore, the method of generating the Sony bank rate from the market rate has the potential for practical use since the method retains the probability structure as the sampling frequency decreases.     

Multi-scale correlations in different futures markets

In the present work we investigate the multiscale nature of the correlations for high frequency data (1 min) in different futures markets over a period of two years, starting on the 1st of January 2003 and ending on the 31st of December 2004. In particular, by using the concept of local Hurst exponent, we point out how the behaviour of this parameter, usually considered as a benchmark for persistency/antipersistency recognition in time series, is largely time-scale dependent in the market context. These findings are a direct consequence of the intrinsic complexity of a system where trading strategies are scale-adaptive. Moreover, our analysis points out different regimes in the dynamical behaviour of the market indices under consideration.     

A nonextensive approach to the dynamics of financial observables

We present results about financial market observables, specifically returns and traded volumes. They are obtained within the current nonextensive statistical mechanical framework based on the entropy . More precisely, we present stochastic dynamical mechanisms which mimic probability density functions empirically observed. These mechanisms provide possible interpretations for the emergence of the entropic indices q in the time evolution of the corresponding observables. In addition to this, through multi-fractal analysis of return time series, we verify that the dual relation q_stat+q_sens=2 is numerically satisfied, q_stat and q_sens being associated to the probability density function and to the sensitivity to initial conditions respectively. This type of simple relation, whose understanding remains ellusive, has been empirically verified in various other systems.     

Correlation and volatility in an Indian stock market: A random matrix approach

We examine the volatility of an Indian stock market in terms of correlation of stocks and quantify the volatility using the random matrix approach. First we discuss trends observed in the pattern of stock prices in the Bombay Stock Exchange for the three-year period 2000–2002. Random matrix analysis is then applied to study the relationship between the coupling of stocks and volatility. The study uses daily returns of 70 stocks for successive time windows of length 85 days for the year 2001. We compare the properties of matrix C of correlations between price fluctuations in time regimes characterized by different volatilities. Our analyses reveal that (i) the largest (deviating) eigenvalue of C correlates highly with the volatility of the index, (ii) there is a shift in the distribution of the components of the eigenvector corresponding to the largest eigenvalue across regimes of different volatilities, (iii) the inverse participation ratio for this eigenvector anti-correlates significantly with the market fluctuations and finally, (iv) this eigenvector of C can be used to set up a Correlation Index, CI whose temporal evolution is significantly correlated with the volatility of the overall market index.     

The effects of behavioral and structural assumptions in artificial stock market

Recent literature has developed the conjecture that important statistical features of stock price series, such as the fat tails phenomenon, may depend mainly on the market microstructure. This conjecture motivated us to investigate the roles of both the market microstructure and agent behavior with respect to high-frequency returns and daily returns. We developed two simple models to investigate this issue. The first one is a stochastic model with a clearing house microstructure and a population of zero-intelligence agents. The second one has more behavioral assumptions based on Minority Game and also has a clearing house microstructure. With the first model we found that a characteristic of the clearing house microstructure, namely the clearing frequency, can explain fat tail, excess volatility and autocorrelation phenomena of high-frequency returns. However, this feature does not cause the same phenomena in daily returns. So the Stylized Facts of daily returns depend mainly on the agents’ behavior. With the second model we investigated the effects of behavioral assumptions on daily returns. Our study implicates that the aspects which are responsible for generating the stylized facts of high-frequency returns and daily returns are different.     

On the probability distribution of stock returns in the Mike-Farmer model

Recently, Mike and Farmer have constructed a very powerful and realistic behavioral model to mimick the dynamic process of stock price formation based on the empirical regularities of order placement and cancelation in a purely order-driven market, which can successfully reproduce the whole distribution of returns, not only the well-known power-law tails, together with several other important stylized facts. There are three key ingredients in the Mike-Farmer (MF) model: the long memory of order signs characterized by the Hurst index H_s, the distribution of relative order prices x in reference to the same best price described by a Student distribution (or Tsallis’ q-Gaussian), and the dynamics of order cancelation. They showed that different values of the Hurst index H_s and the freedom degree α_x of the Student distribution can always produce power-law tails in the return distribution f_r(r) with different tail exponent α_r. In this paper, we study the origin of the power-law tails of the return distribution f_r(r) in the MF model, based on extensive simulations with different combinations of the left part L(x) for x < 0 and the right part R(x) for x > 0 of f_x(x). We find that power-law tails appear only when L(x) has a power-law tail, no matter R(x) has a power-law tail or not. In addition, we find that the distributions of returns in the MF model at different timescales can be well modeled by the Student distributions, whose tail exponents are close to the well-known cubic law and increase with the timescale.     

Leptokurtic portfolio theory

The question of optimal portfolio is addressed. The conventional Markowitz portfolio optimisation is discussed and the shortcomings due to non-Gaussian security returns are outlined. A method is proposed to minimise the likelihood of extreme non-Gaussian drawdowns of the portfolio value. The theory is called Leptokurtic, because it minimises the effects from “fat tails” of returns. The leptokurtic portfolio theory provides an optimal portfolio for investors, who define their risk-aversion as unwillingness to experience sharp drawdowns in asset prices. Two types of risks in asset returns are defined: a fluctuation risk, that has Gaussian distribution, and a drawdown risk, that deals with distribution tails. These risks are quantitatively measured by defining the “noise kernel” — an ellipsoidal cloud of points in the space of asset returns. The size of the ellipse is controlled with the threshold parameter: the larger the threshold parameter, the larger return are accepted for investors as normal fluctuations. The return vectors falling into the kernel are used for calculation of fluctuation risk. Analogously, the data points falling outside the kernel are used for the calculation of drawdown risks. As a result the portfolio optimisation problem becomes three-dimensional: in addition to the return, there are two types of risks involved. Optimal portfolio for drawdown-averse investors is the portfolio minimising variance outside the noise kernel. The theory has been tested with MSCI North America, Europe and Pacific total return stock indices.     

Randomness in denoised stock returns: The case of Moroccan family business companies

In this paper, we scrutinize entropy in family business stocks listed on Casablanca stock exchange and market index to assess randomness in their returns. For this purpose, we adopt a novel approach based on combination of stationary wavelet transform and Tsallis entropy for empirical analysis of the return series. The obtained empirical results show strong evidence that their respective entropy functions are characterized by opposite dynamics. Indeed, the information contents of their respective dynamics are statistically and significantly different. Obviously, information on regular events carried by family business returns is more certain, whilst that carried by market returns is uncertain. Such results are definitively useful to understand the nonlinear dynamics on returns on family business companies and those of the market. Without a doubt, they could be helpful for quantitative portfolio managers and investors.     

Long memory properties in return and volatility: Evidence from the Korean stock market

In this paper, we study the dual long memory property of the Korean stock market. For this purpose, the ARFIMA-FIGARCH model is applied to two daily Korean stock price indices (KOSPI and KOSDAQ). Our empirical results indicate that long memory dynamics in the returns and volatility can be adequately estimated by the joint ARFIMA-FIGARCH model. We also found that the assumption of a skewed Student-t distribution is better for incorporating the tendency of asymmetric leptokurtosis in a return distribution.     

Stock index dynamics worldwide: a comparative analysis

We perform a comparative analysis of twenty-four daily stock indices across the world, encompassing developed and emerging markets. We compute, directly from the return empirical time series, the Kramers-Moyal (KM) expansion coefficients that govern the evolution of the probability density function of returns throughout timelags. Our study discloses universal patterns of the KM coefficients, which can be described in terms of a few microscopic parameters. These parameters allow to quantify features such as deviations from Gaussianity or from efficiency, providing a tool to discriminate market dynamics.     

Fundamental framework for “technical analysis” of market prices

Starting from the characterization of the past time evolution of market prices in terms of two fundamental indicators, price velocity and price acceleration, we construct a general classification of the possible patterns characterizing the deviation or defects from the random walk market state and its time-translational invariant properties. The classification relies on two dimensionless parameters, the Froude number characterizing the relative strength of the acceleration with respect to the velocity and the time horizon forecast dimensionalized to the training period. Trend-following and contrarian patterns are found to coexist and depend on the dimensionless time horizon. The classification is based on the symmetry requirements of invariance with respect to change of price units and of functional scale-invariance in the space of scenarii. This “renormalized scenario” approach is fundamentally probabilistic in nature and exemplifies the view that multiple competing scenarii have to be taken into account for the same past history. Empirical tests are performed on about nine to thirty years of daily returns of twelve data sets comprising some major indices (Dow Jones, SP500, Nasdaq, DAX, FTSE, Nikkei), some major bonds (JGB, TYX) and some major currencies against the US dollar (GBP, CHF, DEM, JPY). Our “renormalized scenario” exhibits statistically significant predictive power in essentially all market phases. In contrast, a trend following strategy and following strategy perform well only on different and specific market phases. The value of the “renormalized scenario” approach lies in the fact that it always selects the best of the two, based on a calculation of the stability of their predicted market trajectories. Received 3 October 1999     

The distribution and scaling of fluctuations for Hang Seng index in Hong Kong stock market

The statistical properties of the Hang Seng index in the Hong Kong stock market are analyzed. The data include minute by minute records of the Hang Seng index from January 3, 1994 to May 28, 1997. The probability distribution functions of index returns for the time scales from 1 minute to 128 minutes are given. The results show that the nature of the stochastic process underlying the time series of the returns of Hang Seng index cannot be described by the normal distribution. It is more reasonable to model it by a truncated Lévy distribution with an exponential fall-off in its tails. The scaling of the maximium value of the probability distribution is studied. Results show that the data are consistent with scaling of a Lévy distribution. It is observed that in the tail of the distribution, the fall-off deviates from that of a Lévy stable process and is approximately exponential, especially after removing daily trading pattern from the data. The daily pattern thus affects strongly the analysis of the asymptotic behavior and scaling of fluctuation distributions. Received 9 August 2000 and Received in final form 28 August 2000     

Understanding the source of multifractality in financial markets

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 multi-scaling. 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. In order to understand the origin of this effect we investigate different simulated time series by means of the Markov switching multifractal model, autoregressive fractionally integrated moving average processes with stable innovations, fractional Brownian motion and Levy flights. Overall we conclude that the multifractality observed in financial time series is mainly a consequence of the characteristic fat-tailed distribution of the returns and time-correlations have the effect to decrease the measured multifractality.     

Multidimensional minimal spanning tree: The Dow Jones case

This paper introduces a new methodology in order to construct Minimal Spanning Trees (MST) and Hierarchical Trees (HT) using the information provided by more than one variable. In fact, the Symbolic Time Series Analysis (STSA) approach is applied to the Dow Jones companies using information not only from asset returns but also for trading volume. The US stock market structure is obtained, showing eight clusters of companies and General Electric as a central node in the tree. We use different partitions showing that the results do not depend on the particular partition. In addition, we apply Monte Carlo simulations suggesting that the tree is not the result of random connections.     

Intra-group and inter-group trading differentiation analysis: A new approach for detecting the effect of trading volume on stock market fluctuations

Analyzing statistical properties of stock market data using statistical physics has received much attention from physicists and economists in recent years. Although some statistical characteristics of stock market data such as power-low tails of stock returns have become established fact, behavior of other related variables such as trading volume are less studied. In this paper, in order to examine the impact of trading volume on statistical properties of stock market returns, different trading behavior of different traders in Tehran Stock Exchange is analyzed. We define a new coefficient which measures the equilibrium between these different forces affecting the market at any given trading day. By adjusting market returns by this coefficient, we also assessed the impact of these forces on the statistical properties of stock market returns.     

Stable power laws in variable economies; Lotka-Volterra implies Pareto-Zipf

In recent years we have found that logistic systems of the Generalized Lotka-Volterra type (GLV) describing statistical systems of auto-catalytic elements posses power law distributions of the Pareto-Zipf type. In particular, when applied to economic systems, GLV leads to power laws in the relative individual wealth distribution and in market returns. These power laws and their exponent α are invariant to arbitrary variations in the total wealth of the system and to other endogenously and exogenously induced variations. Received 31 December 2001