We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.

University. We are especially grateful for suggestions from Geert Bekaert, Bob Hodrick, and Ken Singleton. We also thank an anonymous referee whose comments and suggestions greatly improved the paper.

Abstract. In this paper we clarify dependence properties of elliptical distributions by deriving general but explicit formulas for the coefficients of upper and lower tail dependence and spectral measures with respect to different norms. We show that an elliptically distributed random vector is regularly varying if and only if the bivariate marginal distributions have tail dependence. Furthermore, the tail dependence coefficients are fully determined by the tail index of the random vector (or equivalently of its components) and the linear correlation coefficient. Whereas Kendall’s tau is invariant in the class ofelliptical distributions with continuous marginals and a fixed dispersion matrix, we show thatthis is not true for Spearman’s rho. We also show that sums of elliptically distributed random vectors with the same dispersion matrix (up to a positive constant factor) remain elliptical if they are dependent only through their radial parts. 1.

Stylised facts for univariate high–frequency data in finance are well–known. They include scaling behaviour, volatility clustering, heavy tails, and seasonalities. The multivariate problem, however, has scarcely been addressed up to now. In this paper, bivariate series of high–frequency FX spot data for major FX markets are investigated. First, as an indispensable prerequisite for further analysis, the problem of simultaneous deseasonalisation of high–frequency data is addressed. In the following sections we analyse in detail the dependence structure as a function of the time scale. Particular emphasis is put on the tail behaviour, which is investigated by means of copulas.

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This paper inv estigates the potential for extreme co-mov ements between financial assets by directly testing the underlying dependence structure. In particular, a t-dependence structure, deriv ed from the Student t distribution, is used as a proxy to test for this extremal behav#a(0 Tests in three di#erent markets (equities, currencies, and commodities) indicate that extreme co-mov ements are statistically significant. Moreov er, the "correlation-based" Gaussian dependence structure, underlying the multiv ariate Normal distribution, is rejected with negligible error probability when tested against the t-dependencealternativ e. The economic significance of these results is illustratedv ia three examples: co-mov ements across the G5 equity markets; portfoliov alue-at-risk calculations; and, pricing creditderiv ativ es. JEL Classification: C12, C15, C52, G11. Keywords: asset returns, extreme co-mov ements, copulas, dependence modeling, hypothesis testing, pseudo-likelihood, portfolio models, risk management. # The authorsw ould like to thankAndrew Ang, Mark Broadie, Loran Chollete, and Paul Glasserman for their helpful comments on an earlier version of this manuscript. Both authors arewS; the Columbia Graduate School of Business, e-mail: {rm586,assaf.zeevi}@columbia.edu, current version available at www.columbia.edu\# rm586 1 Introducti7 Specification and identification of dependencies between financial assets is a key ingredient in almost all financial applications: portfolio management, risk assessment, pricing, and hedging, to name but a few. The seminal work of Markowitz (1959) and the early introduction of the Gaussian modeling paradigm, in particular dynamic Brownian-based models, hav e both contributed greatly to making the concept of co rrelatio almost synony...

Properties of risk measures for extreme risks have become an important topic of research. In the present paper we discuss sub- and superadditivity of quantile based risk measures and show how multivariate extreme value theory yields the ideal modeling environment. Numerous examples and counter-examples highlight the applicability of the main results obtained.

Using one of the key property of copulas that they remain invariant under an arbitrary monotonous change of variable, we investigate the null hypothesis that the dependence between financial assets can be modeled by the Gaussian copula. We find that most pairs of currencies and pairs of major stocks are compatible with the Gaussian copula hypothesis, while this hypothesis can be rejected for the dependence between pairs of commodities (metals). Notwithstanding the apparent qualification of the Gaussian copula hypothesis for most of the currencies and the stocks, a non-Gaussian copula, such as the Student’s copula, cannot be rejected if it has sufficiently many “degrees of freedom”. As a consequence, it may be very dangerous to embrace blindly the Gaussian copula hypothesis, especially when the correlation coefficient between the pair of asset is too high as the tail dependence neglected by the Gaussian copula can be as large as, i.e., three out five extreme events which occur in unison are missed.

The infinite source Poisson network model assumes sources begin data transmissions at Poisson time points and continue for random lengths of time. The random transmission times have such heavy tails that the variance is infinite. Transmission rates have been assumed non-random and, usually constant. However, analysis of network data suggests that the transmission rate is also a random variable with a heavy tail. So we consider an infonite source Poisson model with sources transmitting for a random length of time at a random rate. Both the rates and lengths have infinite variance but finite mean and are assumed asymptotically independent, a concept made precise. We carefully discuss equivalent formulations of asymptotic independence and prove a limit theorem for the input process showing that the centered process under a suitable scaling converges to a totally skewed stable Lévy motion in the sense of finite dimensional distributions.

This paper proposes a methodology to provide risk measures for portfolios during extreme events. The approach is based on splitting the multivariate extreme value distribution of the assets of the portfolio into two parts: the distributions of each asset and their dependence function. The estimation problem is also investigated. Then, stress-testing is applied for market indices portfolios and Monte-Carlo based risk measures – Value-at-Risk and Expected Shortfall – are provided.