Note on log-periodic description of 2008 financial crash

We analyse 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.     

Hausdorff clustering of financial time series

A clustering procedure is introduced based on the Hausdorff distance as a similarity measure between clusters of elements. The method is applied to the financial time series of the Dow Jones industrial average (DJIA) index to find companies that share a similar behavior. Comparisons are made with other linkage algorithms.     

Clustering stock market companies via chaotic map synchronization

A pairwise clustering approach is applied to the analysis of the Dow Jones index companies, in order to identify similar temporal behavior of the traded stock prices. To this end, the chaotic map clustering algorithm is used, where a map is associated to each company and the correlation coefficients of the financial time series to the coupling strengths between maps. The simulation of a chaotic map dynamics gives rise to a natural partition of the data, as companies belonging to the same industrial branch are often grouped together. The identification of clusters of companies of a given stock market index can be exploited in the portfolio optimization strategies.     

Local order, entropy and predictability of financial time series

We consider time series of financial data as the Dow Jones Index with respect to the existence of local order. The basic idea is that in spite of the high stochasticity in average there might be special local situations where there local order exist and the predictability is considerably higher than in average. In order to check this assumption we discretise the time series and investigate the frequency of the continuation of definite words of length n first. We prove the existence of relatively long-range correlations under special conditions. The higher order Shannon entropies and the conditional entropies (dynamical entropies) are calculated, characteristic fluctuations are found. Instead of the dynamic entropies which yield mean values of the uncertainty/predictability we finally investigate the local values of the uncertainty/predictability and the distribution of these quantities. Received 19 January 2000     

Cross-correlation dynamics in financial time series

The dynamics of the equal-time cross-correlation matrix of multivariate financial time series is explored by examination of the eigenvalue spectrum over sliding time windows. Empirical results for the S&P 500 and the Dow Jones Euro Stoxx 50 indices reveal that the dynamics of the small eigenvalues of the cross-correlation matrix, over these time windows, oppose those of the largest eigenvalue. This behaviour is shown to be independent of the size of the time window and the number of stocks examined.A basic one-factor model is then proposed, which captures the main dynamical features of the eigenvalue spectrum of the empirical data. Through the addition of perturbations to the one-factor model, (leading to a ‘market plus sectors’ model), additional sectoral features are added, resulting in an Inverse Participation Ratio comparable to that found for empirical data. By partitioning the eigenvalue time series, we then show that negative index returns, (drawdowns), are associated with periods where the largest eigenvalue is greatest, while positive index returns, (drawups), are associated with periods where the largest eigenvalue is smallest. The study of correlation dynamics provides some insight on the collective behaviour of traders with varying strategies.     

Coevolutionary dynamics of opinion propagation and social balance: The key role of small-worldness

We present a (semi-) analytical model of asset fluctuations using the framework of Fokker-Planck equations, together with generalised diffusion coefficients. Allowing for time dependence of the coefficients D ₁ and D ₂ provides a route to the characterization of the long- and short-time nature of autocorrelation functions, as is demonstrated for Dow Jones 1993–2012 financial data.     

Characterizing and modelling cyclic behaviour in non-stationary time series through multi-resolution analysis

A method based on wavelet transform is developed to characterize variations at multiple scales in non-stationary time series. We consider two different financial time series, S&P CNX Nifty closing index of the National Stock Exchange (India) and Dow Jones industrial average closing values. These time series are chosen since they are known to comprise of stochastic fluctuations as well as cyclic variations at different scales. The wavelet transform isolates cyclic variations at higher scales when random fluctuations are averaged out; this corroborates correlated behaviour observed earlier in financial time series through random matrix studies. Analysis is carried out through Haar, Daubechies-4 and continuous Morlet wavelets for studying the character of fluctuations at different scales and show that cyclic variations emerge at intermediate time scales. It is found that Daubechies family of wavelets can be effectively used to capture cyclic variations since these are local in nature. To get an insight into the occurrence of cyclic variations, we then proceed to model these wavelet coefficients using genetic programming (GP) approach and using the standard embedding technique in the reconstructed phase space. It is found that the standard methods (GP as well as artificial neural networks) fail to model these variations because of poor convergence. A novel interpolation approach is developed that overcomes this difficulty. The dynamical model equations have, primarily, linear terms with additive Padé-type terms. It is seen that the emergence of cyclic variations is due to an interplay of a few important terms in the model. Very interestingly GP model captures smooth variations as well as bursty behaviour quite nicely.      

Low q-moment multifractal analysis of Gold price, Dow Jones Industrial Average and BGL-USD exchange rate

An estimate of the low q-moment values of the assumed multifractal spectrum of Gold price, Dow Jones Industrial Average (DJIA) and Bulgarian Lev - USA Dollar (BGL-USD) exchange rate over a 6 1/2 year time span has been made. The findings can be compared to the analysis made on 23 foreign currency exchange rates by Vandewalle and Ausloos but there is a clear indication of some differences. Comparison to fractional Brownian motion is made. The analysis shows that these three financial data are not likely fractal but rather multifractal indeed. Received 17 October 1998 and Received in final form 2 November 1998     

Rényi’s information transfer between financial time series

In this paper, we quantify the statistical coherence between financial time series by means of the Rényi entropy. With the help of Campbell’s coding theorem, we show that the Rényi entropy selectively emphasizes only certain sectors of the underlying empirical distribution while strongly suppressing others. This accentuation is controlled with Rényi’s parameter q$q$. To tackle the issue of the information flow between time series, we formulate the concept of Rényi’s transfer entropy as a measure of information that is transferred only between certain parts of underlying distributions. This is particularly pertinent in financial time series, where the knowledge of marginal events such as spikes or sudden jumps is of a crucial importance. We apply the Rényian information flow to stock market time series from 11 world stock indices as sampled at a daily rate in the time period 02.01.1990–31.12.2009. Corresponding heat maps and net information flows are represented graphically. A detailed discussion of the transfer entropy between the DAX and S&P500 indices based on minute tick data gathered in the period 02.04.2008–11.09.2009 is also provided. Our analysis shows that the bivariate information flow between world markets is strongly asymmetric with a distinct information surplus flowing from the Asia–Pacific region to both European and US markets. An important yet less dramatic excess of information also flows from Europe to the US. This is particularly clearly seen from a careful analysis of Rényi information flow between the DAX and S&P500 indices.     

A model for the evaluation of systemic risk in stock markets

Systemic risk refers to the possibility of a collapse of an entire financial system or market, differing from the risk associated with any particular individual or a group pertaining to the system, which may include banks, government, brokers, and creditors. After the 2008 financial crisis, a significant amount of effort has been directed to the study of systemic risk and its consequences around the world. Although it is very difficult to predict when people begin to lose confidence in a financial system, it is possible to model the relationships among the stock markets of different countries and perform a Monte Carlo-type analysis to study the contagion effect. Because some larger and stronger markets influence smaller ones, a model inspired by a catalytic chemical model is proposed. In chemical reactions, reagents with higher concentrations tend to favor their conversion to products. In order to modulate the conversion process, catalyzers may be used. In this work, a mathematical modeling is proposed with bases on the catalytic chemical reaction model. More specifically, the Hang Seng and Dow Jones indices are assumed to dominate Ibovespa (the Brazilian Stock Market index), such that the indices of strong markets are taken as being analogous to the concentrations of the reagents and the indices of smaller markets as concentrations of products. The role of the catalyst is to model the degree of influence of one index on another. The actual data used to fit the model parameter consisted of the Hang Seng index, Dow Jones index, and Ibovespa, since 1993. “What if” analyses were carried out considering some intervention policies.     

How the financial crash of October 1997 could have been predicted

From the analysis of (closing value) stock market index like the Dow Jones Industrial average and the S&P500 it is possible to observe the precursor of a so-called crash. This is shown on the Oct. 1987 and Oct. 1997 cases. The data analysis indicates that the index divergence has followed twice a “universal” behavior, i.e. a logarithmic dependence, superposed on a well defined oscillation pattern. The prediction of the crash date is remarkable and can be done two months in advance. In the spirit of phase transition phenomena, the economic index is said to be analogous to a signal signature found in a two dimensional fluid of vortices. Received: 23 March 1998 / Revised and Accepted: 23 April 1998     

Power-law autocorrelated stochastic processes with long-range cross-correlations

We develop a stochastic process with two coupled variables where the absolute values of each variable exhibit long-range power-law autocorrelations and are also long-range cross-correlated. We investigate how the scaling exponents characterizing power-law autocorrelation and long-range cross-correlation behavior in the absolute values of the generated variables depend on the two parameters in our model. In particular, if the autocorrelation is stronger, the cross-correlation is also stronger. We test the utility of our approach by comparing the autocorrelation and cross-correlation properties of the time series generated by our model with data on daily returns over ten years for two major financial indices, the Dow Jones and the S&P500, and on daily returns of two well-known company stocks, IBM and Microsoft, over five years.     

Not Fully Developed Turbulence in the Dow Jones Index

The shape of the curves relating the scaling exponents of the structure functions to the order of these functions is shown to distinguish the Dow Jones index from other stock market indices. We conclude from the shape differences that the information-loss rate for the Dow Jones index is reduced at smaller time scales, while it grows for other indices. This anomaly is due to the construction of the index, in particular to its dependence on a single market parameter: price. Prices are subject to turbulence bursts, which act against full development of turbulence.     

Optimal investment horizons

In stochastic finance, one traditionally considers the return as a competitive measure of an asset, i.e., the profit generated by that asset after some fixed time span Δt, say one week or one year. This measures how well (or how bad) the asset performs over that given period of time. It has been established that the distribution of returns exhibits “fat tails” indicating that large returns occur more frequently than what is expected from standard Gaussian stochastic processes [1-3]. Instead of estimating this “fat tail” distribution of returns, we propose here an alternative approach, which is outlined by addressing the following question: What is the smallest time interval needed for an asset to cross a fixed return level of say 10%? For a particular asset, we refer to this time as the investment horizon and the corresponding distribution as the investment horizon distribution. This latter distribution complements that of returns and provides new and possibly crucial information for portfolio design and risk-management, as well as for pricing of more exotic options. By considering historical financial data, exemplified by the Dow Jones Industrial Average, we obtain a novel set of probability distributions for the investment horizons which can be used to estimate the optimal investment horizon for a stock or a future contract. Received 20 February 2002 Published online 25 June 2002     

Mixed-Stable Models: An Application to High-Frequency Financial Data

The paper extends the study of applying the mixed-stable models to the analysis of large sets of high-frequency financial data. The empirical data under review are the German DAX stock index yearly log-returns series. Mixed-stable models for 29 DAX companies are constructed employing efficient parallel algorithms for the processing of long-term data series. The adequacy of the modeling is verified with the empirical characteristic function goodness-of-fit test. We propose the smart-Δ method for the calculation of the α-stable probability density function. We study the impact of the accuracy of the computation of the probability density function and the accuracy of ML-optimization on the results of the modeling and processing time. The obtained mixed-stable parameter estimates can be used for the construction of the optimal asset portfolio.     

Comparing the structure of an emerging market with a mature one under global perturbation

In this paper we investigate the Tehran stock exchange (TSE) and Dow Jones Industrial Average (DJIA) in terms of perturbed correlation matrices. To perturb a stock market, there are two methods, namely local and global perturbation. In the local method, we replace a correlation coefficient of the cross-correlation matrix with one calculated from two Gaussian-distributed time series, whereas in the global method, we reconstruct the correlation matrix after replacing the original return series with Gaussian-distributed time series. The local perturbation is just a technical study. We analyze these markets through two statistical approaches, random matrix theory (RMT) and the correlation coefficient distribution. By using RMT, we find that the largest eigenvalue is an influence that is common to all stocks and this eigenvalue has a peak during financial shocks. We find there are a few correlated stocks that make the essential robustness of the stock market but we see that by replacing these return time series with Gaussian-distributed time series, the mean values of correlation coefficients, the largest eigenvalues of the stock markets and the fraction of eigenvalues that deviate from the RMT prediction fall sharply in both markets. By comparing these two markets, we can see that the DJIA is more sensitive to global perturbations. These findings are crucial for risk management and portfolio selection.     

The geometry of chaotic dynamics — a complex network perspective

Recently, several complex network approaches to time series analysis have been developed and applied to study a wide range of model systems as well as real-world data, e.g., geophysical or financial time series. Among these techniques, recurrence-based concepts and prominently ε-recurrence networks, most faithfully represent the geometrical fine structure of the attractors underlying chaotic (and less interestingly non-chaotic) time series. In this paper we demonstrate that the well known graph theoretical properties local clustering coefficient and global (network) transitivity can meaningfully be exploited to define two new local and two new global measures of dimension in phase space: local upper and lower clustering dimension as well as global upper and lower transitivity dimension. Rigorous analytical as well as numerical results for self-similar sets and simple chaotic model systems suggest that these measures are well-behaved in most non-pathological situations and that they can be estimated reasonably well using ε-recurrence networks constructed from relatively short time series. Moreover, we study the relationship between clustering and transitivity dimensions on the one hand, and traditional measures like pointwise dimension or local Lyapunov dimension on the other hand. We also provide further evidence that the local clustering coefficients, or equivalently the local clustering dimensions, are useful for identifying unstable periodic orbits and other dynamically invariant objects from time series. Our results demonstrate that ε-recurrence networks exhibit an important link between dynamical systems and graph theory.     

Autoregressive conditional duration as a model for financial market crashes prediction

There is an increasing number of studies showing that financial market crashes can be detected and predicted. The main aim of the research was to develop a technique for crashes prediction based on the analysis of durations between sequent crashes of a certain magnitude of Dow Jones Industrial Average. We have found significant autocorrelation in the series of durations between sequent crashes and suggest autoregressive conditional duration models (ACD) to forecast the crashes. We apply the rolling intervals technique in the sample of more than 400 DJIA crashes in 1896–2011 and repeatedly use the data on 100 sequent crashes to estimate a family of ACD models and calculate forecasts of the one following crash. It appears that the ACD models provide significant predictive power when combined with the inter-event waiting time technique. This suggests that despite the high quality of retrospective predictions, using the technique for real-time forecasting seems rather ineffective, as in the case of every particular crash the specification of the ACD model, which would provide the best quality prediction, is rather hard to identify.     

The Anderson-Darling test of fit for the power-law distribution from left-censored samples

Maximum likelihood estimation and a test of fit based on the Anderson-Darling statistic are presented for the case of the power-law distribution when the parameters are estimated from a left-censored sample. Expressions for the maximum likelihood estimators and tables of asymptotic percentage points for the A² statistic are given. The technique is illustrated for data from the Dow Jones Industrial Average index, an example of high theoretical and practical importance in Econophysics, Finance, Physics, Biology and, in general, in other related sciences such as Complexity Sciences.