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.

. Many important probabilistic models in queuing theory, insurance and finance deal with partial sums of a negative mean stationary process (a negative drift random walk), and the law of the supremum of such a process is used to calculate, depending on the context, the ruin probability, the steady state distribution of the number of customers in the system or the value at risk. When the stationary process is heavy--tailed, the corresponding ruin probabilities are high and the stationary distributions are heavy--tailed as well. If the steps of the random walk are independent, then the exact asymptotic behavior of such probability tails was described by Embrechts and Veraverbeke (1982). We show that this asymptotic behavior may be different if the steps of the random walk are not independent, and the dependence affects the joint probability tails of the stationary process. Such type of dependence can be modeled, for example, by a linear process. 1. Introduction In various applied fields...

. We consider a fluid queue with sessions arriving according to a Poisson process. A long--tailed distribution of session lengths induces long range dependence in the system and causes its performance to deteriorate. The deterioration is due to occurrence of load regimes far from average ones. Nonetheless, the extent of this performance deterioration is shown to depend crucially on the average values of the system parameters. 1. Introduction We consider the following fluid queuing model. Sessions (ON periods) are initiated at a network server or multiplexer according to a Poisson process with rate ? 0. Each session is active for a random length of time with distribution F and a finite mean ; during this time it generates network traffic at unit rate. We assume that the lengths of different sessions are independent, and they are also independent of the Poisson arrival process. The service rate is r ? 0 units of traffic per unit time. If X(t) denotes the amount of work (measured in unit...

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.

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The recent increasing interplay between actuarial and financial mathematics has led to a surge of risk theoretic modeling. Especially actuarial ruin models under fairly general conditions on the underlying risk process have become a focus of attention. Motivated by applications to the modeling of operational risk losses in financial risk management, we investigate the stability of classical asymptotic ruin estimates when claims are heavy, and this under variability of the claim intensity process. Various examples are discussed. 1.

. We study the tail behavior of the distribution of certain subadditive functionals acting on the sample paths of L'evy processes. The functionals we consider have, roughly speaking, the following property: only the points of the process that lie above a certain curve contribute to the value of the functional. Our assumptions will make sure that the process ends up eventually below the curve. Our results apply to ruin probabilities, distributions of sojourn times over curves, last hitting times and other functionals. 1. Introduction Both in the theory and in applications of stochastic processes one is often interested in two types of questions: When does the process X = fX(t); t 0g lie above a certain deterministic function (curve) = f(t); t 0g, and given the process exceeds this curve, what are its values? For example, what can be said about the distribution of the biggest excess of the process over the curve and, if both the process and the function are measurable, what is the di...

. For a random walk with negative drift we study the exceedance probability (ruin probability) of a high threshold. The steps of this walk (claim sizes) constitute a stationary ergodic stable process. We study how ruin occurs in this situation and evaluate the asymptotic behavior of the ruin probability for a large variety of stationary ergodic stable processes. Our findings show that the order of magnitude of the ruin probability varies significantly from one model to another. In particular, ruin becomes much more likely when the claim sizes exhibit longrange dependence. The proofs exploit large deviation techniques for sums of dependent stable random variables and the series representation of a stable process as a function of a Poisson process. 1991 Mathematics Subject Classification. Primary 60E07, 60G10; Secondary 60K30. Key words and phrases. Stable process, stationary process, ruin probability, heavy tails, supremum, negative drift, risk. Research supported by NATO Collaborative...

Abstract. Different Monte Carlo copula-based Value-at-Risk (VaR) models are presented and numerical results on real portfolios are reported. The models feature two families of heavy-tailed univariate distributions: Student-t and Stable. A generalized autoregressive conditional heteroskedastic (GARCH) Student-t process is considered and implemented to enhance the static models. Comparisons with other commonly-used VaR models are included. It is numerically demonstrated that the GARCH-Student-t models estimate rather accurately high-quantile VaR measures and outperform the rest of the considered models. The views expressed in this article are solely the authors ’ opinions, and are not supported or shared by The Options Clearing Corporation. § To whom correspondence should be addressed (sivanov@theocc.com) Dynamic Value-at-Risk with Heavy-Tailed Distributions: Portfolio Study 2