Using daily stock returns: The case of event studies

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.

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Empirical properties of asset returns: stylized facts and statistical issues

The distribution of speculative price changes and rates of return data tend to be uncorrelated over time but characterized by volatile and tranquil periods. A simple time series model designed to capture this dependence is presented. The model is an extension of the Autoregressive Conditional Heteroskedastic (ARCH) and Generalized ARCH (GARCH) models obtained by allowing for conditionally t-distributed errors. The model can be derived as a simple subordinate stochastic process by including an additive unobservable rror term in the conditional variance equation. The descriptive validity of the model is illustrated for a set of foreign exchange rates and stock price indices.

A conditionally heteroskedastic time series model for speculative prices and rates of return

http://finance.martinsewell.com/stylized-facts/volatility/Andersen-etal2001a.pdf

The distribution of realized stock return volatility

A new stochastic process, termed the variance gamma process, is proposed as a model for the uncertainty underlying security prices. The unit period distribution is normal conditional on a variance that is distributed as a gamma variate. Its advantages include long tailedness, continuous-time specification, finite moments of all orders, elliptical multivariate unit period distributions, and good empirical fit. The process is pure jump, approximable by a compound Poisson process with high jump frequency and low jump magnitudes. Applications to option pricing show differential effects for options on the money, compared to in or out of the money. Copyright 1990 by the University of Chicago.

The Variance Gamma (V.G.) Model for Share Market Returns

The following sections are included:INTRODUCTIONPROPERTIES OF THE STUDENT AND SYMMETRIC-STABLE DISTRIBUTIONSDefinitions and Properties of the Student and Stable ModelsSome Implications of the Student and Stable Models for Empirical and Theoretical WorkAdditional RemarksMODELS FOR RATES OF RETURNDerivation of the Student and Stable Models: SummaryOther Stock Price ModelsMETHODS FOR MODEL COMPARISONThe Likelihood RatioStabilityESTIMATION OF THE MODEL'S PARAMETERSM.L.E. for the Student DistributionFama-Roll Estimates for the Stable DistributionESTIMATION RESULTSThe Actual DataThe Design of the Monte Carlo StudyDiscussion of the Simulation ResultsResults for Rates of ReturnSUMMARYAPPENDIX A DERIVATIONS OF THE STUDENT AND STABLE MODELSAPPENDIX B PROPERTIES OF THE UNIFORM RANDOM NUMBERS USED IN THE SIMULATIONS

A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices

There has been a great deal of discussion about the statistical distribution of rates of return on common stocks. At an early stage the prevalent belief was that distributions of rates of return on common stocks were adequately characterized by the normal distribution. This belief seemed to be consistent with the pioneering work of Bachelier.1 It was also observed, however, that empirical distributions of such returns had more kurtosis (i.e., "fatter tails") than that predicted by the normal distribution. The evidence provided by Mandelbrot and Fama suggested that one could explicitly account for the observed "fat tails" by using the symmetric-stable distribution.2 This article considers another family of symmetric distributions that can also account for the observed "fat tails" of returns distribution. This alternative is the Student (or t) distribution. It will be indicated that this alternative model has implications for empirical and theoretical work that are quite different from those of the symmetric-stable model. The descriptive validity of the Student model, relative to that of the symmetric-stable model, will be assessed using actual daily rates of return. This article is organized as follows: Section II describes the prop-

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Robert C. Blattberg

Papers

A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices

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