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 ...

The distribution of returns in financial time series exhibits heavy tails. In empirical studies, it has been found that gaps between the orders in the order book lead to large price shifts and thereby to these heavy tails. We set up an agent based model to study this issue and, in particular, how the gaps in the order book emerge. The trading mechanism in our model is based on a double-auction order book, which is used on nearly all stock exchanges. In situations where the order book is densely occupied with limit orders we do not observe fat-tailed distributions. As soon as less liquidity is available, a gap structure forms which leads to return distributions with heavy tails. We show that return distributions with heavy tails are an order-book effect if the available liquidity is constrained. This is largely independent of the specific trading strategies.

The instability of the financial system as experienced in recent years and in previous periods is often linked to credit defaults, i.e., to the failure of obligors to make promised payments. Given the large number of credit contracts, this problem is amenable to be treated with approaches developed in statistical physics. We introduce the idea of ensemble averaging and thereby uncover generic features of credit risk. We then show that the often advertised concept of diversification, i.e., reducing the risk by distributing it, is deeply flawed when it comes to credit risk. The risk of extreme losses remains due to the ever present correlations, implying a substantial and persistent intrinsic danger to the financial system.

In structural credit risk models, default events and the ensuing losses are both derived from the asset values at maturity. Hence it is of utmost importance to choose a distribution for these asset values which is in accordance with empirical data. At the same time, it is desirable to still preserve some analytical tractability. We achieve both goals by putting forward an ensemble approach for the asset correlations. Consistently with the data, we view them as fluctuating quantities, for which we may choose the average correlation as homogeneous. Thereby we can reduce the number of parameters to two, the average correlation between assets and the strength of the fluctuations around this average value. Yet, the resulting asset value distribution describes the empirical data well. This allows us to derive the distribution of credit portfolio losses. With Monte-Carlo simulations for the Value at Risk and Expected Tail Loss we validate the assumptions of our approach and demonstrate the necessity of taking fluctuating correlations into account.

We consider random vectors drawn from a multivariate normal distribution and compute the sample statistics in the presence of stochastic correlations. For this purpose, we construct an ensemble of random correlation matrices and average the normal distribution over this ensemble. The resulting distribution contains a modified Bessel function of the second kind whose behavior differs significantly from the multivariate normal distribution, in the central part as well as in the tails. This result is then applied to asset returns. We compare with empirical return distributions using daily data from the NASDAQ Composite Index in the period from 1992 to 2012. The comparison reveals good agreement, the average portfolio return distribution describes the data well especially in the central part of the distribution. This in turn confirms our ansatz to model the nonstationarity by an ensemble average.