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260
Empirical properties of asset returns: stylized facts and statistical issues
 Quantitative Finance
, 2001
"... 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 des ..."
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Cited by 335 (4 self)
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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.
A multifractal wavelet model with application to TCP network traffic
 IEEE TRANS. INFORM. THEORY
, 1999
"... In this paper, we develop a new multiscale modeling framework for characterizing positivevalued data with longrangedependent correlations (1=f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the mo ..."
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Cited by 215 (35 self)
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In this paper, we develop a new multiscale modeling framework for characterizing positivevalued data with longrangedependent correlations (1=f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the model provides a rapid O(N) cascade algorithm for synthesizing Npoint data sets. We study both the secondorder and multifractal properties of the model, the latter after a tutorial overview of multifractal analysis. We derive a scheme for matching the model to real data observations and, to demonstrate its effectiveness, apply the model to network traffic synthesis. The flexibility and accuracy of the model and fitting procedure result in a close fit to the real data statistics (variancetime plots and moment scaling) and queuing behavior. Although for illustrative purposes we focus on applications in network traffic modeling, the multifractal wavelet model could be useful in a number of other areas involving positive data, including image processing, finance, and geophysics.
Multiscale Modeling and Estimation of Poisson Processes with Application to Photonlimited Imaging
 IEEE TRANS. ON INFO. THEORY
, 1999
"... Many important problems in engineering and science are wellmodeled by Poisson processes. In many applications it is of great interest to accurately estimate the intensities underlying observed Poisson data. In particular, this work is motivated by photonlimited imaging problems. This paper studies ..."
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Cited by 72 (10 self)
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Many important problems in engineering and science are wellmodeled by Poisson processes. In many applications it is of great interest to accurately estimate the intensities underlying observed Poisson data. In particular, this work is motivated by photonlimited imaging problems. This paper studies a new Bayesian approach to Poisson intensity estimation based on the Haar wavelet transform. It is shown that the Haar transform provides a very natural and powerful framework for this problem. Using this framework, a novel multiscale Bayesian prior to model intensity functions is devised. The new prior leads to a simple, Bayesian intensity estimation procedure. Furthermore, we characterize the correlation behavior of the new prior and show that it has 1/f spectral characteristics. The new framework is applied to photonlimited image estimation and its potential to improve nuclear medicine imaging is examined.
LogInfinitely Divisible Multifractal Processes
, 2002
"... We define a large class of multifractal random measures and processes with arbitrary loginfinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined lognormal Multifractal Random Walk processes (MRW) [33, 3] and the logPoisso ..."
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Cited by 68 (5 self)
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We define a large class of multifractal random measures and processes with arbitrary loginfinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined lognormal Multifractal Random Walk processes (MRW) [33, 3] and the logPoisson “product of cynlindrical pulses” [7]. Their construction involves some “continuous stochastic multiplication” [36] from coarse to fine scales. They are obtained as limit processes when the finest scale goes to zero. We prove the existence of these limits and we study their main statistical properties including non degeneracy, convergence of the moments and multifractal scaling.
Multifractal Measures and a Weak Separation Condition
, 1999
"... We define a new separation property on the family of contractive similitudes that allows certain overlappings. This property is weaker than the open set condition of Hutchinson. It includes the wellknown class of infinite Bernoulli convolutions associated with the P.V. numbers and the solutions of ..."
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Cited by 61 (13 self)
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We define a new separation property on the family of contractive similitudes that allows certain overlappings. This property is weaker than the open set condition of Hutchinson. It includes the wellknown class of infinite Bernoulli convolutions associated with the P.V. numbers and the solutions of the twoscale dilation equations. Our main purpose in this paper is to prove the multifractal formalism under such condition.
Multifractality in Asset Returns: Theory and Evidence
 REVIEW OF ECONOMICS AND STATISTICS
, 2001
"... This paper investigates the Multifractal Model of Asset Returns, a class of continuoustime processes that incorporate the thick tails and volatility persistence exhibited by many financial time series. The simplest version of the model compounds a Brownian Motion with a multifractal timedeformatio ..."
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Cited by 58 (6 self)
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This paper investigates the Multifractal Model of Asset Returns, a class of continuoustime processes that incorporate the thick tails and volatility persistence exhibited by many financial time series. The simplest version of the model compounds a Brownian Motion with a multifractal timedeformation process. Prices follow a semimartingale, which precludes arbitrage in a standard twoasset economy. Volatility has long memory, and the highest finite moments of returns can take any value greater than two. The local variability of the process is highly heterogeneous, and is usefully characterized by the local Hölder exponent at every instant. In contrast with earlier processes, this exponent takes a continuum of values in any time interval. The model also predicts that the moments of returns vary as a power law of the time horizon. We confirm this property for Deutsche Mark/U.S. Dollar exchange rates and several equity series. We then develop an estimator, and infer a parsimo...
Scaling Analysis of Conservative Cascades, With Applications to Network Traffic
, 1999
"... Recent studies have demonstrated that measured widearea network traffic such as Internet traffic exhibits locally complex irregularities, consistent with multifractal behavior. It has also been shown that the observed multifractal structure becomes most apparent when analyzing measured network tr ..."
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Cited by 57 (7 self)
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Recent studies have demonstrated that measured widearea network traffic such as Internet traffic exhibits locally complex irregularities, consistent with multifractal behavior. It has also been shown that the observed multifractal structure becomes most apparent when analyzing measured network traffic at a particular layer in the welldefined protocol hierarchy that characterizes modern data networks, namely the transport or TCP layer. To investigate this new scaling phenomenon associated with the dynamics of measured network traffic over small time scales, we consider a class of multiplicative processes, the socalled conservative cascades, that serves as a cascade paradigm for and is motivated by the networking application. We present a waveletbased time/scale analysis of these cascades to determine rigorously their global and local scaling behavior. In particular, we prove that for the class of multifractals generated by these conservative cascades the multifractal formal...
Fractional Brownian motion and data traffic modeling: The other end of the spectrum
 Fractals in Engineering
, 1997
"... Introduction Fractal analysis of computer traffic has received considerable attention since the seminal work of Leland and al. [11] who provided experimental evidence that some traces of data traffic exhibit long range dependence (LRD). This is a typical fractal feature which is not found with the ..."
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Cited by 49 (13 self)
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Introduction Fractal analysis of computer traffic has received considerable attention since the seminal work of Leland and al. [11] who provided experimental evidence that some traces of data traffic exhibit long range dependence (LRD). This is a typical fractal feature which is not found with the classical Poisson models. An important issue since then has been to propose "physical" models that lead to such fractal behavior. A popular model [27] is based on the superposition of simple i.i.d ON/OFF sources which ON and/or OFF periods follow a heavy tailed law (P r(X ? ) ¸ c \Gammaff ; 1 ! ff ! 2). When properly normalized, the resulting traffic is a fractional Brownian motion (fBm) of LRD exponent H = (3 \Gamma ff)=2. Several practical implications of LRD traffic have consequently been investigated, e.g. the queuing behavior [15] (see
Multifractal Processes
, 1999
"... This paper has two main objectives. First, it develops the multifractal formalism in a context suitable for both, measures and functions, deterministic as well as random, thereby emphasizing an intuitive approach. Second, it carefully discusses several examples, such as the binomial cascades and sel ..."
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Cited by 42 (6 self)
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This paper has two main objectives. First, it develops the multifractal formalism in a context suitable for both, measures and functions, deterministic as well as random, thereby emphasizing an intuitive approach. Second, it carefully discusses several examples, such as the binomial cascades and selfsimilar processes with a special eye on the use of wavelets. Particular attention is given to a novel class of multifractal processes which combine the attractive features of cascades and selfsimilar processes. Statistical properties of estimators as well as modelling issues are addressed.
The Markovswitching multifractal model of asset returns: GMM estimation and linear forecasting of volatility
 Journal of Business and Economics Statistics
, 2008
"... Multifractal processes have recently been proposed as a new formalism for modelling the time series of returns in finance. The major attraction of these processes is their ability to generate various degrees of long memory in different powers of returns a feature that has been found in virtually al ..."
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Cited by 42 (10 self)
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Multifractal processes have recently been proposed as a new formalism for modelling the time series of returns in finance. The major attraction of these processes is their ability to generate various degrees of long memory in different powers of returns a feature that has been found in virtually all financial data. Initial difficulties stemming from nonstationarity and the combinatorial nature of the original model have been overcome by the introduction of an iterative Markovswitching multifractal model in Calvet and Fisher (2001) which allows for estimation of its parameters via maximum likelihood and Bayesian forecasting of volatility. However, applicability of MLE is restricted to cases with a discrete distribution of volatility components. From a practical point of view, ML also becomes computationally unfeasible for large numbers of components even if they are drawn from a discrete distribution. Here we propose an alternative GMM estimator together with linear forecasts which in principle is applicable for any continuous distribution with any number of volatility components. Monte Carlo studies show that GMM performs reasonably well for the popular Binomial and Lognormal models and that the loss incurred with linear compared to optimal forecasts is small. Extending the number of volatility components beyond what is feasible with MLE leads to gains in forecasting accuracy for some time series.