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.

/pdf/empirical-properties-of-asset-returns-stylized-facts-and-3gbrgrs0vm.pdf

Empirical properties of asset returns: stylized facts and statistical issues

This book provides the most comprehensive treatment of the theoretical concepts and modelling techniques of quantitative risk management. Whether you are a financial risk analyst, actuary, regulator or student of quantitative finance, Quantitative Risk Management gives you the practical tools you need to solve real-world problems. Describing the latest advances in the field, Quantitative Risk Management covers the methods for market, credit and operational risk modelling. It places standard industry approaches on a more formal footing and explores key concepts such as loss distributions, risk measures and risk aggregation and allocation principles. The book's methodology draws on diverse quantitative disciplines, from mathematical finance and statistics to econometrics and actuarial mathematics. A primary theme throughout is the need to satisfactorily address extreme outcomes and the dependence of key risk drivers. Proven in the classroom, the book also covers advanced topics like credit derivatives. Fully revised and expanded to reflect developments in the field since the financial crisis Features shorter chapters to facilitate teaching and learning Provides enhanced coverage of Solvency II and insurance risk management and extended treatment of credit risk, including counterparty credit risk and CDO pricing Includes a new chapter on market risk and new material on risk measures and risk aggregation

/pdf/quantitative-risk-management-concepts-techniques-and-tools-afazrwn4wm.pdf

Quantitative Risk Management: Concepts, Techniques, and Tools

Non-Gaussian processes of Ornstein–Uhlenbeck (OU) type offer the possibility of capturing important distributional deviations from Gaussianity and for flexible modelling of dependence structures. This paper develops this potential, drawing on and extending powerful results from probability theory for applications in statistical analysis. Their power is illustrated by a sustained application of OU processes within the context of finance and econometrics. We construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive OU processes, and we study these models in relation to financial data and theory.

Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics

Three processes reflecting persistence of volatility are initially formulated by evaluating three Levy processes at a time change given by the integral of a mean-reverting square root process. The model for the mean-reverting time change is then generalized to include non-Gaussian models that are solutions to Ornstein-Uhlenbeck equations driven by one-sided discontinuous Levy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general mean-corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean-corrected exponential model is not a martingale in the filtration in which it is originally defined. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered filtrations consistent with the one-dimensional marginal distributions of the level of the process at each future date.

/pdf/stochastic-volatility-for-levy-processes-2bofrpm84e.pdf

Stochastic Volatility for Lévy Processes

http://faculty.baruch.cuny.edu/lwu/papers/timechangeLevy_JFE2004.pdf?origin=publication_detail

Time-Changed Levy Processes and Option Pricing ⁄

The goal of this paper is to show that normality of asset returns can be recovered through a stochastic time change. Clark (1973) addressed this issue by representing the price process as a subordinated process with volume as the lognormally distributed subordinator. We extend Clark's results and find the following: (i) stochastic time changes are mathematically much less constraining than subordinators; (ii) the cumulative number of trades is a better stochastic clock than the volume for generating virtually perfect normality in returns; (iii) this clock can be modeled nonparametrically, allowing both the time-change and price processes to take the form of jump diffusions. The relations among trading volume, stock prices, and price volatility, the subject of empirical and theoretical studies over many years, have lately received renewed attention with the increased availability of high frequency data. A vast amount of research has focused on issues such as news arrivals, volume, and price changes or volatility moves, usually outside any framework of general or even partial equilibrium. Is the normality of returns-a key issue, for example, in the mean-variance paradigm for portfolio choice, or the recent study of the problems of risk management (e.g., in Value at Risk)-verified at any time horizon? The evidence accumulated from a number of studies that document the presence of leptokurtosis and skewness in the distribution of returns of a wide variety of financial assets suggests that the answer is no. Studies as early as, for example, Fama (1965), showed that daily returns are more long tailed than the normal density, with the distribution of returns approaching normality as the holding period is extended to one month. In the same manner, volatility smiles and other observed

Thierry Ané

Papers

Order Flow, Transaction Clock, and Normality of Asset Returns

Stochastic volatility and transaction time: an activity-based volatility estimator