Thomas Mikosch

Nonstationarities in Financial Time Series, the Long-Range Dependence, and the IGARCH Effects

We give the theoretical basis of a possible explanation for two stylized facts observed in long log-return series: the long-range dependence (LRD) in volatility and the integrated GARCH (IGARCH). Both these effects can be explained theoretically if one assumes that the data are nonstationary.

Handbook of financial time series

Recent Developments in GARCH Modeling.- An Introduction to Univariate GARCH Models.- Stationarity, Mixing, Distributional Properties and Moments of GARCH(p, q)#x2013 Processes.- ARCH(#x221E ) Models and Long Memory Properties.- A Tour in the Asymptotic Theory of GARCH Estimation.- Practical Issues in the Analysis of Univariate GARCH Models.- Semiparametric and Nonparametric ARCH Modeling.- Varying Coefficient GARCH Models.- Extreme Value Theory for GARCH Processes.- Multivariate GARCH Models.- Recent Developments in Stochastic Volatility Modeling.- Stochastic Volatility: Origins and Overview.- Probabilistic Properties of Stochastic Volatility Models.- Moment#x2013 Based Estimation of Stochastic Volatility Models.- Parameter Estimation and Practical Aspects of Modeling Stochastic Volatility.- Stochastic Volatility Models with Long Memory.- Extremes of Stochastic Volatility Models.- Multivariate Stochastic Volatility.- Topics in Continuous Time Processes.- An Overview of Asset-Price Models.- Ornstein-Uhlenbeck Processes and Extensions.- Jump-Type Levy Processes.- Levy-Driven Continuous-Time ARMA Processes.- Continuous Time Approximations to GARCH and Stochastic Volatility Models.- Maximum Likelihood and Gaussian Estimation of Continuous Time Models in Finance.- Parametric Inference for Discretely Sampled Stochastic Differential Equations.- Realized Volatility.- Estimating Volatility in the Presence of Market Microstructure Noise: A Review of the Theory and Practical Considerations.- Option Pricing.- An Overview of Interest Rate Theory.- Extremes of Continuous-Time Processes..- Topics in Cointegration and Unit Roots.- Cointegration: Overview and Development.- Time Series with Roots on or Near the Unit Circle.- Fractional Cointegration.- Special Topics - Risk.- Different Kinds of Risk.- Value-at-Risk Models.- Copula-Based Models for Financial Time Series.- Credit Risk Modeling.- Special Topics - Time Series Methods.- Evaluating Volatility and Correlation Forecasts.- Structural Breaks in Financial Time Series.- An Introduction to Regime Switching Time Series Models.- Model Selection.- Nonparametric Modeling in Financial Time Series.- Modelling Financial High Frequency Data Using Point Processes.- Special Topics - Simulation Based Methods.- Resampling and Subsampling for Financial Time Series.- Markov Chain Monte Carlo.- Particle Filtering.

Regular variation of GARCH processes

We show that the finite-dimensional distributions of a GARCH process are regularly varying, i.e., the tails of these distributions are Pareto-like and hence heavy-tailed. Regular variation of the joint distributions provides insight into the moment properties of the process as well as the dependence structure between neighboring observations when both are large. Regular variation also plays a vital role in establishing the large sample behavior of a variety of statistics from a GARCH process including the sample mean and the sample autocovariance and autocorrelation functions. In particular, if the 4th moment of the process does not exist, the rate of convergence of the sample autocorrelations becomes extremely slow, and if the second moment does not exist, the sample autocorrelations have non-degenerate limit distributions.

Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach

This paper studies the quasi-maximum-likelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form Xt = σt Zt , where the unobservable volatility σt is a parametric function of (Xt−1 ,...,X t−p ,σ t−1 ,...,σ t−q ) for some p, q ≥ 0, and (Zt ) is standardized i.i.d. noise. We assume that these models are solutions to stochastic recurrence equations which satisfy a contraction (random Lipschitz coefficient) property. These assumptions are satisfied for the popular GARCH, asymmetric GARCH and exponential GARCH processes. Exploiting the contraction property, we give conditions for the existence and uniqueness of a strictly stationary solution (Xt ) to the stochastic recurrence equation and establish consistency and asymptotic normality of the QMLE. We also discuss the problem of invertibility of such time series models. 1. Introduction. Gaussian quasi-maximum-likelihood estimation, that is, likelihood estimation under the hypothesis of Gaussian innovations, is a popular method which is widely used for inference in time series models. However, it is often a nontrivial task to establish the consistency and asymptotic normality of the quasi-maximum-likelihood estimator (QMLE) applied to specific models and, therefore, an in-depth analysis of the probabilistic structure generated by the model is called for. A classical example of this kind is the seminal paper by Hannan [18] on estimation in linear ARMA time series. In this paper we study the QMLE for a general class of conditionally heteroscedastic time series models, which includes GARCH, asymmetric GARCH and exponential GARCH. Recall that a GARCH(p, q) [generalized autoregressive conditionally heteroscedastic of order (p, q)] process [4 ]i s def ined by

Non-Life Insurance Mathematics

Non-life insurance mathematics , Non-life insurance mathematics , کتابخانه دیجیتال جندی شاپور اهواز

Large deviations of heavy-tailed sums with applications in insurance

First we give a short review of large deviation results for sums of i.i.d. random variables. The main emphasis is on heavy-tailed distributions. We stress more the methodology than the detailed calculations. Large deviation techniques are then applied to randomly indexed sums and shot noise processes. We also indicate the close relationship between large deviation results and the modeling of large insurance claims.

Explosive Poisson shot noise processes with applications to risk reserves

process and investigate their asymptotic properties. Our main result is a functional central limit theorem with a self-similar Gaussian limit process which, in the classical case, is Brownian motion. The theorems are derived under regularity conditions on the moment and covariance functions of the shot noise process. The crucial condition is regular variation of the covariance function which implies the self-similarity of the limit process. The model is applied to delay in claim settlement in insurance portfolios. In this context we discuss some specific models and their properties. We also use the asymptotic theory for studying the ruin time and ruin probability for a risk process which is based on the Poisson shot noise process.

The extremogram: A correlogram for extreme events

We consider a strictly stationary sequence of random vectors whose finite-dimensional distributions are jointly regularly varying with some positive index. This class of processes includes among others ARMA processes with regularly varying noise, GARCH processes with normally or student distributed noise, and stochastic volatility models with regularly varying multiplicative noise. We define an analog of the autocorrelation function, the extremogram, which only depends on the extreme values in the sequence. We also propose a natural estimator for the extremogram and study its asymptotic properties under �-mixing. We show asymptotic normality, calculate the extremogram for various examples and consider spectral analysis related to the extremogram. 1. Measures of extremal dependence in a strictly stationary sequence The motivation for this research comes from the problem of choosing between two popular and commonly used families of models, the generalized autoregressive conditional heteroscedastic (GARCH) process and the heavy-tailed stochastic volatility (SV) process, for modeling a particular financial time series. Both GARCH and SV models possess the stylized features exhibited by log- returns of financial assets. Specifically, these time series have heavy-tailed marginal distributions, are dependent but uncorrelated, and display stochastic volatility. The latter property is manifested via the often slow decay of the sample autocorrelation function (ACF) of the absolute values and squares of the time series. Since both GARCH and SV models can be chosen to have virtually identical behavior in the tails of the marginal distribution and in the ACF of the squares of the process, it is difficult for a given time series of returns to decide between the two models on the basis of routine time series diagnostic tools. The problem of finding probabilistically reasonable and statistically estimable measures of ex- tremal dependence in a strictly stationary sequence is to some extent an open one. In classical time series analysis, which mostly deals with second order structure of stationary sequences, the ACF is a well accepted object for describing meaningful information about serial dependence. The ACF is sometimes over-valued as a tool for measuring dependence especially if one is only interested in extremes. It does of course determine the distribution of a stationary Gaussian sequence, but for non-Gaussian and non-linear time series the ACF often provides little insight into the dependence structure of the process. This is particularly the case when one considers heavy-tailed non-linear time series such as the GARCH model. In this case, the estimation of the ACF via the sample ACF is also rather imprecise and even misleading since the asymptotic confidence bands are typ- ically larger than the estimated autocorrelations, see for example the results in Basrak et al. (1) for bilinear processes; Davis and Mikosch (12), Mikosch and Stuaricua (26) and Basrak et al. (2) for ARCH and GARCH processes, Resnick (32) for teletraffic models. 1.1. The extremal index. The asymptotic behavior of the extremes leads to one clear difference between GARCH and SV processes. It was shown in Davis and Mikosch (12), Basrak et al. (2), Davis and Mikosch (13) (see also Breidt and Davis (7) for the light-tailed SV case) that GARCH processes exhibit extremal clustering (i.e., clustering of extremes), while SV processes lack this

Functional large deviations for multivariate regularly varying random walks

We extend classical results by A. V. Nagaev [Izv Akad. Nauk UzSSR Ser Fiz.-Mat. Nauk 6 (1969) 17-22, Theory Probab. Appl. 14 (1969) 51-64, 193-208] on large deviations for sums of i.i.d. regularly varying random variables to partial sum processes of i.i.d. regularly varying vectors. The results are stated in terms of a heavy-tailed large deviation principle on the space of cAdlAg functions. We illustrate how these results can be applied to functionals of the partial sum process, including ruin probabilities for multivariate random walks and long strange se-ments. These results make precise the idea of heavy-tailed large deviation heuristics: in an asymptotic sense, only the largest step contributes to the extremal behavior of a multivariate random walk.

Modeling teletraffic arrivals by a Poisson cluster process

In this paper we consider a Poisson cluster process N as a generating process for the arrivals of packets to a server. This process generalizes in a more realistic way the infinite source Poisson model which has been used for modeling teletraffic for a long time. At each Poisson point Γj, a flow of packets is initiated which is modeled as a partial iid sum process