## Empirical properties of asset returns: stylized facts and statistical issues (2001)

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Venue: | Quantitative Finance |

Citations: | 164 - 3 self |

### BibTeX

@ARTICLE{Cont01empiricalproperties,

author = {Rama Cont},

title = {Empirical properties of asset returns: stylized facts and statistical issues},

journal = {Quantitative Finance},

year = {2001},

volume = {1},

pages = {223--236}

}

### Years of Citing Articles

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### Abstract

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.

### Citations

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Citation Context ... for which the scaling in equation (30) holds exactly, the Legendre transform (31) may be inverted to obtain D(α) from ζ(q). The technique was subsequently refined [62,99] using the wavelet transfor=-=m [92]-=-, who proposed an algorithm for determining the singularity spectrum from its wavelet transform [66, 67]. 7.3. Singularity spectra of asset price series These methods provide a framework to investigat... |

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Citation Context ...markets and time periods, has been observed by independent studies and classified as ‘stylized facts’. We present here a pedagogical overview of these stylized facts. With respect to previous revi=-=ews [10, 14, 16, 50, 95, 102, 109]-=- on the same subject, the aim of the present paper is to focus more on the properties of empirical data than on those of statistical models and introduce the reader to some new insights provided by me... |

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Citation Context ...r the estimator from which the asymptotic normality is obtained. This central limit theorem can be obtained by assuming that the noise terms (innovations) in the return process are ‘weakly’ depend=-=ent [30]-=-. In order to obtain confidence intervals for finite samples, one often requires the residuals to be IID and some of their higher-order (typically fourth order) moments to be well defined (finite). As... |

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Citation Context ...market analysts has been and remains an event-based approach in which one attempts to ‘explain’ or rationalize a given market movement by relating it to an economic or political event or announcem=-=ent [27]. -=-From this point of view, one could easily imagine that, since different assets are not necessarily influenced by the same events or information sets, price series obtained from different assets and—... |

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Citation Context ...markets and time periods, has been observed by independent studies and classified as ‘stylized facts’. We present here a pedagogical overview of these stylized facts. With respect to previous revi=-=ews [10, 14, 16, 50, 95, 102, 109]-=- on the same subject, the aim of the present paper is to focus more on the properties of empirical data than on those of statistical models and introduce the reader to some new insights provided by me... |

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Citation Context ... in which one computes the likelihood of a model to hold given the value of a statistic and rejects/accepts the model by comparing the test statistics to a threshold value. With a few exceptions (see =-=[8, 87, 88]-=-), the large majority of statistical tests are based on a central limit theorem for the estimator from which the asymptotic normality is obtained. This central limit theorem can be obtained by assumin... |

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Citation Context ...pastime: there are dozens of parametric models proposed in the literature, starting with the normal distribution, stable distributions [80], the Student distribution [9, 72], hyperbolic distributions =-=[37, 104]-=-, normal inverse Gaussian distributions [7], exponentially truncated stable distributions [11,21] are some of the parametric models which have been proposed. From the empirical features described abov... |

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Citation Context ...ge rate returns. Time scale: ticks. attraction but the tail index is found to be larger than two— which means that the variance is finite and the tails lighter than those of stable Lévy distributio=-=ns [41], -=-but compatible with a power-law (Pareto) tail with (the same) exponent α(T ) = 1/ξ. These studies seem to validate the power-law nature of the distribution of returns, with an exponent around three,... |

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Citation Context ... precisely with the probabilities of extreme events. To our knowledge, the first application of extreme value theory to financial time series was given by Jansen and de Vries [70], followed by Longin =-=[76], Dacorogn-=-a et al [28], Lux [77] and others. Given a series of n non-overlapping returns r(t, �t), t = 0,�t,2�t,...,n�t, the extremal (minimal and maximal) returns are defined as: mn(�t) = min{r(t + k... |

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Citation Context ...tive autocorrelations at the first lag in bid or ask prices themselves, suggesting a fast mean reversion of the price at the tick level. This feature may be attributed to the action of a market maker =-=[47]. -=-229sR Cont Q UANTITATIVE F INANCE The absence of autocorrelation does not seem to hold systematically when the time scale �t is increased: weekly and monthly returns do exhibit some autocorrelation.... |

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Citation Context ...esent the time the market takes to react to new information. This correlation time is typically several minutes for organized futures markets and even shorter for foreign exchange markets. Mandelbrot =-=[85] exp-=-ressed this property by stating that ‘arbitrage tends to whiten the spectrum of price changes’. This property implies that traditional tools of signal processing which are based on second-order pr... |

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Citation Context ...e way one can study autocorrelation functions of various powers of the returns: Cα(τ) = corr(|r(t + τ, �t)| α , |r(t,�t)| α ). (16) Comparing the decay of Cα for various values of α, Ding a=-=nd Granger [34, 35] rem-=-arked that, for a given lag τ, this correlation is highest for α = 1, which means that absolute returns are more predictable than other powers of returns. Several authors [11,20–22,54,55,59,105] h... |

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Citation Context ...unately, this turns out not to be the case: it has been shown that, for large classes of stochastic processes, the singularity spectrum is the same for almost all sample paths. Results due to Jaffard =-=[68] -=-show that a large class of Lévy processes verifies this property. As defined above, the singularity spectrum of a function does not appear to be of any practical use since its definition involves fir... |

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Citation Context ...on may have a finite tail index α without being a power-law distribution. Measuring the tail index of a distribution gives a measure of how heavy the tail is. A simple method, suggested by Mandelbrot=-= [80,89]-=-, is to represent the sample moments (or cumulants) as a function of the sample size n. If the theoretical moment is finite then the sample moment will eventually settle down to a region defined aroun... |

30 |
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Citation Context ...eir covariance is zero: covariance does not measure the correlation of extremes. Some recent theoretical work has been done in this direction using copulas [108] and multivariate extreme value theory =-=[64, 112, 113]-=-, but a lot remains to be done on empirical grounds. For a recent review with applications to foreign exchange rate data see Hauksson et al [64]. 7. Pathwise properties The risky character of a financ... |

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Citation Context ...ity factor whose dynamics should be specified to match the empirically observed dependences. Examples of models in this direction are GARCH models [10,39] and long-memory stochastic volatility models =-=[20,59,100]. Note howe-=-ver that in this decomposition the volatility variable σ (t, �t) is not directly observable, only the returns r(t,�t) are. Therefore, the definition of ‘volatility’ is model dependent and ‘... |

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16 |
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Citation Context ...oposed in the literature, starting with the normal distribution, stable distributions [80], the Student distribution [9, 72], hyperbolic distributions [37, 104], normal inverse Gaussian distributions =-=[7]-=-, exponentially truncated stable distributions [11,21] are some of the parametric models which have been proposed. From the empirical features described above, one can conclude that, in order for a pa... |

16 |
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Citation Context ...�t) and Mn(�t) as the sample size n increases. If such a limit exists, then it is described by the Fisher–Tippett theorem in the case where the returns are IID. Extreme value theorem for IID seq=-=uence [38]. Assume-=- the log returns (r(t, �t))t�0 form an IID sequence with distribution F�t. If there exist normalizing constants (λn,σn) and a non-degenerate limit distribution H for the normalized maximum ret... |

16 | Characterization of self-similar multifractals with wavelet maxima
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(Show Context)
Citation Context ...n the case of multifractal processes for which the scaling in equation (30) holds exactly, the Legendre transform (31) may be inverted to obtain D(α) from ζ(q). The technique was subsequently refine=-=d [62,99]-=- using the wavelet transform [92], who proposed an algorithm for determining the singularity spectrum from its wavelet transform [66, 67]. 7.3. Singularity spectra of asset price series These methods ... |

16 | non-linearities can ruin the heavy tailed modeler’s day, In A practical guide to heavy tails : statistical
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Citation Context ...meaning and relevance of such confidence intervals. As we will discuss below, this can have quite an impact on the significance and interpretation of commonly used estimators (see also discussions in =-=[1, 31, 107]-=-). 4. The distribution of returns: a tale of heavy tails Empirical research in financial econometrics in the 1970s mainly concentrated on modelling the unconditional distribution of returns, defined a... |

15 |
Extreme financial returns and their comovements
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(Show Context)
Citation Context ...eir covariance is zero: covariance does not measure the correlation of extremes. Some recent theoretical work has been done in this direction using copulas [108] and multivariate extreme value theory =-=[64, 112, 113]-=-, but a lot remains to be done on empirical grounds. For a recent review with applications to foreign exchange rate data see Hauksson et al [64]. 7. Pathwise properties The risky character of a financ... |

13 | Nonlinear time series, complexity theory, and finance
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(Show Context)
Citation Context ...markets and time periods, has been observed by independent studies and classified as ‘stylized facts’. We present here a pedagogical overview of these stylized facts. With respect to previous revi=-=ews [10, 14, 16, 50, 95, 102, 109]-=- on the same subject, the aim of the present paper is to focus more on the properties of empirical data than on those of statistical models and introduce the reader to some new insights provided by me... |

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12 |
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(Show Context)
Citation Context ...ity factor whose dynamics should be specified to match the empirically observed dependences. Examples of models in this direction are GARCH models [10,39] and long-memory stochastic volatility models =-=[20,59,100]. Note howe-=-ver that in this decomposition the volatility variable σ (t, �t) is not directly observable, only the returns r(t,�t) are. Therefore, the definition of ‘volatility’ is model dependent and ‘... |

12 |
Multifractal Formalism for Functions: I. Results Valid for all Functions
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(Show Context)
Citation Context ...sey et al [58] who, in different contexts , proposed a formalism for empirically computing the singularity spectrum from sample paths of the process. This formalism, called the multifractal formalism =-=[58, 66, 67, 101], enab-=-les the singularity spectrum to be computed from sample moments (‘structure functions’) of the increments. More precisely, if the sample moments of the returns verify a scaling property 〈|r(t,T)... |

11 | Modeling economic randomness: statistical mechanics of market phenomena - Cont - 1999 |

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10 |
de Vries (1991): “On the Frequency of Large Stock Returns: Putting Booms and Busts into Perspective
- Jansen, G
(Show Context)
Citation Context ...robability theory dealing precisely with the probabilities of extreme events. To our knowledge, the first application of extreme value theory to financial time series was given by Jansen and de Vries =-=[70], follow-=-ed by Longin [76], Dacorogna et al [28], Lux [77] and others. Given a series of n non-overlapping returns r(t, �t), t = 0,�t,2�t,...,n�t, the extremal (minimal and maximal) returns are defined... |