Non-stationarity in financial time series: Generic features and tail behavior

Abstract

Financial markets are prominent examples for highly non-stationary systems. Sample averaged observables such as variances and correlation coefficients strongly depend on the time window in which they are evaluated. This implies severe limitations for approaches in the spirit of standard equilibrium statistical mechanics and thermodynamics. Nevertheless, we show that there are similar generic features which we uncover in the empirical return distributions for whole markets. We explain our findings by setting up a random matrix model. The great success of statistical mechanics and thermodynamics is borne out by their ability to characterize, in the equilibrium, large systems with many degrees of freedom in terms of a few state variables, for example temperature and pressure. Ergodicity (or quasi-ergodicity) is the prerequisite needed to introduce statistical ensembles. Systems out of equilibrium or, more generally, nonstationary systems still pose fundamental challenges [1][2][3][4]. Complex systems -the term "complex" is used in a broad sense -show a wealth of different aspects which can be traced back to non-stationarity [5,6]. Financial markets are presently in the focus, because they demonstrated their non-stationarity in a rather drastic way during the recent years. To assess a financial market as a whole, the correlations between the prices of the individual stocks are of crucial importance [7][8][9][10]. They fluctuate considerably in time, e.g., because the market expectations of the traders change, the business relations between the companies change, particularly in a state of crisis, and so on. The motion of the stock prices is in this respect reminiscent of that of particles in many-body systems such as heavier atomic nuclei. Depending on the excitation energy, the motion of the individual particles can be incoherent, i.e., uncorrelated in the above terminology, or coherent (collective), i.e., correlated, or even somewhere in-between [11][12][13]. This non-stationarity on the energy scale leads to very different spectral properties, [12,13]. Such an analogy can be helpful, but we do not want to overstretch it.Here, we want to show that the non-stationarity, namely the fluctuation of the correlations, induces generic features in financial time series. These become visible when looking at quantities which measure the stock price changes for the entire market. We have four goals. First, we carry out a detailed data analysis revealing the generic features. Second, we set up a random matrix model to explain them. Third, we demonstrate that the non-stationarity of the correlations leads to heavy tails. Fourth, we argue that our approach maps a noninvariant situation to an effectively invariant one. For an economic audience we discuss the consequences for portfolio management elsewhere [14].Consider K companies with stock prices S k (t), k = 1, . . . , K as functions of time t. The relative price changes over a fixed time interval ∆t, i.e., the returnsare well-known to have distributions with heavy tails, the ...

Constructing analytically tractable ensembles of stochastic covariances with an application to financial data

“…(4) and for the financial data in Ref. [16], the superposition of the amplitudes, in the present case the returns, drives the multivariate distribution towards a Gaussian, provided that the covariances are sufficiently constant. Second, as observed in Ref.…”

“…II C. Here, we derive the approach for the general case, for sake of illustration, the reader is referred to Ref. [16] and Sec. III…”

“…-As in Ref. [16], we model this by random matrices. As the covariance matrix is different at each time t where it is analyzed, we replace the covariance matrix in the distribution (4) by the expression…”

“…-Can we set up statistical models for these non-stationarities? -In the context of finance, we recently put forward a random ma- * Electronic address: frederik.meudt@uni-due.de † Electronic address: martin.theissen@stud.uni-due.de ‡ Electronic address: rudi.schaefer@uni-due.de § Electronic address: thomas.guhr@uni-due.de trix approach to tackle these issues [16]. We also succesfully applied it in a study of credit risk and its impact on systemic stability [17].…”

Detrended fluctuation analysis made flexible to detect range of cross-correlated fluctuations

“…Standard correlation measures like the Pearson correlation coefficient and the cross-correlation function require stationary data in order to provide reliable results, which is a requirement that is hard to fulfill in many real-world situations (the financial and physiological data are the negative examples here [1][2][3][4][5][6][7][8][9][10]). (By stationarity we mean stability of the probability distribution functions of the data over time; from this perspective nonstationarity can be produced both by the long-range autocorrelations and by the pdf's heavy tails that make any signal length effectively insufficient.)…”